C R I T I C A L P O I N T T H E O R Y F O R S M O O T H F U N C T I O N S O N H I L B E R T M A N I F O L D S W I T H S I N G U L A R I T I E S A N D ITS A P P L I C A T I O N T O S O M E O P T I M A L C O N T R O L P R O B L E M S

S. A. V a k h r a m e e v UDC 517.977.1+517.974+515.164.174

Introduct ion

The purpose of the present paper is to reinforce and to define more precisely the preceding results ob- tained by the au thor (see [70, 72]) in order to make them more suitable for the investigation of some optimal control problems associated with smooth control systems of a constant rank [68]. In [70, 72] some versions of Morse and Lusternik-Schnirelman theories have been presented for Ca- func t ions defined on Hilbert man- ifolds with singularities, the so-called Hilbert submanifolds with corners, and Hilbert transversally convex subsets. However, for the basic optimal control problem (P)

m

U i .

i = 1

x e M, x(0) = z (T) V,

T

= / f ~ ~ inf, u(.) e L~[O,T], J(u) . 1

0

(where M is a smooth Ca-mani fo ld , V is a finite-dimensional submanifold with corners, or a transversally convex subset of M, f , g~, i = 1 , 2 , . . . , m , are Ca-vec to r fields on M) these theories had only limited applicability, since the assumption of the Ca-smoothness of the function - J on the corresponding path space

p(T) = {u(.) e L~[O, Tllx(T ) = x(T;xo,u(.)) e V} XO,V

leads to a very artificiM condition for the integrand f0 to be quadratic with respect to the variable u (for differentiability properties of integral functionals see [146, 173]).

The prime objective of the present article is to remove the quadrat ic condition mentioned above (which is unrelated to the question under consideration), and to present versions of Morse and Lusternik- Schnirelman theories that would be more suitable for studying the problem (P) under most natural con- ditions. To this end, on the one hand, a version of the Lusternik-Schnirelman theory for Cl-functions on Hilbert transversally convex subsets is presented. The possibility of such a variant is connected with the simple observation that the smoothness of a pseudogradient vector field (with respect to the function under consideration) is greatly related with the Smoothness of the base manifold and scarcely with the smoothness of that function. Hence it is sufficient to consider Cl-functions on a C2-Hilbert manifold containing the corresponding transversally convex subset. One can further reinforce this result with the application of nonsmooth analysis techniques (in this connection see [100, 951,895]).

Slightly different is the situation with Morse theory. Here the most essential point is that the func- tionM - Y must possess some additional smoothness properties in order to define, at least, the notions of nondegeneracy for critical points, and of their (co)indices. in the case of the main problem (P) an analysis

(T) of extremal controls (i.e., of critical points of the functional - Y on P~o,V) shows that every extremal control coincides on a set of.full measure in [0, T] with some function from W~[0, T], where W~[0, T] is the space

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika iee Prilozheniya. Tematicheskiye 0bzory. Vol. 1, Analysis-i, 1993.

0090-4104/93/6701-2713512.50 (~) 1993 Plenum Publishing Corporation 2713

of all absolutely continuous m-dimensional functions with square ~ntegrabJ.e deri.vatives equipped, with the

inner product (u, v)w~t0,~ = (u(0), v ( 0 ) ) + (~, v)L~I0X"

Therefore it would seem to be natural to study the problem (P) on the space W2~[0, T] sim:e the functional - J on the pa th space

(T) W2 [0, T] x(T) V} �9 0 , v = e I e

belongs to C~ However, this does not lead to any result: one is confronted here with the loss of compactness effect, which means that under no conditions does the functional - J satisfy the generalized (C) condition or its analogs. These two phenomena, namely the loss of smoothness in the L~[0, T]-setting and the loss of compactness in the W~[0, T]-setting, appear in slightly different contexts, e.g. when studying boundary value problems via variational methods. In general, M. Gromov remarks [95] that there exist very few interesting variational problems to which the Morse theory based on the Palais-Smale condition can be directly applied. To overcome the difficulties mentioned above, an approach closely related with the approach of [52] has been chosen in the present paper. Namely, instead of a single pa th space the pair

of spaces ~b(T) :p(T) and a function - J on these spaces are considered; these objects form the so-called ~xo~Y' xo~Y Morse triple. The axiomatization of this Morse triple is nothing but the postulat ion of regularity properties

of the problem (P). On the space ~b(T) the function - J has the required smoothness properties, and on the ~xo~Y

space 7 )(T) it has those of compactness. All this, together with the approach of I. V. Skrypnik [173, 174], ~0,v permits one to present a version of Morse theory that is suitable for the investigation of the problem (P) under most natural assumptions imposed on the integrand f0 that do not even provide the twice Gateaux

-D(T) differentiability of the functional - J on r~0,y. In conclusion, the author expresses his sincere grat i tude to A. A. Agrachev and R. V. Gamkrelidze for

their at tention to the present paper. I also thank V. M. Tikhomirov for support and N. N. Petrov for some critical remarks that have accelerated the rate of my work on this article.

1. H i l b e r t S u b m a n i f o l d s w i t h C o r n e r s a n d T r a n s v e r s a l l y C o n v e x S u b s e t s

In this section some classes of Hilbert manifolds with singularities are defined, for which Morse and Lusternik-Schnirelman theories will be constructed below; see also [70, 72] for details.

Let M be an n-dimensional smooth (of class C ~ manifold. 1 A closed subset V C M is called a submanifold with corners if, for every point x0 E V, there exist a coordinate chart (O, qa) of the manifold M and a polyhedral cone K C R'* with vertex at the origin such that qa(x0) = 0 and ~(O N V) = K.

Generally speaking, in the present definition one does not suppose a solid property of the cone K. It follows from the definition that submanifolds with corners are given locally by systems of smooth

inequalities gi(x) < 0, i = 1 , . . . , l ,

where l may be greater than n -= dim M. In a natural way, the tangent cone T~V to V at a point x C V is defined. Namely, a tangent vector

X~ E T~M is said to be tangent to V at a point x E V if there exists a smooth curve, a(c), 0 < ~ < ~0, such that a(0) = x, a(E) E V, 0 < ~ < r and

da(6) de ]~=0 = Xx.

A tangent cone, T,V, to V at a point x C V is a collection of all tangent vectors to V at that point. This cone is naturally isomorphic to the cone K, modeling V; the isomorphism is stated by the corresponding coordinate chart (O, qa):

qa.,,T,V = ToK "~ K.

1 One can also consider the case where the base manifold M has finite smoothness.

2714

Hence T~V is a polyhedral cone in T~M with vertex at the origin. Moreover, the corresponding multivalued

mapping x ~-~ T . V : V --~ 2 T M

has the following basic properties: (1) the mapping x ~ T,:V �9 V ~ 2 T M is lower semicontinuous;

(2) the support planes Tx~'~ = TxV - T~V have a constant ( independent of x) dimension, and there exists a regular completely integrable distr ibution H~, x E U, which is defined over some open neighborhood U of the set V, such tha t

II~ = T~V Vx E V.

it follows from condition (2) that for a given submanifold with corners there exists a natural support of that submanifold, i.e., a maximal integral manifold N of distr ibution II,:, x E U. The la t ter is natural

since dimT~V = dim T~N Vx E V,

so that T,~V is a solid cone in T~N. For a submanifold V with corners of M, one can define an open face F as a maximal , with respect

to inclusion, smooth connected submanifold F of M lying entirely in V. A family {F} of all open faces of the submanifold V wi th corners defines on V the s t ructure of a Whi tney stratified space; a k-dimensional s t ra tum of this stratification (a k-dimensional face) allows for a simple description: it is a connected

component of the set {x E V I dim(T~V M ( - T ~ V ) ) = k}.

We shall also consider smooth mappings between submanifolds with corners. Let N be a finite- dimensional smooth manifold. A map f : V -* N is said to be smooth if f is a restrict ion to V of some smooth map f : M --~ N. In what follows we shall not distinguish between f and f , and shall denote them by the same letter.

Let W be a submanifold with corners of N. A smooth mapping f : V --~ N is transversal to W ( f ch W) if, for all x E V with f ( x ) E W, the following relation holds true:

- f , , ~ T ~ V + Tf(~)W = TS(~)N (1.1)

Generally speaking, in contrast to the smooth case, when V and W are smooth submmlifolds of M and N respectively, one does not assert that the inverse image f-1 (W) for f ch W is a submanifold with corners. The set f - ~ ( W ) is an impor tant example of the object that we shall call a transversal!y convex subset. Note that transversali ty may also be defined for smooth mappings between more general objects than submanifolds with corners if only for the lat ter the tangent cones are well defined.

Let K be a closed convex cone in ]im with vertex at the origin. There exist different ways to define a tangent cone T~K to K at a point x C K. For example, one can consider T z K as a tangent cone in the sense of F. Clarke [105]. One can directly verify that in this case T~K coincides with the closure of the set of all tangent vectors to K at a point x defined by smooth curves: a vector X~ E T~]R m is said to be tangent to K at a p o i n t x C K if there exist a s m o o t h c u r v e d ( e ) , 0 _< e <_ e0, such that G(0) = x, G(c) E K,

O<e<eo, and da(~) [ = X~.

dc ~=0

By this definition, a tangent cone T~K to K at a point x is a closed convex cone in T~R m with vertex at the origin; moreover, dim T~K = const , the mapping

x ~ T ~ K : K ~ 2 T ~ "

is lower semicontinuous, and there exists a regular completely integrable distr ibution II~, x C U, defined over some open neighborhood U of K such that

i

II~ = T~K Vx E K.

2715

Now let ~: 1R n ~ !R r~ be a smooth (of class C ~ ) mapping preserving the origin (~(0) = 0) and being transversal to a cone K , i.e., Vz: O(x) E K,

+ = (1.2)

Let us consider an inverse image W = r of the cone K under this mapping. It is natura l to define a tangent cone T, W to W at a point x by the formula

- 1 i.e., by T ~ W = g2.,r According to this definition, the tangent cone T z W inherits the properties

of the tangent cone T z K mentioned above. N a V y , T x W is a closed convex cone in T~R n with vertex at the origin, the dimension of support planes T ~ W = T ~ W - T ~ W is independent of x, the map x T~W: W ~ 2 :r~" is lower semicontinuous, and there exists a regular c o ~ e t e l y integrable distribution II~, x E U, defined over some open neighborhood U D W, such that IIx = T ~ W Vx E W .

Let M be again an n-dimensional smooth (of class C ~176 manifold. A closed subset V C M is said to be transversally convex if, for every point x0 E with (p(x0) = O, a closed convex cone K C R m that preserves the origin, such that

V, there exist a coordinate chart (0 , ~) of the manifold M with vertex at the origin, and a Coo-mapping (I): N n ~ ]I( m

N V) = qs- l (K) .

For a transversally convex subset V C M it is natura l to define a tangent cone T~V at a point x E V as a set

r r0(o-l(K))}. According to this definition, T~V has properties similar to those of tangent cones to submanifolds with corners, namely

(1) T~V is a closed convex cone with vertex at the origin in the space T~M; support planes T x V = T~V - T~V are of a constant dimension ( independent of x), and there exists a smooth regular completely in tegrable distr ibution II~, x E U, which is defined over some open neighborhood U D V, such that T~V = II~ Vx E V.

(2) the mult ivalued mapping x ~-~ T~ V: V ~ 2 T M is lower semicontinuous. For a transversally convex subset V C M, as in the case of a submanifold with corners, there exists a

natural support, i.e., a maximal integral manifold N of the distribution II~, x E U, ment ioned in (1). As in the case of a submanifold with corners, one can define (by formula (1.1)) transversality of a

smooth mapping f : V ~ N to a transversally convex subset W of a smooth manifold N. In this case the inverse image f - 1 (W) is also a transversally convex subset of M. In particular, from this assertion it follows that the transversal intersection 2 of two transversally convex subsets V1, V2 is also transversally convex, close convex subsets of N '~ are transversally convex, etc.

Transversally convex subsets are topological manifolds with a boundary, hence they have the ANR- property (i.e., they are absolute neighborhood retracts); the same conclusion also holds for the submanifolds with corners.

Now let us define Hilbert analogs of the notions introduced above. Let .M be a Hilbert manifold (of class Coo), let M be a finite-dimensional Coo-manifold, and let

5r: Ad --* M be a surjective submersion (of class Coo). An inverse image .~ ' - I(V) of a submanifold with corners (a transversally convex subset) V C M is called a Itilbert nubraanifold with corners (a Itilbert transversally convex subset): P = , ~ ' - I ( V ) .

For Hilbert submanifolds with corners (Hilbert transversally convex subsets) 1) a tangent cone T~V at a point z E 1) is natural ly defined, namely

% v = e e T x V } .

2 We say that Va and V2 have a transversal intersection if the inclusion mapping ia: V1 '-* M is transversal to V2.

2716

The tangent cone T~12 to 12 at a point x inherits the propert ies of the tangent cone to V, tha t is: (1) the mult ivalued mapping x ~ T.12:12 ~ 2 T'ta is lower semicontinuous;

(2) T.12 is a closed convex cone in T . M , support planes Tx12 = T.12-T.12 have a constant ( independent of x) codimension in T . M .

Moreover, for a Hilbert submanifold with corners (a Hilbert transversally convex subset) there exists a natural ~upport, namely a smooth (of class C ~176 Hilbert submanifold .M C M with propert ies Af D 12 and

i

T,12 = T.Af Yx E 12. (1.3)

In fact, let N be the na tura l support of the submanifold with corners, ( the transversally convex subset) V C M, a model for 12; then one can assume Af = ~--1 (N).

If 12 is a Hilbert submanifold with corners, then one can define an open face g of that manifold as a maximal, with respect to inclusion, smooth connected submanifold G C 34, which lies in 12.

A family of open faces {g} sets on 12 the s t ructure of a Whi tney stratified space and admits of a simple description in terms of open faces of the finite-dimensional submanifold V with corners, a model for V.

One can prove (see [70, 72]) that for Hilbert submanifolds with corners (Hilbert transversally co__vex subsets) the following assertion on their local representat ion holds true: for Vx E 12, there exists a coordinate chart (O, r of a Hilbert manifold 34 with r = 0, such that

r n12) = H | W,

where H is a Hilbert subspace of a finite codimension in a Hilbert space 7-g, a model for 3 4 ( 0 ( 0 ) = H), W is a polyhedral finite-dimensional cone in ~ with vertex at the origin, or a finite-dimensional transversally convex subset, containing the origin, depending on whether 12 is a Hilbert submanifold with corners or a transversally convex subset, respectively.

From this it follows that 12 is a topological manifold with a boundary (in a well-defined sense), and, in particular, that it has the ANR-property.

From the local representat ion ment ioned above one can obtain the decomposit ion of the tangent cone T~12 to )2 at a point x:

T~12 = 7G (91s c T~: M ,

where 7-/, is a Hilbert subspace in Tx.M smoothly dependent on x and of a constant codimension in TJL4, K:~ is a polyhedral finite-dimensional cone with vertex at the origin or a finite-dimensional closed convex cone with vertex at the origin; moreover, support planes of that cone have a constant dimension, and the multivalued mapping x ~-+/C,: 12 --+ 2 T'aa is lower semicontinuous.

Concluding this section, we note that if X , , x C 34, is a smooth vector field with the property X , C T,12, x E 12 (12 is a Hilbert transversally convex subset or a Hilbert submanifold with corners), then the t ra jectory ~t(x), t C IR, of this field, s tart ing from a point x for t = 0, remains in 12 for all t for which it is defined. See in this Connection [595].

2. L u s t e r n i k - S c h n l r e l m a n T h e o r y for C 1 - F u n c t i o n s on Hi lbert Transversal ly C o n v e x S u b s e t s

In this section, we shall consider smooth (of class C 1) functions f : 12 --~ IR defined on a Hilbert transver- sally convex subset V of acomplete Riemannian manifold Ad (of class C~176 a

Let (-, .}~ be a Riemannian structure on M , let II" N~ be the corresponding norm on a tangent space T~Ad and let p be a.Riemannian metric, i.e.,

1

p(x, v) = i n f d t ,

0

x ,y E M ,

3 One can also consider the case when a base manifold .~4 belongs to class C 2.

2717

where the infinum is taken over all smooth curves 7: [0, 1] ~ .M connecting two points x and y. If f : l; ~ R is a function of class C 1 (i.e., a restriction to ]) of some smooth Cl- funct ion f : .M --~ R;

as we have already noticed, we do not distinguish between f and ] in their notat ion), then a gradient V f of that function (with respect to the Riemannian s t ructure under consideration) is defined by

(V f ( x ) ,X~)~ = (df(z) ,X~) VX~ E T~AA.

Let us connect another lower semicontinuous nonnegative function my: V ~ ]R with a smooth function

f: 'P ~ R by the formula ms(x ) = sup (Vf (x) ,X~)=.

IIX=ll.<l X=eT=~)

A point x0 E 1) is a critical point of a smooth Cl- funct ion f : V ~ R if mS(x0 ) = 0. Equivalently, a point x0 is said to be critical if

df(xo) eT;oV= {~,o e T;0z41 (~,0,X,o)<0 vx,0 e T, oV},

and hence the definition is independent of a Riemannian structure. We shall say that a Cl- funct ion f : ]2 ~ N satisfies the generalized (Cs)-condition or the generalized (C)-

condition in the sequential form if every sequence {x,~} C 1), with {f (xn)} bounded and with ms(xn ) ~ 0 as n --* ec, contains a convergent subsequence.

For a Cl- funct ion f : l; ~ R let us introduce s tandard notations:

K~ = K ~ ( f ) = {x E V l f (x ) = c, m y ( x ) = O}

is the critical level; K = K ( f ) is the set of all critical points, i.e.,

K = {x e ]21 my(x) = 0},

and fr = {x E ~) I f ( x ) _> c},

f - l ( c ) = {x E ]) I f (x ) = c},

f~ = {x e V I f (x) _< c},

f~,b = {x E Via <__ f(x) <<. b}

are typical level sets. Our goal now is to construct a Lusternik-Schnire lman type theory [774, 851], which gives, in particular,

a lower bound to a number of critical points of a Cl-funct ion f : 1) --* ]R by category cat (])) and permits one to make conclusions about the nonvoidness of the set K of critical points (an existence theorem). The

following main assertion holds true. D e f i n i t i o n a n d P r o p o s i t i o n 2.1. Let P be a ttilbert transveraally convex subset of a C~176

manifold Jr4 and let f: )) --+ N be a function of class C 1 defined on )2. Then there exists a smooth (of class C ~176 vector field X~, x E G, which is said to be a paeudogradient for f , defined on an open domain,

G = U G(k), k = l

G (k) = {= �9 v Ires(=) > ' M , k = 1 , 2 , . . . ,

that is free of critical points of that function f , where {~k} is some sequence of positive real numbers with 0 < ~k < ~k-1, k = 2, 3 , . . . , ~k ~ 0 as k --~ co, such that X~ E T~]), IIX~ll~ < 1, x E G, and the condition x E G (k) implies (V f ( x ) ,X~)x >_ ~k+~. In particular, an integral trajectory ~ot(x), t E R, of this field starting from a point x E G, for t = O, is defined for all t E R such that ~ ( x ) E G, and the function f increases along this trajectory, i.e., the function t ~ f ( v t (x ) ) increases.

2718

We stress that al though the function f is of class C 1, a pseudogradient vector field of that function has smoothness defined by those of a base manifold .44 D V; this is equal to oo in our case and is equal to k (k _> 0) in the case of finite smoothness C k+l.

In essence, the proof of this proposition is given in [70, 72]; nevertheless, in view of its great importance, we shall present here the principal steps of that proof.

P r o o f o f D e f i n i t i o n a n d P r o p o s i t i o n 2.1. Let {Sk}, 0 < 5k < 5k-1, k = 2, 3 , . . . , 5k ---* 0 as k ~ oo, be an arbitrary sequence of real numbers. From the lower semicontinuity of the function x ~-~ my(x): V ~ R, it follows that the set

C (k) = {x e V Ires(x) > 5k}

is open for all k = 1, 2 , . . . ; moreover, r C G (k+l) and the domain

o o

G = U G(k) k=l

is free of critical points of f . Since G is open in V it is a Hilbert manifold of class Coo, and, hence, by [122] there exists a C~176 of unity on G. We shall construct in each domain G (k) a smooth C~-vector

field X (k), x C G (k), with following properties:

(1) X (k) E T,'I; when x E G(k);

(2) IIXs _< 1, x e G(k);

(3) ( V f ( x ) , X ( k ) ) . >_ 5k, x e G (k). To this end, let us consider the multivalued mapping

p : G (k) __, 2 T3a

defined by P ( x ) = {X. c TWIlIX~!I~ 1, ( V f ( x ) , X . ) . > 5k}.

This mapping has nonvoid closed convex values in T..M and is lower semicontinuous. The support planes

P(z) = span~P(x ) of this map coincide with the support planes T.'P of the cone T.V (T.V = T .F - T . F ) when x E G (k). Replacing, if n~ecessary, M for the natural support Af of the transversally convex subset V C .M, one can assume that T . P = T . M for Vx E G (k). Then the multivalued mapping

x ~ P(x): G (k) -+ 2 T M

has a smooth (of class C ~ ) single-valued selector by the following assertion: If H1 and H2 are two Hilbert spaces and Q: H1 --+ 2 H2 is a lower semicontinuous mapping with closed

convex values such that int Q(x) # 0 Vx E H1, then there exists a smooth (of class C ~ single-valued selector g: H1 -+ H2 of that mapping: g(x) E Q(x) Vx E HI.

Let us give the proof of the preceding statement (see [184] for its finite-dimensional version). It is sufficient to construct a single-valued selector g: H1 --* H2 locally in a neighborhood of an arbitrary point x0 E HI; this selector is globally constructed by means of the corresponding Coo-partition of unity.

Le t / ? , be a closed ball in H2 of radius r > 0 with center at the origin: /3, {z E H2 I IixI]2 _< e} (l[" ]li, i = 1, 2, is a norm on Hi). Since the mapping Q: H1 --~ 2 H2 is lower semicontinuous for all r > 0, there exists 5 : 5(e) > 0 such that - xolll < ~ implies Q(xo) c Q(z) + [1,. By the condition int Q(x0) r 0 , for any point Y0 C int Q(xo), there exists a # > 0 such that

vo + B. c Q(xo).

Hence assuming s = # we have

yo + / 3 , c Q(zo) c Q(x) + B, when Ilx - xolll < 6; (2.1)

2719

this implies that yo e Q ( x ) w h e n 1127-z0]ll ( ~, and therefore we can assume g(27) = yo when iix-270iil '( ~. One can check the above statement by means of the Hahn-Banach theorem given in terms of support

functions [146]. Namely, if crQ is a support function of a set Q c / / 2 , i.e., if aQ(p) = sup,~eQ(p, x)2, p � 9 then (2.1) implies that, for Vp � 9

(P, Y0)2 < (rQ(~)(p);

from this, by [t46], we conclude that yo �9 Q(x) when IIx - x01ll < ~. So the mapping P(27), x �9 G (k), has a smooth Coo-selector X (k), x �9 G (k), and therefore the smooth

vector field with required properties (1)-(3) is constructed. From the system { X (k), x �9 G (k)} of vector fields we now construct a single smooth C~176 field

X~, x �9 G, according to the following inductive procedure. Let V (1) be an arbitrary nonvoid domain with I )(1) C G (1). Assume y(1) = X(1) in G 0). Define a

smooth vector field y(2), 27 �9 G(2), by the formula

y(2) = cq(27)y(1) + (1 - O~l(27))Xz (2), 27 �9 C (2),

where O~l(X), a7 �9 a (2), is a s m o o t h Coo-funct ion wi th oQ(27) --_ 1 on lTrl, suppoq C a (1), 0 < O/I(X) __.~ 1,

z E G (2). Then y(2) is well defined on G (2), and, moreover,

y ~ 2 ) = y(1), 2; �9 T~rl;

y(2) < ,~(~)llY~(1)ll= + (1 -,~l(~)l lX~(~)l l~ < ,~1(~) + (1 - Oel(X)) = 1; 2 ; 27 - -

(Vf(x),y(2)\~ ,~ =Oel(X)(Vf(x,) , y(1)}~ +(1-Oel (X)) (Vf(x) ,X(2) )x _>

> ~l(X)al + (1 -~ l (X) )a= = as + ~1(27)(el - e = ) >__ e~.

Now, suppose that the smooth Coo-vector fields y(k), x �9 G (k), k _> 2, and the domains V (k-l), ~(k-1) C G (k-l) , V (k-D D G (k-2), such that

(i) y~(~) = y ( ~ - l ) on ~-(~-1);

(ii) IIYff)ll~ _< 1, 27 �9 G(k); y(k) \ > Sk, x �9 G (k) (iii) (Vf(x), x ,,~ _

have been constructed. Consider a domain V (k) such that ~(k) C G (k), V (k) D G(k-1); the possibility of the choice of such a domain can be provided by properly choosing the corresponding sequence {6k}. Let

u (k+l) =Olk(X)Y (k) + ( 1 - e e k ( x ) ) X (k+l), x �9 G (k+l),

where ak is a smooth Coo-function with ak(x) -- 1 on I 7"(k), suppak C G (k), 0 _< ak(z ) <_ 1, x E O (k+~). Then y(k+l) is a well-defined smooth Coo-vector field on G (k+l) and

y f f + l ) = y f f ) , �9 c g(k).

A calculation which is similar to that given above shows that

Ilgff+l)ll= < 1, ( V f ( x ) , y ( k + l ) ) x ~ r X e a (k'l-1).

By the construction, V (1) C V (2) C . . . ,

and o o

U V(k) = G. k=l

2720

C~-ps eudog rad i en t vec tor field. x 6 1 ) ,

Moreover, y(k) = y(k-~), x E l ) (k-~) . Thus the smoo th C ~ - v e c t o r field X~, x E G, is well defined by the formula

X~ = Y(~) for k sufficiently large.

The lat ter is the desired C~ vector field. Definit ion and Propos i t ion 2.1 is proved, Thus for a C L f u n c t i o n f defined on a Hilbert t ransversal ly convex subset V C Ad there exists a

In the const ruct ions in [72] we also used a normalized vector field Y~,

~k(~)X. (v/(z),X.)~, x E G (k+l),

Y~ = O, x c V k g (k+~),

where ak(x) , x E P, is a smoo th C~~ with 0 <_ ak(x) _< 1, ak(X) -- 1 on G (k), ak(z) =- 0 on {x E P lrnf(z) <_ 5k+~ }, which in the case under considerat ion is only cont inuous since it involves the mult ipl ier

1

(Vf (x ) , X~)~"

Nevertheless, we shall show tha t integral curves of this field exist and are unique, and this enables us to use all subsequent cons t ruc t ions f rom [72] (as is known, see, e.g., [942], in the inf ini te-dimensional case these curves can fail to exist). We begin wi th the proof of their uniqueness. Let ~ ( x ) , t C N, be a not necessarily unique t ra jec tory of the vector field y k , s tar t ing from a point x for t = 0. Note tha t if x E G = U~=IG(k) , then r C G for all t since Y) = 0 in some subdomain of G. If x r G, i.e., if x E V \ G C V \ G (k+i), then r = x, and there are no o ther trajectories. Taking this into account , it is sufficient to prove that if x E G, then there exists at most one t ra jec tory r of the vector field y k , s ta r t ing f rom a point x C G for t = 0. Denote by ~ ( x ) , t E R, the unique t ra jec tory of the smooth vector field

z) = { -k(~)x~ , x e a (k+~), 0, x E ~) \ G (k+l).

Let a(t) , t 6 N, be the solut ion of the scalar ordinary differential e q u a t i o n

: . (~( , ) (~)) , ~(o) : o,

where ot(z) : 1/(Vf(x),X~)~. According to [33], there exists a unique solut ion of this equat ion since c~(Fk(0)(x)) = c~(x) > 0, x 6 G; moreover, for this solution one has &(t) > 0, t E N, hence the inverse

function, ( r - l ( r ) , v 6 R, is defined. Let ~ptk(x); t 6 R, be an arbi t rary t ra jec tory of the vector field y k s tar t ing f rom x E G for t = 0. Let

~(~) = r r E R. Then

J~Or(X) : d@kc-~(r)(X) dff-l(T) k 1 k k dt dr : Y;~o_,(~)(.)~,(~) - Z~L~(~)(, ) : Z~(.),

thus ~a~(x), r E R, is a t ra jec tory of the field Z k, s tar t ing f rom x for t = 0. By the uniqueness, we have

~,,-(x) - ~,k,.(x), ~ e R .

So any t ra jec tory Ctk(x), t E N, of the vector field y k (if it exists) satisfies the relat ion

= ~ _ l ( ~ ) ( z ) ,

which provides the uniqueness. The above formula also gives the existence of the t ra jec tory of vector field yk , s tar t ing f rom x E G for t = 0.

2721

By having proved the existence of trajectories of the normalized vector field and their uniqueness, the subsequent constructions are completely similar to those of [72], and so we limit ourselves to the formulation of the corresponding results.

P r o p o s i t i o n 2.2 (Existence theorem). Let f be a C~-function bounded from above defined on a Hilbert transversally convex subset ]2 of a complete Riemann-Hilbert manifold 24, which satisfies the generalized (C) condition in the sequential form. Then it assumes on ]2 its greatest value: there exists a point xo E Y, such that

f ( x o ) = s u p f ( x ) . zEY

Moreover, the critical set of that function is compact on every critical level, i.e., the set

K c ( f ) = {xE V l f ( x ) = c , m y ( x ) = 0 }

is compact. Let cat (A; ]2) = cat (A) be a Lusternik-Schnirelman category of a closed subset ,4 C ]2 (see [16, 20]).

Let us recall that cat (A) is the least integer k > 0 such that there exist k closed subsets A1 , . . . ,Ak C ]2, each of them being contractible in P to a point, with property ,4 C Uk=lAk. If such k fails to exist, one assumes cat (.2.) = +oc.

For a smooth CLfunct ion f: P --* IR, let us denote

Cm(f) = inf sup(-f(x)). c a t ( . A ) > r n xEA

The following main theorem holds true. T h e o r e m 2.1 (Minimax principle). Let f be a Cl-function defined on a Hilbert transversally convex

subset ]2 of a complete Riemann-Hilbert C~176 24, which satisfies the generalized (6") condition in the sequential form. I f m <_ n < cat (]2) = cat (]2;]2), and

-o0 < - c = cm(f) . . . . . Cn(f) < CX~,

then c is the critical value o f f , and cat(Kc) > n - m + 1 . In particular, if n = m, then Kc 7 ~ ~. If at least one number cj( f ) is equal to +0% then f assumes arbitrarily small values on the set K of its critical points, which, hence, is infinite.

C o r o l l a r y 2.1. Let f be a Cl-function bounded from above defined on a Hilbcrt transversally convex subset ]2 of a complete Riemann-Hilbert Coo-manifold J~, that satisfies the generalized ( C) condition in the sequential form. Then it has at least cat (]2) = cat (]2; ]2) critical points.

3. T h e Basic O p t i m a l P r o b l e m (P) a n d t h e G e n e r a l i z e d (C) C o n d i t i o n

in the Sequential Form

In this section, the basic optimal control problem (P) will be considered and the general conditions of the applicability to it of the Lusternik-Schnirelman theory of Sec. 2 will be formulated.

Let M be a smooth (of class Coo) n-dimensional manifold realized as a closed submanifold of the Euclidean space R d. Let us consider a smooth control system

IT&

= f (x ) + E uigi(x)' x e M, x(O) = Xo, (3.1) i-----1

and suppose that the smooth C~176 fields f , gi, i = 1, 2 , . . . , m, which in the case under consideration are identified with the smooth d-dimensional functions tangent to M at its every point, satisfy the following continuation condition: for every x0 E M and T > 0 there exists a bounded, on bounded subsets of N+, function z/,0,T:N + ~ N+ such that , for any admissible control u(-) E L~[0, T], for the corresponding trajectory x(t) = x(t; x0, u(.)) of the system under consideration, the following inequality holds true:

IIx(.)llc,,to, --- o,T(llu(')llLrtO, ) (3.2)

2722

This condition is valid, e.g., if

(x,f(x)) <_ k1(1 + Ixl ), < k~(x + i = 1 , . . . ,rn, kl,k2 = eonst _> 0, (3.3)

where (., .) is the standard inner product on R d, l" I is the corresponding (Euclidean) norm. In what follows we shall consider a particular class of systems of form (3.1), which we shall call the

smooth control systems of constant rank [68]. These systems are distinguished by the following condition: a natural map (the so-called input-output map)

F,,o,T:U(" ) ~ x(T;xo,u(.)):L'~[O,T]-+ M

has a constant rank for every x0 C M, T > 0 (i.e., the rank of the d i f f e r e n t i a l F;o,T(U ) = (Fzo,T).,u(.) of this map is independent of u(.) e L•[0, T]).

Note that the map F~o,T: L~[0, T] ~ M belongs to class CO~ moreover, the map

u(.) ~ x(.;Xo,U(.)):L~[O,T] ~ wd[o, T]

also belongs to the same class and, in particular, it is continuous; hence, by the simplest embedding theorem wd[0, T] ~-~ cd[o,T], the convergence in Lr[0 , T]-topology of a sequence {un(.)} implies the uniform convergence of the corresponding sequence {Xn(')}, Xn(t) = x(t; Xo,Un(')), 0 < t < T, of trajectories of system (3.1).

We shall suppose henceforth that the smooth control system (3.1) satisfies the following condition of finitely definiteness [68]: in a neighborhood of any point x0 E M and for some integer s > 0,

adS+:fg i = ~ ~ a~i~ad~fgz, ~=0 8=1

i = 1,. . ,,% (3.4)

with smooth functions a si in that neighborhood. Here and below by ad X we denote the operator ad X Y -

IX, Y], where [X, Y] X o Y - Y o X are the Lie brackets of vector fields X, Y and ad kX are iterations (powers) of that operator:

"ad~ = Id, a d ~ X Y = a d ~ - l X ( a d X Y ) = a d X ( a d ~ - l x y ) .

There exist sufficient conditions under which a finitely defined system (3.1) is a system of constant rank [68]; these bang-bang conditions are as follows: in a neighborhood of any point z0 E M,

k m

[adkfgi'gJl=EE i j k . , ~ . a ~ act .r g~, a=0 f l=l

i , j = 1 , . . . , m ; k = 0 , 1 , 2 , . . . , (3.5)

ijk with smooth functions a z in that neighborhood. If (3.1) is a finitely defined control system of constant rank, then a reachable set

9d~:o(T) = {z(T;zo,U('))lu(.) e L~[O,T]} = F,:o,T(L~[O,T])

of that system from a point x0 in time T > 0 is an (immersed) submanifold of the base manifold M; more- over, PJ~0(T) is an open subset of the derived orbit O,(T)(T) (see [86]) of the system under consideration, which passes through the point x(t) = x(T; xo, u(.)) with arbitrary u(-) e Lr[0 , T].

Now if V is a transversally convex subset in M which is transversal to a reachable set 9.1,0(T), then, since F,o,T: Lr[0 , T] --* 9"l*0(T) is a surjective submersion and 92,0(T) f'l V is transversally convex, the path space

p(T) L~[O, T] z(T; V} �9 0,y = {u(.) e I xo,U(.)) 6 : FZol, T(V)

2723

is a Hilbert transversally convex subset in L~[0, T]. Let f0: iRd x IR m ---* R be a smooth (of class C cr integrand which satisfies the following conditions:

.f~ _> klul 2, k = const > 0; (3.6)

If~ <~ a(x)lul 2 -4- b(x)lul + c(x); (3.7) If~ <_ d(x)tul-4- e(x); (3.8)

( f ~ > ~(x)lyl 2, (3.9)

where a, b, c, d, e are nonnegative continuous functions on ]R d, # is a nonnegative function bounded away from zero on compact subsets of• d, i.e., for any compact K C ]Ra there exists a #k > 0 such that #(x) > #k when x E K.

Under conditions (3.7)-(3.8) a functional

T

J(u)= f f~ 0

is well defined over the set of all trajectories of system (3.1). This functional is Gateaux differentiable as a function

T

LF[0, T]-~ ~t, u(.) ~ J (~ )= ff~ 0

and for its Gateaux derivative, by, e.g. [72], one has the representation

J ' (u) ( t ) = f ~ r 0 < t < T, (3.1o)

where G(x) = ( g l ( x ) , . . . , g i n ( x ) ) , r = r is a solution of the adjoint system along a trajectory x(t) = z(t; x0,~( . )):

= fo(x(t),~(t)) - r + G~(z(t))u(t)), o < t < T, (3.11)

with the "target" condition r : 0 . (3.12)

Let us show that the conditions (3.7)-(3.8) together with the continuation condition (3.2), provide that the operator

u(.) ~ J ' (u ) : f ~ - r --~ L~[0, T]

is continuous, and hence they give the Fr~chet continuous differentiability of the functional

T T

u(.) ~ y(u)= f f~ = f f~ o o

To this end, let us begin with the proof of continuity for the operators

u(.) ~ f ~ = f~ L~n[0, T] ~ Lla[0, T],

u(.) ~ f ~ f~ : L~[0, T]--) L~[0, T],

(the fact that these operators map from L~[0, T] into /d[0, T] and into 5~'[0, T] respectively follows from (3.7) and (3.S)).

2724

Let us prove, for example, the continuity of the first operator (the continuity of the second one is stated in the same manner).

Let u0(') E L~[O, T] be an arbitrary control, and

We must prove that

un(.) ~ u0(-) in L•[0,T] (n ~ co).

f o ( ~ , ( . ) , ~ ( . ) ) o f~(xo(.),Uo(.)) in L~[O,T],

where, as above, xn(-) = x(.; x0, un(-)). To this end, let us write the following representation:

f~0(x~(-), u n ( - ) ) - fO(xo(.), Uo(.)) = [fO(xo('), Un(')) - g(X0('),U0('))] + [fO(xn('), U~(')) - fO(Xo('), Un('))].

The first term in this expansion,

f ~ f~ ,

tends to zero in L~[0, T] by the Krasnosel'skii theorem (see [117, 146]) since the operator u(.) f~ [x0(-)is fixed] maps, by (3.7), from the space L~'[0, T] into L~[0, T].

Let us prove that f ~ f~ , 0

in L~[0, T] as n ---* c~. Denote by ~n(t), 0 < t < T, the function

~ n ( t ) = I f O ( z n ( t ) , U n ( t ) ) -- f O ( x o ( t ) , u , , ( t ) ) l .

Then w,(t) ~ 0 as n --~ c~ in measure since the convergence in L~'[0, T] of the sequence (u,~(.)} to u0(') imp!ies the convergence . . ( . ) ~ u 0 ( ) in measure, ~nd in this case

xn(t) ~ x0(t), 0 < t < T.

Further, by the convergence u,,(-) ~ u0(') in L~'[0, T] and by (3.7), the family {~a,,(.)) has absolutely equicontinuous integrals, hence for this family all the conditions of the Vitali theorem [140] are valid; according to the latter, for every bounded measurable function ~: [0, T] ~ I~, we have

T

f~a~(t )~( t )dt ~ as n ~ c~. 0

0

In particular, assuming ~(t) = 1, we obtain the required convergence

f~ - f~ --~ 0

in LId[O, T]. Having in mind the continuity of the operators

u(.) ~ f~~ u(.) ~-~ fo(x( . ) ,u( . ) )

which we have already proved, now for the proof of continuity of the operator

u(.) ~ J ' ( u ) = f~162 L~[0, T]

it is sufficient to prove the continuity of the map

u(.) ~-~ r L~[0, T] ~ wd[0, T].

2725

By the Cauchy formula,

t

r162162 f s~ 0 < t < T , 0

where F(.) is the fundamental matrix of solutions of the homogeneous adjoint system

i~ = -~(f~(x(t)) + a~(x(t))u(t)),

that is normalized by the condition P(0) = I. From this expansion it follows, on one hand, that the operator u(.) ~ r u(.)) maps from Lr[0, T ] into W~[0, T], by (3.7). Further, the operator

r(.; u(.)): Lr[0, w xd[0, rl

is continuous; hence, by the continuity of the superposition operator

~(.) ~ f~ Lf[0,T],

we get the continuity of the operator

0(.; u(.)): Lr[0, r] -~ w~[0 , T].

In particular, if u,,(-) --* u0(.) in Lr[0, r ] , then, by the embedding wd[0, T] ~ cd[0, TI, we have

O(.; u~(.)) = 0~( . ) ~ r = r ~0(.)).

This proves that the operator

u(.) ~ f~ r L~"[0, T] ~ n~"[0, T]

is continuous, and hence the functional

T

J :u( . ) �9 L~"[0, T] ~ / f ~ a ]

0

is continuously differentiable in the Frechet sense. The basic optimal control problem (P) consists in mini- mizing the functional

T

J(u) = f f~ 0

over the set of all trajectories of the finitely defined system

m

dt

x E M , i = l

of constant rank under conditions x(O) = xo, x(T) �9 V, where V is a transversally convex subset of M, which is transversal to 9.1~o(T), in the class L~[0, T] of admissible controls.

This problem has an equivalent description as the problem of maximizing the Cl-function - J defined on the path space

p(T) L'~iO, TIIx(T) x(T;xo,u(.)) �9 V} �9 o,v : {u(.) �9 which, as we already know, is a Hilbert transversally convex subset of L~ n [0, T].

2726

For an application of the Lusternik-Schnirelman theory developed in Sect. 2 to the problem stated above, it is necessary to check that - J satisfies the generalized (C) condition in the sequential form. Before presenting the precise formulation of the corresponding result let us note that critical points of the function

"D(T) coincide with extremal controls. -J on - - z o , V

Here, as usual, by an extremal control is meant a control that satisfies Pontryagin's maximum principle the latter can be stated (see [72]) for the problem under consideration in the following strong version:

P r o p o s i t i o n 3.1. An admissible control u(t), 0 <_ t <_ T, is eztremal if and only if there exists a * t solution r 0 < t < T, r 6 T;(t)O~(t)( ), 0 < t < T, of the adjoint system

(~ = f~ - r + a~(x(t)u(t)) (3.14)

with the "target" condition

such that for almost all t, 0 < t < T,

r e V, (3.15)

f~ u(t)) = r (3.16)

The main result of this section is the following T h e o r e m 3.1. Let M be an n-dimensional smooth of class C ~ closed submanifold of R d, fo :

R d x R "~ ~ IR be a smooth integrand (of class C~176 which satisfies conditions (3.6)-(3.9), and (3.1) be a smooth finitely defined system of a constant rank, which satisfies the continuation condition (3.2). Let V be a transversaUy convex subset of M, which is transversal to a reachable set f21~o(T ) of the system under consideration. Then the function - J is smooth of class C 1 on the Kilbert transversally convex subset,

p(r) z0,v : {u(.) E L~'[O,T]Ix(T) : x(T;xo,U(.)) 6 V},

and for this functio n the generalized (C) condition in the sequential form is valid. Let us give the proof of this theorem, noting first that [70] and [72] present its weaker forms: the

corresponding result was proved for the case when f~ u) has a sublinear growth in u, and, in addition, f~ u) and f~ u) both satisfy a uniform Lipshitz condition in x.

However, we must recognize that the proof given in [70] has some gaps, in particular, it requires an additional Lipshitz condition in u for f~ u); this was pointed out to the author by N. N. Petrov.

Let (., ")u be the natural Riemannian structure on L~'[0, T], i.e., the inner product on the tangent space Tu(.)L~'[O,T] the latter is identified with u ( . )+ Lr[0, T]:

(v.(.), w.(.)>. = (v(.), w(.)) rEom,

where v~(.) = u(.) + v(.), w,(.) = u(.) + w(.) e T~(.)L~'[O,T]. Let 1) p(r) be the path space associated x0~g with the problem (P). To verify the generalized (C) condition in the sequential form we must prove that

"re(T) with {-J(un)} bounded, and with every sequence {u~,(.)} C--z0,V

m - j ( u n ) = sup II ~,,,, (')11 ~ . _<1

( - v j ( u . ) ( . ) , v,,, (.)).~ --,o,

contains a convergent subsequence. From the growth condition (3.6), it follows that every sequence {un(.)} with the properties mentioned above is necessarily bounded:

Ilu-(')llL EOm Ca = const .

Hence, by the weak compactness of a closed ball in a Hilbert space, we can assume that

u,( .) ~ Uo(') weakly in L~[O,T].

2727

Let us prove that some subsequence of the sequence {u,,(.)} strongly converges to u0('). By the continuation condition (3.2), we have that the sequence {x,~(.)}, x,,(t) = x(t;xo,u,( .)) , of trajectories of the system (3.1) corresponding t o un(-) and to the initial condition x0 is equibounded. Moreover, this sequence is equicontinuous since

[ ~ . ( t ) l < If(Zn(t))l + Ia(z.(t))u.(t)l < Ca + c41u.(t)l,

and, for all t', t" E [0, T], t' _< t",

Ix(t') - x ( t") l _< fL .(t)ldt <_ ca l t ' - t" l+c4 / ,~o( , )Ld , t ~ t ~

(I )'" <_ c.lt' - t"l + dr < It'- t"l ' / ' (caT ' / ' + t I

Thus, without loss of generality, we can assume that

CLC4).

�9 xo(t), o < t < T,

and x0(') is a trajectory of the system (3.1) corresponding to a control u0(') and to the initial condition x0 since the weak convergence un(-) -+ u0(-) and the uniform convergences

zn(t) ~ xo(t), f (xn(t)) ~ f(xo(t)), gi(x,,(t)) ::=t gi(xo(t)), i = 1 , . . . , m , 0 < t < T,

imply that t

xn(t) = xo + / ( f ( x ~ ( 7 ) ) + G(x~(r))un(r))d7 o

t

Xo + ](f(zo(T)) + G(xo(T) )Uo(r )) d-r. . 1

0

.D(T) Consider the sequence {xn (T)} of the right end points of these trajectories; by the choice of Un(') E --zo,V' we

have x,,(T) C V. Let "D(T) = F~ol, T(Xn(T)). Since F,o,T is a surjective submersion, for all n = 1,2,. . . --~0,~,(T) p(T) is a closed submanifold of L~' [0, T] of class C ~ completely lying in the path space ~(T) xo,xn(T) zo,V"

Denote by (T) (T) P2o,,~.(T)(U,O an orthogonal projection map from L~'[0, T] onto L o,~.(T)(Un ), where

L (T) tun~ u(.)rxo,xn(T) at a point �9 0,~(T)~ j is the closed subspace in the representation of the tangent space T ,n(T)

U n ( ' ) : T "r~(T) L (T) tu u, , ( ' )rxo,x , , (T) = Un(") + x0,z, , (T)k n/ .

The orthogonal projection map p(T) tu ~ also acts on T,,.(.)L~[O,T] in a natural way, and its action Zo,xn(T)k n/ on an arbitrary element v(-) C L~[0, T] is equivalent to the addition to v(-) of a general element from the kernel of the map F~0,T in L~" [0, T]; thus, in particular,

p(T) (T)(U.)VJ(u,~)(t ) = u~(t) + f~ r 0 < t < T,

where ~b,,(t) = ~b,,(t; u,,(.)) is a solution of the adjoint system

(~n = f~ u~(t)) - Cn(f~(x.(t)) + G~(xn(t))u,,(t)), 0 < t < T. (3.17)

2728

Hence, by the condition m - j ( u n ) --* 0 as n --* c~, we have

o , - s u p ( - V J ( u , ) ( . ) , v , . _ II ~,,,, ( ')tl~,n <1

vun ( ' )ETun(. ) "p(T) --=:0IV

The basic optimal control problem (P) consists in minimizing the functional

T

J(u) : / f~

0

dt

over the set of all trajectories of the finitely defined system

m

= f (x ) + E uigi(x), i = 1

x E M ,

of constant rank under conditions x(O) = Xo, x(T) 6 V, where V is a transversally convex subset of M, which is transversal to 92~0(T), in the class L~[0, T] of admissible controls.

This problem has an equivalent description as a problem of maximizing the Cl-function - Y defined on the path space

p(T) Lr[0 ' x(T; V} .o,V = {u(.) 6 T][x(T) : Xo,U(.)) 6

which, as we already know', is a Hilbert transversally convex subset of LT[0 , T]. For an application of the Lusternik-Schnirelman theory developed in Sect. 2 to the problem stated

above, it is necessary to check that - J satisfies the generalized (C) condition in the sequential form. Before presenting the precise formulation of the corresponding result, let us note that critical points of the function

p(T) coincide with extremal controls. -J on xo,V Here, as usual, by an extremal control is meant a control that satisfies Pontryagin's maximum principle

[143]; the latter can be stated (see [72]) for the problem under consideration in the following strong version: P r o p o s i t i o n 3.1. An admissible control u(t), 0 <_ t <_ T, is extremal if and only if there ezists a

solution r 0 < t < T, r e T~*(t)O~(t)(t), 0 < t < T, of the adjoint system

= f~ - r + G,(x(t)u(t)) (3 .14)

with the "target" condition

such that for almost all t, 0 < t < T,

r e --T~(T)V, (3.15)

f~ = r (3.16)

The main result of this section is the following T h e o r e m 3 .1 . Let M be an n-dimensional smooth (of class C ~ closed submanifold of R 4, fo :

R d x R ~ --* R be a smooth integrand of class C a , which satisfies conditions (3.6)-(3.9), and (3.1) be a smooth finitely defined system of constant rank, which satisfies the continuation condition (3.2). Let V be a transversalIy convex subset of M which is transversal to a reachable set 92,o(T ) of the system under consideration. Then the function - J is smooth of class C 1 on the Hilbert transversalIy convex subset,

p(T) LT[0 ' x(T; V}, �9 0,v = {u(.) e T]Ix(T) : xo,U(')) e

and for this function the generalized (C) condition in the sequential form is valid.

2729

Let us give the proof of this theorem, noting first that [70] and [72] present its weaker %rms: the corresponding result was proved for the case when f~ u) has a sublinear growth in u, and, in addition, f~ and f~ both satisfy a uniform Lipshitz condition in x.

However, we must recognize that the proof given in [70] has some gaps, in particular, it requires an additional Lipshitz condition in u for f~ u); this was pointed out to the author by N. N. Petrov.

Let (., ")u be the natural Riemannian structure on L~[0, T], i.e., the inner product on the tangent space T,4.)L~[O,T]; the latter is identified with u(-) + Lr[0, T]:

< ~ ( . ) , ~ ( . ) ) ~ = (v( . ) ,~( . ) )Lrt0,~ ,

where v~(.) : u(-) + v(.), w,,(.) = u(.) + w(.) 6 T~(.)L~'[O,T]. Let ]2 : p(T) be the path space associated zo,V with the problem (P). To verify the generalized (C) condition in the sequential form we must prove that

every sequence {un(-)} C ~ D(T) with { -J (u~)} bounded, and with a~01V

~_j(~) : sup ( -Va(un) ( . ) ,%, ( . ) )~ . --+0, I 1 ~ (')t1~= <_l

~~176 (.)~T~o<.,,,,<~o?,.

contains a convergent subsequence. From the growth condition (3.6), it follows that every sequence {un(.)} with properties/nentioned above is necessarily bounded:

Ilun()llL~t0,~ < C l = c o n s t .

Hence, by the weak compactness of a closed ball in a Hilbert space, we can assume that

u~,(.) --+ u0(.) weakly in LT[O , T].

Let us prove that some subsequence of the sequence {un(.)} strongly converges to u0(.). By the continuation condition (3.2), we have that the sequence {xn(.)}, xn(t) = x(t;xo,u~(.)), of trajectories of the system (3.1) corresponding to u,,(.) and to the initial condition x0 is equibounded. Moreover, this sequence is equicontinuous since

lx,(t)l __ If(x,~(t))l + Ia(x,( t))u,( t) l < ca + c41u,(t)l,

and, for all t',t" E [0, T], t' < t",

t t ' t ,

- ~(t")l < / l ~ ( t ) l Iz(t') * / l t

dt _< c31t' - t"l + c , / 1 < ~ ) 1 <_ t '

�9 n(t) ~ xo(t),

imply that

( j )'" < e d i t ' - t"l + c,t"4VC-:e-t ' lu,(~)l 2 dr <_ I t ' - t"l'12(cam '12 + CLC4). t '

Thus, without loss of generality, we can assume that

xn(t) ~ xo(t), 0 < t < T,

and z0 (.) is the trajectory of the system (3.1) corresponding to the control uo(.) and to the initial condition x0 since the weak convergence un(.) ~ uo(') and the uniform convergences

f(xn(t)) ~ f ( zo ( t ) ) , gi(x,,(t)) =~ gi(zo(t)) , i = 1 , . . . , m , 0 < t < T,

t

xn(t) = Xo + J ( f ( xn ( r ) ) + G(xn(r))un(r)) dr ::~ o

2730

t

xo + / ( f ( x o ( r ) ) P

+ G(xo(r) )uo(r) )dr .

0

..D ( T ) Consider the sequence {xn(T)} of the right end points of these trajectories; by the choice of un(') 6 --~0,v, we

have ~ ( T ) 6 V. Set p(r) = FL~r(~,(T)) . Since F,o,r is a su,jective submersion, for an ~ = 1,2, z o , z . ( T ) "" "'

~D( T) xo,z,~(T) is a closed submanifold of L~[0, T] of class C ~ completely lying in the path space "D(T) ~' ~ 0 ~ V "

(T) Denote by P~o,z~(T)(Un) an orthogonal projection map from L~[0, T] onto ~x0,z~(T)~r'(T) (unj, ~ where

L(T) (u u(.)t"zo,z.(T) *0,*=(r)v n) is the closed subspace in the representation of the tangent space T .n(T) at the point u~(.):

T .r~(T) ~~ ),- o,~~ : u ~ L~ l~~

The orthogonal projection map p(T)xo,x.(T)(Itn) also acts on T~.(.)L~[O,T] in a natural way, and its action

on an arbitrary element v(.) E L~[0, T] is equivalent to the addition to v(-) of a general element from the kernel of the map F" 0,r in L~[0, T]; thus, in particular,

P~:) (T ) (U. )VJ(u . ) ( t ) = u . ( t ) + f~ -- ~b.(t)a(x,~(t)), 0 < t < T,

where g&(t) = r un(')) is a solution of the adjoint system

~n 0 X = f ~ ( . ( t ) , u . ( t ) ) - ~b.(f~(x.( t)) + a=(z . ( t ) )u . ( t ) ) , 0 < t < T . (3.17)

Hence, by the condition rn_z(un) --+ 0 as n --+ c~, we have

0 ~- m _ , ( ~ . ) = sup (-VJ(u,~)( . ) ,vu.( . ) ) ,~. >_

v,,,~ (.) eTu. (.)'P(v) V UI

> 1 ~ - - 7 " f t" \ t" \ / "~ k sup ~ - w t u . a . j , v . . ~ . ~ . . =

II ,,,,. (011 ,,. < 1

= sup (-P~:)(T)(U.)VJ(u.)(.), v,,.(.)>,~. = Ibu. (-)llu. <~

v~.(.)eT~.(.)Lg[O,T]

= IIpir)~.(r)VJ(u-)( ') l l , ,~ --I Ip~r)~.(m)VJ('~,,)()- '*,~(')llLr[o,:q =

= IIf~ u = ( - ) ) - ~,(-)a(x=(.))llLr[0,:q.

So we have the following convergence in L~[0, T]:

w,(.) = fo(~, ( . ) , u . ( . ) ) - r176 --, o as ,~ + ~ . (3.18)

By the growth condition (3.8), we get

If~ <_ d(x . ( t ) )b . ( t ) l + e(x,~(t)), 0 < t < T;

this gives, by the equiboundedness of {x.(.)} and by the condition l lu-( ')l lnrt0m ~ Cl, the bound

IIfO(~n('), ~.( '))IILr[0,~ __< c5.

Hence the convergence (3.18) implies the boundedness in L~"[0, T] of the sequence {~,.( .)G(x.(-))}:

II~,(.)G(~,(.))It ~m[om <_~. (3.19)

2731

Let P n(-) = F~(.; u,,(.)) be a fundamental matrix of solutions of the homogeneous adjoint system

r = - ~ , . ( h ( ~ . ( t ) ) + G(~.(t))u.(t)), (3.20)

normalized for t = 0 by Ca(0) = I, where I is the identity matrix. Then ~,,(t) = r 0 < t < T, is the solution of (3.20), and the solution G,(t), 0 < t < T, of the inhomogeneous adjoint system (3.17), by the Cauchy formula, is represented in the form

t

r = r + ff~ = 0

where

=~,.(t)+G,(t)r,,(t),

t

e . ( t ) = f fo (x . (~ - ) ,u . ( , - ) ) rx ' ( , - ) a , , o < t < r 0

It is easy to see that both sequences {Pn(')} and {F~I(.)} are equibounded:

(3.21)

llr~-'(.)llc,,,,,,[o,~, llr,-,(.)llc,,,,,,[o,:~ < c~;

moreover (see [70]) they are equicontinuous, so, without loss of generality, we can assume that

r . ( t ) ~ r0(t), r ;~ ( t ) ~ rob(t) , 0 < t < T,

as n ~ ec, P0(-) being a fundamental matrix of solutions of the system

4 . = -~,0(f~(x0(t) + G(xo( t ) )u0( t ) ) .

By the growth condition (3.7), we obtain

T

II'~r,(')llC'to,T] ~ f lf3(~.(t),u.(t))llrz~(t)ldt < 0

T

< ~ f(a(~.(t))l~,,(t)l ~ 0

+ b(z,~(t))lu.(t)[ + c (x . ( t ) ) )d t < cs = c o n s t

since {xn(.)} is bounded in cd[O,T]; a, b, c are continuous functions, and

II~,,,(')llLrto,~ -< ,~1 = c o a s t

Thus, by the representation (3.21), and by the boundedness condition (3.19), we obtain that the sequence vn(') = ~,( ')G(xr,(-)), n = 1 ,2 , . . . , is bounded in L~'[0, T]:

II",,()llL~to,~ = II~,.,(')G(;:,,,,(-))ll,r.,rto,:q -< e,, : r (3.22)

Now literally repeating the argumentation given in [70], we deduce that the sequence {vn(-)} is uniformly convergent to vo(') = 9zo(')G(xo(')):

v,,(t) ~ vo(t), o < t < T. (3.23)

2732

Consider the family of functions

t

= - uo(r))dr, 0

0 < t < T .

This family is bounded in cd[o, T],

[l%(')t lc, to,T] ~ e l 0 = c o a s t ,

and is equicontinuous; the latter fact follows from the boundedness of the weakly convergent sequence {u,~(.)}, from the equicontinuity of {xn(.)}, and the equiboundedness of {P,(.)}. Since, for every t, by the weak convergence un(.) --+ u0(') in L~[0, T], we have %~(t) --+ 0, then, without loss of generality, we can assume that ~n(t) ~ 0, 0 < t < T. Hence

T

f ~.(t)r.(t)a(~.(t))(~.(t)- u o ( t ) ) 0

dt ~ O. (3.24)

In fact, integrating by parts, we have

T

/ On(t)Fn(t)G(z,~(t))(u,~(t) - uo(t))dt

o

T

= ,~n(T)qI,,(T) - / ~, , ( t )~, , ( t )dt =

o

T

: O n ( T ) g n ( T ) - ff~ 0

Since ~n(t) --+ 0 and {0n(T)} is bounded,

dt.

r --+ 0 as n ~ ~a.

Further, we have T

f f~ <_ 0

T

< o~ta~Tl%,(t)l/[a(zn(t))lu.(t)l ~ + b(xn(t))[u.(t)[ + c(x.(t))] d t . c r < 0

<_ II%(.)llcqo,~crc8 --, o, which already prove.s (3.24). Moreover, by the weak convergence un(') "-+ uo(') we have the convergence

T

f fg(..(t),~o(t))(~.(t) - u o ( t ) ) o

d t ~ 0 a s n ~ c , o .

Thus, by (3.18), (3.21), (3.23), (3.24), (3.25), we obtain that

(3.25)

T

[(f~ u , , ( t ) ) - f ~ u o ( t ) ) ) ( u . ( t ) - ,~o(t) ) ] ,~t =

o

2733

T

= f [ ( f ~ r uo(t))]dt+

T

+ f r - uo(t))

T

T

d t - f uo(t)) 0

= - uo(t))dt

T

+ f ~ . ( t ) p . ( t ) a ( ~ . ( t ) ) O , . ( t ) - uo(t)) o

T

+ fv . ( t ) (u . ( t ) - uo(t))dt+ 0

T

d t - / f ~ - uo(t)) 0

Since, by the ellipticity condition,

dt =

dt 4 0 . (3.26)

T

0 _ 0 [f~,(xn(t),u,~(t)) f~,(x,~(t),uo(t))](u,~(t)- uo(t))

o

dt =

T 1

0 0

T

+ s ( u . ( t ) - uo(t))(u.(t) - uo(t), ~,,(t) - ~0(t)) dt >_

T

dt > #o f lu.(t) - uo(t)? dt, 0

because #(x , ( t ) ) >_ #o by boundedness in cd[o,T] of the sequence {x,(.)}, from the convergence (3.26) we obtain

Ilu,(.) - u0(')llL~[0,n --' 0 as n -* ~ .

The theorem is proved.

4. W~[0, T]-Regular i ty of t he Basic O p t i m a l Con t ro l (P)

In this section, we shall investigate some regularity properties for the basic optimal control problem (P) associated with the finitely defined smooth control system (3.1) of constant rank, which motivate the construction of a Morse type theory in Sect. 5.

We need some additional conditions for the smooth integrand f0 : Rd • ]Rm ~ ]R concerning growth properties of its second derivatives, namely we shall claim the following conditions to be valid:

If~ u)l, IfX~(x,~ u)l ~ d(x)lut + e(x), (4.1)

If~ u)l ~ a(x)lul 2 + b(x)lul + c(x), (4.2) with continuous a, b., c, d, e : ]R d ~ R+, which, without loss of generality, we can assume to be equal to the corresponding functions in the bounds (3.7)-(3.8), thus

IfO(x,u)l <d(x)lul+e(x), If~ <a(x)lu[ 2 +b(x)lul+c(x).

The following assertion holds true. P r o p o s i t i o n 4.1. Let the first inequality (4.1) be valid, and suppose that conditions (3.7)-(3.9), (3.2)

also hold. Then every extremal control u(t), 0 < t < T, for the problem (P) belongs to the class w2rn[0, T] (i.e., there exists v(.) e W2~[0, T] with v(t) = u(t) for almost all t). Moreover, if {un(')} is a sequence

2734

of extremal control~ converging to an extremaI control uo(t), 0 < t <_ T, in L~[0, T], then this ~equence on erge to uo(') in W [0, T].

Proof . Let u(.) be an arbitrary extremal control, let x(t) = x(t; xo, u(.)) be a trajectory of the system (3.1) corresponding to that control, and r = r be a solution of the adjoint system (3.14) (see Proposition 3.1). Let us prove that the equation

f ~ = r 0 < t < T, (4,3)

has a unique solution in the class W~[0, T], and hence, by Proposition 3.1, the first statement is valid. This can be done with the help of the Hadamard theorem (a global version of the inverse function

theorem), which states that if F : R t -* W is a Cl-mapping with property

det OF(x) ~ l _>a>O,

then there exists a unique Cl-mapping G : N t ~ 1R l such that F o G = G o F = idx, (see [141]). In fact, consider two sequences {xn(-)}, {r of Cl-functions that converge to x(-) and to ~(.)

respectively in the space W~ [0, T]. Then for every mapping

F~ : ( ~ ) ~-~ ( t , f ~ - r T : RI--* R I, l = r n + l ,

one can apply the Hadamard theorem stated above, since from (3.9) and from the boundedness in cd[o, T] of the sequence {xn(.)} it follows that

>_ olyl 2

thus, in particular,

for some a > O.

V y E R '~, uEIR m,

det OFn(t,v) o x Ox = d e t IfS~( . ( t ) , v ) l > a > 0,

(4.4)

2735

Hence, for all n = 1, 2, . . . , there exists a Cl-function vn(t), 0 < t < T, that satisfies the equation

0 < t < T . (4.5)

Let us prove that some subsequence of the sequence {vn(-)} converges in W2~[0, T] to a function v(.) which, by the convergences

xn(t) ~ x(t), r ~ r C(x.( t)) =2; C(x(t)), 0 < t < T,

is a unique solution of (4.3); the uniqueness follows directly from (3.9). By (4.4), we have that the matrix f~ is invertible, moreover,

i fo~i(Xn(t) ,vn( t ) ) [ <_ --,1 0 < t < T. (4.6) #o

By differentiation of (4.5), we have

iJ,(t) = f ~ 1 7 6

+r + Cn(t)G(x,,(t))) , 0 < t < T. (4.7)

Hence, by the first inequality (4.1) and by (4.6), we have

d[vn( t ) l 2 = 2(v~(t),i)~(t)) < 2 (If~ #o

where

+(l(b,,(t)llG(z.(t))l + Ir < 2

< - - ( d ( x . ( t ) ) l ~ . ( t ) l l v . ( t ) l 2 + e (x . ( t ) ) l Jc . ( t ) l l v . ( t ) l+ tZo

+(l~b.(t)lla(x~(t))l + Ir

< - - w

2r.(t)~0 (Iv"(t)lZ + 21v"(t)l) ~ 4r"(t)~ (Iv"(t)12 + l),

r.(t) = max{d(x.(t))lSz.(t)l + e(x.(t))l&.(t)], I(P.(t)lla(x.(t))l + ir 0 < t < T.

Since xn(.) ~ x(-), Ca(-) ~ r in W2m[0, T], we have

~.(.) �9 z~[0, T], T

f r2n(t) dt < c = const .

o

Moreover, the family { r 2 (-) } has absolutely equicontinuous integrals since { r . (.) } converges to some function in L2 [0, T]. Thus we have the inequality

dlv.(t)l 2 dt

d 4rn(t) dt(Ivn(t)12 + 1) < (IVn(t)l 2 + 1)

#0

which, by the Gronwall lemma, gives the bound

t

4/.0 f ~.(,-) d,- lv,(t)l 2 _< Iv,~(t)l 2 + 1 <_ (Iv,(0)l 2 + 1)e 0 <

< cle 1/t~~176 ~ c2 = const .

(The boundedness of the sequence {Iv,,(.)l 2 } follows from (3.9), and from the boundedness of {r and of,{xn(.)}.) Thus the family {v,(.)} is bounded in C'~[0, T]. Therefore, by (4.7), we have

16,(t)l < 4/tZor.(t)ca, c3 = const ; (4.s)

[;;2aence the sequence {~)n(')} is bounded in L~[0, T]. From the weak compactness of a ball in L~[0, T], it 'follows that passing to a subsequence, if necessary, ~),(.) ~ ~)(.) weakly in L~[0, T]. Then we have

v. ( t ) :~ v(t), 0 < t < T,

for some function v(-) 6 W~[0, T]. Hence, by (4.7), and in view of the convergences

�9 . ( t ) ~ ~(t), r ~ r o < t < T,

we have 6,~(.) ~ ~3(-) in measure. Since the bound (4.8) holds and in view of absolutely equicontinuous integrals of the family {r~(.)}, by the Vitali theorem [140], we have 6,,(-) --~ 6(.) in L~[0, T] and, finally,

vn(.) ~ v ( . ) in W~[0, T].

The proof of the second statement of Proposition 4.1 is based on similar compactness reasons: if {un(-)} is a sequence of extremal controls that converges to an extremal control u0(') in L~[0,T], the following convergences

= . ( t ) --, =(t), r --, r

2736

hold in wd[0, T] both for the solutions r : r xn(.) = x(.;Xo,Un(.)) of the adjoint system and for trajectories of the system (3.1). Thus further reasoning is absolutely analogous to that given above (in the case under consideration it is not necessary to pass to a subsequence because the convergence dn(') + /~(') in measure immeadiately follows from (4.7) with vn(.) = un(.) and from the convergence u,~(-) --+ u0(-) in L~[0, T]).

The proposition is proved. Proposition 4.1 implies further results on the regularity of extremal controls - - one can even prove its

Cm-regularity - - but we do not need them, so they are not formulated here. The set of conditions (4.1)-(4.2) gives the "amoothnea~ of an order > 1" of the basic optimal control

problem (P) in question. However, we note that the conditions (4.1)-(4.2) do not even provide for the existence of the second Gateaux derivative of the functional

T

u( ) f y~ o

dr: L['[0, T] + R.

This functional is infinitely differentiable on the space W~ [0, T]. Thus in view of the W2 m [0, T]-regularity mentioned above, it seems natural to consider the problem (P) in the W2 ~ [0, T]-setting, i.e., to study critical points of the function - J on the path apace

~( T) �9 o,V = {u(.) 6 W~[O,T]Ix(T;xo,u(.)) E V}.

The reachable act ~xo(T) of the system (3.1) in the class W~[0, T],

2{=0(T) = {x (T;xo ,u( . ) ) ;u ( . ) �9 W~n[0, T]},

coincides with P2~0(T ). In fact, it is obvious that

c

On the other hand, let X T �9 Primo(T). Then one can consider the following extremal problem:

T 1 f (u , u) dt ~ inf,

0

= f ( x ) + Ei%1 utgi(x)' x(0) = x0, x(T) = XT,

x E M , m r u(.) �9 tO, T].

The solution of this problem exists by Proposition 2.2, and from Proposition 4.1 it follows that this solution u(-) �9 W~[0, T], hence, x T �9 ~[z0(/); this proves the opposite inclusion 9[=0(T ) C 9Iz0(T).

So 21~0(T ) is the smooth manifold which coincides with 9.1~0(T). The mapping

F=0,T = F~o,TIW7[o,TI : u ( . ) ~ x (T;xo ,u ( . ) ) : W2m[0, T]-+ ~=0(T)

is a surjective submersion of class C r162 and therefore the path space 75(T) zo,V = -P-~2,T(V) is a Hilbert transversally convex subset of W2 ~ [0, T] provided that V is a finite-dimensional transversally convex subset of M that is transversal to ~,0(T) = 91,0(T ).

Henceforth we shall suppose that V is a submanifold with corners in P.I,0(T ). Then the path spaces )(T) and ~(T) ,o,V --,o,Y are both Hilbert manifolds with corners in L~'[0, T] and W~[0, T] respectively. The

~(T) belongs to the class C 1 and satisfies the generalized (C) condition in the sequential function - Y on --,o,V form, provided the conditions (3.2), (3.6)-(3.9) hold. The conditions (4.1)-(4.2) do not imply the C 2- smoothness of the functional

T

u(.) ~-+ / f ~ = J(u) t "

c a [

0

2737

o n p ( T ) ; thus, with the exception of the case when f0 is quadratic in u, the abstract Morse theory developed 2~o~V

in [70] is not applicable. On the other hand, the functional J belongs to the class C ~ on T dT) However, in this case the x o , V "

abstract theory of [70] is also inapplicable. In fact, under no reasonable conditions imposed on the integrand

of the functional - J does the latter satisfy the generalized (C) condition or its analogs on 75(T)z0,g since the

functional T

f f ~ W~[0, T]---* IR (4.9) u ( - ) . /

0

is noncoercive while the coercivity condition, as we know, is the necessary condition for the (C) condition to hold (see [443]). Moreover, there exists one more obstruction to the W2"[0, T]-theory of the problem (P), related with the complete continuity of the mapping (4.9), and hence with the same property of its differential and of Frechet's second derivative. Namely, all critical points (extremal controls) are degenerate since a completely continuous operator cannot have a bounded inverse. For example, the problem

T 1/ J(u) : ~ u 2 dt

o

inf,

~? = u, u(.) 6 L2[0, T],

x(0) = 0, x(T) = 0,

has a unique nondegenerate (in the L2[0, T]-setting) extremal control u(t) = 0 since the Hessian of J on

p(r) is the quadratic form 0,0 T

(v(.),,(.)) o

T rr .~(T) on ~0ro,o~(r) = {v(.) E L2[O,T]l fv(r)dr = 0}. Integrating by parts, we have that, for all v(.) E "o"o,o =

o T

{v(.) e W=[O,T]If v( )dT = 0}, this quadratic form is equal to o

T t T

f 0 0 r

so it is generated by the completely continuous operator A : W2[0, T] -~ W2[0, m],

t t

0 r

0 < t < T ,

hence u(t) -- 0 is the degenerate extremal control in the W2[0, T]-setting. Nevertheless, the conditions (4.1)-(4.2) permit one to find the equilibrium between the loss of smooth-

ness (of class C 2) in the L~'[0, T]-setting and the loss of compactness (the generalized (C) condition in the sequential form) and nondegeneracy with the acquisition of the necessary smoothness in the W2 m[0, T]- setting. Thus with some blurriness we call the conditions (4.1)-(4.2) "the smoothness conditions of an order > 1 . "

A natural embedding map i : W~"[0, T] ~ n~'[0, T] is compact, and the space W~"[0, T] is dense in "D(T) "D(T) between submanifolds L~'[0, T]. This embedding defines a C~-smooth injective map i : r~0,y ~ --~0,y

2738

with singularities; its differential is also injective at every point. Even if V is a smooth submanifold, one

cannot say that 75(T) is an ( immersed) submanifold in p(T) since the differential i , could not split a xo~V xo~V

'~r "b(T) and "D(T) modeI space for --.0.V'D(T) (see [122] for definitions). However, both differential s tructures .~. -- .0.v - . 0 . v are compatible in some reasonable sense.

Let u0(.) be an arbi t rary point of p(T) let N be an open face with maximal dimension of a submanifold x0~g' with corners V that contains the point xo(T) = x(T;xo,uo(.)) in its interior, and let A/" = F[ol, T(T),

_F~-~,T(N) be open faces of Hilbert submanifolds "D(T) and 75(T) respectively. Suppose that locally, ~xo~V xo~W in a neighborhood of the point x0(T), the face N is given by the equation

= 0,

where ff is a k-dimensional smooth function of class C ~ , and gradients of the components of that function are linearly independent in that neighborhood. Denote by (T) P~o,N(U) the orthogonal projection map from

the space L~[0, T] onto the subspace, (T) L,o,N(uo ) = Ker (<I' o F ,0 ,T) , , , 0. By the Lusternik theorem (see

[103]), L,o,N(UO ) is the tangent space T~o(.)Af to A/', t ranslated to zero, at the point uo(.):

(T) T.o(.)Af = uo(.) + L.o,N(uo ).

Consider the map o Fxo, (u h

Y: u ( . ) ~ \ p(T)N(uo ) j : LT[0, T ] ~ R k • Lxo,g(Uo ). (4.10)

A direct calculation shows that Y is a local diffeomorphism of class C ~ from a neighborhood }42 of the point u0(') onto Y0A)); a restriction qo of the map 9 r to U = A/" Q 14) defines a chart (b/, qo) of a Hilbert

(T) submanifold A/" C L~'[0, T] at the point u0(-). The orthogonal projection map P~o,g(Uo) acts on an element u(-) e L~n[O,T] by the formula

p x T) o,g(Uo )u(t) = u(t) -- A(u(. ))~2,(xo(T))xo(T) o Ad pT 1 (u0)Ad pt(uo )G, (4.11)

0 < t < T, (4.12)

where A: L~[0, T] ~ R k* is a linear continuous map, and O,(xo(T)) is the differential of the map if: R '~ Rk at the point xo(T),

--1 Ad p,(uo)X = pt(u0) o X o p, (u0),

x o X = X , ( x E M ) f o r a smooth vector field X on M, G = ( g l , . . . ,gin), and

t

pt(uo ) = e ~ p / ( f

0

+ Guo(r))dr, t e R,

is the flow generated by the smooth control system (3.1), see [86]. Note that the function

C(t) = ~(xo(T) )xo(T) o Ad pTl(U0)Ad pt(uo)G, 0 < t < T,

is absolutely continuous (in fact, it belongs to WkXm[0, T]). Then T(u) e W2~[0, T] if u(-) e /~ = U f3 ,-(T) , W2~[0, T]. If G is an inverse map to Y, then r = ~(0, .) is the inverse map to ~: he --, n~o,N[UO) , defined

(T) on the neighborhood ]) = qa(Lt) C L~o,N(uo ). Hence, for all v(.) E Y, we have

v(t) : r - A(r 0 < t < T. (4.13)

2739

This representation implies that if v(.) �9 ~) = V f'l W~[0, T], then r �9 W~[0, T]. Thus the chart (/,(, q0) defines on ./~ a differential structure compatible with those of ./V" in a natural

way: the chart (/~,~) of the manifold iV" with/~ = U N./~f appears, and, moreover, ~ = ~0tt~, ~--1 = ~0-119,

9 = ~(])) n W~[0, T]. By representation (4.13), we have, in view of the linearity of ,~, that for the differential r and for

the second Frechet derivative r the chart r at the point v �9 V = T(L/) the relations

w(t ) = r - ~ ( r o < t < T,

1"(T) w(.) e ~o,N,

0 = r /~(r

(T) Vwl('),w2(') �9 L~o,N(Uo)

hold. From (4.15) it follows that the quadratic map

O < t < T ,

(4.14)

(4:15)

~ r w)

is completely continuous. In fact, by (4.15), this map is factored through a finite-dimensional, and therefore completely continuous, linear map A: L~[0, T] --+ ]~k.:

r = ~(r

so the complete continuity of r is a direct consequence of the following elementary proposition: If X, Y, Z are all Banach spaces, A : X --~ Y is a completely continuous nonlinear operator, and

A : Y ~ Z is a linear completely continuous operator, then the composition A o A : X --+ Z is a completely continuous operator.

By the same reasons, the linear map

(T) w ~ )~(r L~o,g(Uo ) --+ L~[0, T]

is completely continuous. The following assertion holds true. P r o p o s i t i o n 4.2. Let V be a submanifold with corners in the reachable set 92~o(T ) of the smooth

finitely defined control system (3.1) of constant rank, satisfying the continuation condition (3.2), and let f0: Rd x]~m ~ R be a C a smooth integrand that satisfies the conditions (3.7)-(3.9), (4.1)-(4.2). Then, for

Yuo(.~ e p(T) . , there exists a Ca-char t (N, qo) of an open face Af = 7 9(T) of the Hilbert submanifoId 79~T) y k / X o ~ V O~ Oy

with corners (IV is the open face of V of maximal dimension containing the point x 0 ( T ) : x(T;~0, u(-)) in its interior) such that:

1. ~ maps from/g = U n W~[0, T] onto ]) = V n W~[0, T], V = ~(/,/), and r = qo -1 maps from 9 onto &

2. the functional I = ] o r : V ~ ]R belongs to C 1 and its restriction, i , to ~ = ]2 N W~[O, T] belongs to C ~ ;

3. the second Frechet derivative I"(v) of the functional I at a point v e ~2 admits a continuation to a

continuous Legendre quadratic form I"(v) on ~o,gkr(T) t no) ~ = ,r,luo(.)r~o,N~(T) _ Uo;

4. if ~r2i(v): Lzo.N(uO --~ ~Uxo,Nkr(T) ru0) a is the aelf-adjoint operator that generates this form, i.e., if

(v 2r(v)w, w)L~c0,n = I"(v)(~, ~),

then for the corresponding decomposition

V2I(v) = P(v) + Q(v) (,, e 9)

2740

of this operator into a positively defined operator

P(v) (T) L(T) , �9 Lxo,N(UO) ~ zo,N[UO)

(T) (T) L zo,N(uO ) the conclusions true: and a completely continuous operator Q(v): L .o,N(uo ) --~ following hold

(a) the convergence v,( . ) ~ v(.) in L~[0, T] of a sequence vn(.) e 9 to v(.) E 9 implies the convergence

(T) P(vn)h --~ P(v)h (n --+ oo) in LT[0, T ] Vh e Lxo,N(uo),

(b) the map -T(T) (Uo , (T) Q(~): 9 ~:(L~o,N, ) v ---+ Lzo,N(UO) )

is continuous in the strong L~[0, T]-operator topology; (c) if'[:) C ~; is bounded in L~'[0, T], then, for ally E D,

p(v)(w, w) = (P(v)w, W)L~[o,~ ~ cllwll~to,T~

(T) Yw E Lzo,N(uO ), c : const > 0.

P roof . Assertions 1-2 are already proved. Let us _prove assertion 3. If v(.) e 9 and u(-) = r then for the second Frechet derivative of the functional I = J o r at the point v(-) we have

/"(v)(w, w) = (J o r w) = J"(u)(r r

L (T) [U \ -~-J t (u)(~) t t (v) ) (w,w)) Vw e xo,Nk o)

For the second Frechet derivative JIr(u) of the functional

(T) =-- L zo,N(uo ) f7 w2m[0, T]. (4.16)

T

J : u ( . ) ~ / f~ : W~[0, T] ~ ]R

0

we have T

y " ( u ) ( w , w ) =

o

T

+2/f~ u(t))(F'o,,(u)w,w(tl)dt+ 0

T

+ ff%(x(t),u(t))(F'o,,(,,)w,F'~o,,(u)w)dt+ 0

T

+ u(t))F;'o,t(u)(w,w)dt o

Vw(-) E W~[0, T],

where x(t) = x(t;xo,u(.)) . Since u(.) = r e W~'[0,T] for v(.) e V, we have

]lf~ IIf~

(4.17)

2741

IIJ~ ~x<.], < consl~ IIf;.(x('),u('))iic.,[o,~, u( ' ) ) l c,[o,T] _ ,

and, therefore, the form (4.17) admits a continuation to a conti:anous quadratic form on L~'[0, T]. Hence ~(T) , ,

the continuation of I"(v) to the continuous form I"(v) on ~0,N[u0) follows. Let us prove that the quadratic form I"(v) has the Legendre property, so that for the self-adjoint

operator V~I(~) given by (v~i(,,)w,w)Lr[o,~ = r ' (v ) (w,~)

the decomposition into a positive definite part P(v) and into a completely continuous part Q(v) of the type

v~i(v) = P(~) + Q(v)

is valid. To this end, we note that

(T) ~ J'(u)(r : L~0,N(~0 ) -~

is a completely continuous quadratic form since, as we mentioned above, the operator

(T) w ~ ~b"(v)(w,w): L~o,N(uo) --~ L~[0, T]

is completely continuous and since the form under consideration is a superposition of this operator with the continuous linear functional

J ' ( u ) : L~'[0, T] --, R.

Denote by O(v) the completely continuous operator Q(v): (T) (T) L~o,N(uo ) ~ L~o,g(Uo ) generating this form:

(T) (O(v)w,w)L,~[o,T] = ~](v)(w,w) --~ Jt(u)(~Y'(v))(w,w) Vw E Lxo,N(uO ). (4.18)

Let us prove the Legendre property of the quadratic form w ~-~ J '(u)r w), which will imply, by (4.16), that the form I"(v) has the same property.

To this end, we note that all terms in (4.17), except for the first one, are completely continuous forms on L~[0, T]. Therefore it is sufficient to prove the Legendre property of the quadratic form w 7"(u)(~'(~)~, r

T

=

0

By (4.14), we have r = w + a(r

Thus

fl '(u)(r ~/(v)w) = J"(u)(w, w) + 27"(u)(w,~(r

+ 7" (u)(~(r (v)~)c , ~(r (4.19)

The first term in (4.19) defines, by the ellipticity condition (3.9), the positively defined form p(v) on L (T) I u

x0,Nk 0): p(v)(w, w) = (P(v)w, w)L~'[0,T] = 7"(u)(w, w), (4.20)

where P(v) is the self-adjoint operator generating this quadratic form. The other terms in (4.19) are completely continuous quadratic forms. In fact, let us prove, for example, that the quadratic form

w ~ 2 7 " ( u ) ( ~ ( r

2742

is completely continuous. As mentioned above, w ~ ,~(r is the completely continuous operator L(T) ru , Lr[0 , T] (moreover, it is finite dimensional). Thus, w ~ 2,]"(u)(A(r w) is completely x o , N ~, O) -'+

continuous as a superposition of a continuous quadratic form J"(u) with a finite-dimensional operator. This proves assertion 3 of Proposition 4.2.

Assertions 4(a)-(c) are direct consequences of the "smoothness conditions of an order > 1," i.e., of the conditions (4.1)-(4.2) and of the ellipticity condition (3.9).

Let us prove, for example, assertion 4(a). By definition, the operator P(v) is generated by the quadratic form

T

a,, 0

(T) /5(u) of the orthogonal projection map P;o,N(UO) : L~[0, T] --* i.e., it is the superposition P~o,N(Uo) o (r) L(T) , zo,N[UO) with tile self-adjoint operator/~(u) in L~[0, T] given by

P(u)w( t ) = w(t)T f~ 0 < t < T.

If ~n(') e ~) and vn(') --~ , ( ' ) e ~) in L~[0, T], then u,,(.) ~ u(.), n ~ ~ , in Lr[0, T], where ~n(') = r Since, by (4.1),

Jf~ < d (x~( t ) )Jun( t )J+c(x , ( t ) ) , 0 < t < T, (4.21)

and the convergence un(-) -* u(-) in L~[0, T] implies the uniform convergence

x~(t) = x( t ;xo,u~( . ) ) ~ ~(t) = ~(t;~0,~( . ) ) , 0 < t < T,

and also the convergence J f~ f (x( t ) ,u( t ) )J ~ -* 0

in measure, all conditions of the Vitali theorem [140] are valid for the sequence

g~(t) = IhT(t)(f~ - f~ , 0 < t < T, Yh(.) e L~[0, T],

i.e., gn(t) ~ 0 in measure and {g,,(.)} has absolutely equicontinuous integrals (the latter follows from the convergence un(.) --~ u(.) in L~[0, T] and from (4.21)). Thus, according to this theorem,

T

/ gn(t)

0

dt ~ 0 as n --* oc;

hence P(vn)h --~ P(v )h Vh e L~[0, T], and, therefore,

P(vn)h = P~o,g(Uo)o(T> P ( v , ) h ~ p,o,N(uo) P(v) = P(v )h

(T) Vh C L~o,N(uo ).

Some remarks on assertion 4(b) are neccessary. The operator Q(v) in the decomposition of V2I(v) into positively defined and complete continuous parts involves, in particular, the operator Q(v), which generates the quadratic form

{(v): w ~ 2J"(u) (A(r L~o,g(Uo ~ N.

(T) (T) The continuity of the map v ~-~ Q(v): ]) --~ s with respect to the strong operator L~ [0, T]-topology may be stated in the following way.

2743

Let v , ( . ) e 9 converge in L~[O,T] to v(.) e 1); then u , ( . ) = r ~ r = u(.) in L~[O,T], and we mus t show tha t

I lO(v,,) - O( , , ) l lLr IO,m ---' 0 a s n ~ oo.

Since Q(v) is a self-adjoint operator , its no rm is

I IQ(v)I ILr[O,T] = s u p Iq(v) (w, w) l . II,-.llL~ro,r]<l

Thus

I I Q ( v . ) - Q ( * ' ) I I L r [ o , ~ = s u p I ( q ( v . ) - q ( , , ) ) ( ~ , , , - , , ) l = llwll~'[o,rl_ <1

T

=2 sup I / at- Ib~176 0

T

- ff~,(x(t),u(t))(a(r 0

Let us use the fact tha t the opera tor w ~ ,~(r maps from L~ n [0, T] into a f ini te-dimensional subspace of W~[O,T]. We have

T

f f~ u,(t))(A(r w(t)) dt- 0

T

=

o

T

= /(f~ f~ 0

T

+ ff~ a(r < 0

T

< Ila(C'(v.)w)C(.)llcm[o,m - f ~

o

T

+ l l a ( C ' ( v . ) w ) c ( - ) - a(C'(v)w)C(.)llcm[o,m /IJg~(~(t), u ( t ) ) l l w ( t ) l dt <_ 0

' II f~(=.( . ) , u . ( . ) ) - fX.(x('),u('))IILT[O,T]IIw(')IILT[O,T]+ _< ll.X(',/, ( v , , )w )c ( . ) l l c . , [o ,~ 0 0 �9

+ I I ,XC'(v , , ) , - , , )c( - ) - ,~(r [o,'~-

In a f inite-dimensional space all norms are equivalent, thus, there exists M1 = const > 0 such tha t

I IIA(r (v . )w)C() l lc~.[o ,Tl <_ MlllA(r <_ M, M2llw(')IILg'Io,TI

2744

(where II~(r ~ M~llw(')llLrtO,~ by the convergence v~ --* v as n --, oo in Lr[0, T ] and by the smoothness of class C ~176 of the chart tP), and

ll~(r ~(r

Mlll~(r ~(r ~ MIM~II(v, - V)IILrtO,HIIwilLrtO,Tq" Thus we obtain

IIC)(vn)- Q(v)llLr[O,~ _< 2M~Mlll f~ -

- f~ u(.))llLr to,~ + 2M~M~ IIv,,(') - v(')llLr to,~-

We have proved above that

(4.22)

I I s s ~ 0

as u~(.) --~ u(-) in L~[0, T]. Thus, from (4.22) it follows that

I I Q ( v n ) - 0 ( v ) l t L r t O , ~ --' 0 as n ~ ~ .

Similarly using bounds in intermediate norms and the conditions (4.1)-(4.2), we can prove the continuity with respect to a strong operator topology of all other terms involved in the operator Q(v), except for the term Q(v), see (4.20). The continuity of the map v ~-~ ~)(v), in the latter case follows merely from the C~-smoothness of the chart qp and from the continuity of u ~ J'(u).

Proposition 4.2 is proved. From the line of reasoning used when proving assertion 4 of Proposition 4.2, we also get the following

statement, whose proof is omitted here. P r o p o s i t i o n 4.3. Let all conditions of Proposition 4.2 be valid. Then, for every point ~ 6 ~ and for

~(T) (T) any finite-dimensional subspace H C Lzo,N(UO ) = M L~,o,N(UO ) W~[0, T], there exists a continuous (in the L~[0, T]-topology) functional a~,H : Y ---* 1R such that

t~)O,H(V) ~ 0 as v ~ v ,

and, for all v E ]), "we have

I I " ( v ) ( w , h ) - Z"(~)(w, h)l < ~

(T) Vw E Lxo,N(uo), h E H.

7 )(T) By Proposition 4.1, u0(.) E w2m[0, T]; Now, let u0(-) be a critical point of the function - J on ,0,y"

hence u0(-) is a crit'ical point of the restriction of the function - J to A)" = Af gl w2m[0, T], where Af = F~ol, T (N), N C V is the open face of Y with maximal dimension, containing the point xo(T) = x(T; x0, u0(')) in its interior.

Since u0(-) is a critical point of the function - J on f f ~(T) the second differential - ] " ( u 0 ) is = ~ a g 0 ~ g ,

defined at that point as an invariant object, and it is a quadratic form on T~o(.)A/" = T,,0(.)Af fq W~[0, T].

"D(T) at a critical point u0(') we mean an extension By the Hessian H~,o(.)(-J ) of the function - d on --,o,N

of the quadratic form - J " ( u 0 ) to a continuous Legendre form - J " on T,,o(.)Af. By Proposition 4.2 this extension exists and is a Legendre form, moreover,

Tttt 'u 5{v - ~ ~ oj~ : , V 2 Z , v o j v , v,Lrto,H, ito , V uo ) ~--* I" (vo)(v ,v ) (4.23)

(T) for all v~,0(. ) C T~0(.)Af = u0(') + L~o,g(uo ), where I"(vo) is the second Frechet differential at a point v0,

(T) r r = u0, of the function I = J o r : L~o,y(Uo) ---, N, ~ = is the chart of the manifold A/" that we have constructed above.

2745

Thus, the Hessian H~o(.)(-d ) is a Legendre quadratic form provided that the conditions of Proposi- tion 4.2 hold. In particular, an index of that form, i.e., a maximal dimension of a subspace in T=o]( where this form is negatively defined, is finite and is called a coindex of a critical point uo(') of the function - J

~[)(T) on ~o,V, or a Morse index of an extremal control uo(t), 0 < t < T; hence, the latter is equal to the index (T)

of the quadratic form I"(vo) on L~o,N(uo ). A critical point u0(') is said to be non&generate if - V J ( u 0 ) E rel int ~r* ~(T) and if the Hessian ~uo(.) --xo,V

(T) H~o(.)(-J ) is nondegenerate, i.e., if the quadratic form I"(vo) on Lxo,N(uo ) is nondegenerate; accordingly, we also speak about a nondegenerate eztremal control u0(t), 0 < t < T.

One can also present a more convenient form of the Hessian, which was already obtained in [72]:

T

H~o(-J)(v~o,%o) = ]{ t I~( t ) (v ( t ) ,v ( t ) ) + 2[-I~(t)(v(t),y(t))+ o

+-f-Iz~(t)(y(t),y(t))} dt+ ~(Uo)(px~(xo(T))(y(T),y(T)) V%o(. ) e T~oA/,

where y(t) = y(t; v(.)) is a trajectory of the linear control system

(4.24)

f y = A(t)y + B(t)v, 0 < t < T, y(0): 0 , r : 0,

(4.25)

II,~(t) = H~(xo(t),r ~I~x(t)= H~(xo(t),r

_f-Ix~(t) = H~(xo(t),%bo(t),uo(t)), 0 < t < T,

H(x, ~, u) is the Hamilton-Pontryagin function of the problem (P), i.e.,

H(x,%b,u) = - f ~ + r +G(x)u), G(x) =(gl(z) , . . . ,gm(x)) , (4.26)

r 0 < t < T, is the solution of the adjoint system

(~o = f~ - r + Gx(xo(t))uo(t)), 0 < t < T, (4.27)

with the "target" condition r : ~(uo(.))r (4.28)

~(u0(.)) : A(VJ(u0(.)), A(t) = f~ ~- G~(xo(t))uo(t),

B( t ) :G(xo( t ) ) .

In [72] the necessary and sufficient conditions of the nondegeneracy of an extremal control uo(t), 0 < t < T, and the formula of its Morse index were also obtained; they are as follows.

Let Tl(t,p,y,v) = 9 ~ ( t ) ( v , v ) + 2H~(t)(v ,y) + 9~x(t)(y,y) + p(A(t)y + B(t)v) (4.29)

and let ~ = ~(t,p, y) be a unique solution of the equation

2vTH~=(t) + 2yTH~=(t) + pB(t) = 0 (4.30)

and 7~(t,p,y) = 7-[(t,p,y,O(t,p,y)),

~p(t ,p,y) = Tlp(t,p,y,O(t,p,y)), 7~u(t,p,y) = Tlu(t,p,y,O(t,p,y)),

2746

A point T is called a focal point for an extremal control u0(t), 0 < t < T, if there exists a covector

~o(T) e T~(T)N = {~ e T;o(T)MI(~,X.o(T)) : 0 VX.o(v ) e T.o(T)N }

such that the Hamiltonian system

[~ = -7~y(t ,p,y), ~ = 7:Lp(t,p,y)

has a solution (p(t), y(t)), 0 < t < T, with y(t) ~ O, that satisfies the boundary conditions

p(T) = -~vo(T) + 2yT(T)~(uo(.))c~**(x(T)), y(O) = O, r = O.

(4.31)

A point t, 0 < t < T, is called a conjugate point of an extremal control u0(t), 0 < t < T, if there exists a solution (p(t),y(t)), 0 < t < i, of the Hamiltonian system (4.31), with y(t) ~ O, that satisfies the boundary conditions

y(0) = 0, y(i) = 0. (4.33)

The following theorem holds true. I n d e x T h e o r e m . Let all conditions of Proposition 4.2 be valid. Then an extremal control no(t),

_ _ "1:) ( T ) 0 < t < T, is a nondegenerate critical point of the function - J on --xo,v if and only if for the solution r 0 < t < T, of the adjoint system (4.27) corresponding to that control, one has

- r e tel int T*o(T)V

and the point T is a nonfocal point for that control. If, in addition, the bang-bang conditions (3.5) are valid, then a Morse index Ind~,o(.)(-J ) of the extremal control is finite and is calculated according to the formula

Ind~o(')(-Y) = E I(i) + Ind,0(.)(]), (4.34) 0</<T

where Ind~o(.)(Y ) is the index of the quadratic b r m

(y,p) ~ J(uo)(y,p) = ~p(T)y(T) - ~(uo(.))c~**(xo(T))(y(T),y(T)) (4.35)

on the finite-dimensional subspace of Wd[O, T] • wd[o, T], consisting of the solutions (p( t ), y( t ) ), 0 < t < T, of the Hamiltonian system (4.31) with the boundary conditions

y(0) = 0, r = O,

here l(i) is the multiplicity of a conjugate point i of i.e., the dimension of the solution pace of the boundary value problem (4.31)-(4.33), and the sum in (4.34) is taken over all conjugate points on the interval (0, T).

Concluding this section we shall mention the last result concerning the W~[0, T]-regularity of the problem (P).

Let (T) L ( T) , l:xo,N(uO) C xo,N[uo) be a subspace on which the form I"(vo), r = u0, is negatively defined, i.e., the subspace

(T) uo(.) + s C T,,o(.)Af

is a "coindex subspace" of the Hessian 7-/,,o(.)(-J ) of the function - J at a critical point u0(.). (T) P r o p o s i t i o n 4.4. Let all conditions of Proposition 4.2 be valid. Then the subspace s C

(T) Lzo,N(uo ) is of finite dimension and ~.xo,N(Uo)(T) C W2m[0, T].

2747

(4.32)

Proof . The finite dimensional property of (T) •xo,N(uo) follows directly from the Legendre property of quadratic form I"(vo)(see Proposition 4.2).

To prove the inclusion s ) C W~[0, T] it is sufficient to check that all eigenfunctions of the self-

adjoint operator V2I(vo) �9 L(T)~(Uo) ~ L(T)N(uO ), generating the quadratic form I"(vo), that correspond to their negative eigenvalues belong to W2m[0, T]. From [72] ~t follows that

(T) v21(v0) = - ( M S ( u 0 ) +

where M~To,)(uo) = p~To,)N(uO)o M(uo), M(uo) is the self-adjoint operator in Lr[0 , T] defined by the formula

M(uo)v(t) = --vT(t)f~ 0 < t < T,

(T) (T) L• [0, T] defined by P•o,N(UO) K(u0), K(uo) i s the self-adjoint operator on Kzo,N(UO) = o

t

- ( . 0 ) v ( , ) :

0

T

+ fvr(r)H.x(~)r(T)drr-l(t)a(xo(t))+ t

T r

T ^ 1

§ i t 0

7"

+(fr(T)r-l(t)G(xo(t))v(t) dr) T~(uo(.))%.(zo(T))r(T)r-l(t)G(zo(t)), 0

0 < t < T ,

where F(t) is the fundamental matrix of solutions of the system ~ = A(t)z, F(0) = I, I is the identity matrix. Therefore, the eigenvalue problem for this operator is equivalent to the solution of the integral equation

- (K (uo ) + M(uo))v(t) = #v(t) " A(K(uo)v + M(uo)v)C(t), 0 < t < T,

where C(t) : "~,(xo(T))xo(T) o Ad pTl(u0)Ad pt(uo)G =

: a),(xo(T))F(T)F-l(t)G(xo(t)) , 0 < t < T.

If # < 0, then - (M(uo) + M) is a positively defined operator; hence it is invertible, and

v(t) = - ( M ( u o ) + # I ) - l (K(uo ) - A(K(uo)v + M(uo)v)C(t))v(t) , 0 < t < T.

The right-hand side of this equation belongs to W~[0, T] since the extremal control u0(-) C W~ [0, T], and this proves Proposition 4.4.

5. Morse T h e o r y for C1-Funct ions on Hi lbe r t Mani fo lds w i t h Corne r s

In this section, a variant of the Morse theory for Cl-functions on Hilbert manifolds with corners will be constructed. The corresponding definitions are based on the facts established in Sect. 4 for the problem (P) and formulated in Propositions 4.1-4.2.

2748

Let 7-{ and ~ bo th be Hilbert spaces, ~ be cont inuously e m b e d d e d into 7-I and ~ be dense in 7-/. Suppose tha t 5- is a C~-sur jec t ive submers ion from 7-I onto a C ~ - s m o o t h finite dimensional manifold M such tha t ~ = ~'lhq is also a C~-sur jec t ive submers ion onto tha t manifold.

For a finite dimensional submanifold V with corners of M let V = ~ - - I (V) and 1) = ~ - ~ ( V ) be Hilbert submanifolds wi th corners of 7-[ and 7~ respectively.

If 3.4 = 9 r - l ( M ) is a na tura l suppor t of the submanifold V with corners (see Sect. 1), then a natura l Pdemannian s t ruc ture is considered on M tha t is induced by the s t andard Riemannian s t ruc ture {.,-), of a Hilbert space 7-{ genera ted by the inner p roduc t on (., .) on tha t space. In what follows, by p we will denote the Riemannian metr ic on M genera ted by the above-ment ioned R iemann ian s t ructure . Wi th in the metric p the space M is a complete metr ic space.

Let f : 7-/--+ IR be a C l - func t ion such tha t its restr ict ion flTi belongs to the class C ~ . We call (~, 1), f ) a HiIbert tr iple if the following condit ions hold:

I. All critical poin ts of the funct ion f on 12 belong to 1) = 12 n ~ , and the convergence x,, -+ x0 of a sequence {x~} of critical points in the topology of 12 implies the convergence of tha t sequence to x0 in the topology of 12;

II. For any point x0 E 12 there exists a C ~ - c h a r t (O, ~0) of the open face A/" of the Hilbert submanifold !2 with corners conta in ing x0 and having a minimal codimension in 7-{, such tha t ~: O --+ H (H is a closed subset in 7-{) and the following condit ions hold:

(1) qp maps f rom 0 = 19 f'l ~ on to /~ = / g M ~ , / . / = ~(O), and the inverse m a p p i n g r = ~-1 also has the same property , i.e., it maps f r o m / ~ onto ~ thus some chart (O, ~5) compat ib le wi th the differential s t ruc ture of the Hilbert submanifo ld A~" = A/" Cl 7-I is defined:

(2) the second Frechet derivative F " ( u ) of the C ~ - s m o o t h funct ional /w = - f o ~b -1 It9 at a point u E admits an extension to a cont inuous Legendre quadrat ic form F"(u) on H;

(3) if V 2 F ( u ) is the self-adjoint opera tor on H generat ing the quadrat ic form F"(u) on H, i.e., if

F"(u)(w, = ( v

where (., .) is the inner p roduc t on H and

V Y(u) = P(u) + #(u)

is a decompos i t ion of this opera tor into a positive definite opera tor P(u) : H --~ H and into a completely continuous o p e r a t o r Q ( u ) : H ~ H, then

(a) all e igenfunct ions of the opera tor V2F(u) tha t correspond to negative eigenvalues belong to H = H N T / ;

(b) tile convergence of a sequence {u,~} C g) to u0 E g) in the topology of H implies the convergence in the same topology of {P(u , )h} to P(uo)h for all h E H;

(c) the m a p p i n g u ~ Q(u) : U --* s H) is cont inuous in the s t rong H-ope ra to r topology; (d) if 79 C U is b o u n d e d in H, then there exists a constant c > 0 such tha t

p(u)(w,w) ~f (P(u)w,w) > cllwll 2

Vw E H, u E D;

(e) for any v E /~ and for any finite dimensional subspace L C H = H fq 7~, there exists a continuous functional W~,L : L/ ~ N in the topology of H such tha t oJ~,L(U) --* 0 as u ~ v in H, u E/~, and

I Y " ( u ) ( w , h) - F"(v)(w,h)l < ,~,z(u)llwllllhl[

Vw C H, h E L.

2749

Let (1), ~), f ) be a Hilbert triple. Let us recall (see Sect. 1) that a point x0 is called a critical point of a function f on 1) if ms(x0 ) = 0 or equivalently if

df(xo) �9 T;oV = �9 T2oUl@.o,X.o) < 0 VX.0 �9 T.oV}.

If z0 is a critical point of f , then x0 �9 1) is a critical point of the restriction f[.~" of the function f to the open face 24" of the Hilbert submanifold V with corners of 7-I containing x0 in its interior and having minimal codimension in 7-/, x0 is also a critical point of fiN', where AT/" = A f (-1 ~ . The second differential -d2f(xo ) of the function - f : J~.---~ IR is defined at the point x0 as invariant object, and since (12,1), f ) is a Hilbert triple, this second differential admits a unique extension to a continuous Legendre quadratic form on the tangent space T~0Af. That quadratic form 7-/~ 0 ( - f ) is called the Hessian of f on 1) at the critical point x0. Obviously, 7-/~0(-f) does not depend on the choice of a coordinate system, thus it is defined as an invariant object.

A critical point x0 of the function f on 12 is said to be a nondegenerate if T * " (1) df(zo) �9 rel int ~0 V,

(2) the Hessian 7-/~0(-f) is the nondegenerate quadratic form on the tangent space Tz0Af. The index of the quadratic form 7-/z0(-f) is called the eoindez of the critical point x0. A Hilbert triple (1), ]), f ) is called a Morse triple if f satisfies the generalized (C) condition in the

sequential form and has only nondegenerate critical points. As in Sect. 2, let us consider the following level sets for the Cl-smooth function f : ]) ~ R:

f ~ = { x � 9 f ~ , b = { x � 9

The Morse theory relates topological characteristics of these level sets with the number of critical points of the function f of corresponding coindices.

According to Proposition and Definition 2.1 (see Sect. 2) for the Cl-function f on the Hilbert sub- manifold 13 C 7-/with corners there exists a smooth (of class C a ) pseudogradient vector field X~, x �9 G, defined on the domain G which is free of critical points. In what follows, the necessary deformations of level sets will be performed by means of trajectories of that field, ~t(x), t �9 R.

P r o p o s i t i o n 5.1. Let f be a Cl-function on a Hilbert submanifold 1) C 7-l with corners. Then, for all a, b �9 R, a <_ b, a level set fb is a deformational retract of f~ provided that

f~,b = {x �9 Via < f ( x ) <_ b}

is free of critical points of that function, in particular, f~ and fb are of the same homotopy type. Proof . Since f~,b is free of critical points of the function f , we have fa,b C G, where

oo

G = U G(k)' k = l

a (k) = {x �9 > },

is the domain that is free of critical points of this function, and {bk} is the sequence of real numbers from the definition of the pseudogradient vector field X~, x E G, for f .

Let r(x) be the smallest instant of time t > 0 such that f (~ (~ ) (x ) ) = b, where qpt(x), t >_ O, is the trajectory of the pseudogradient vector field X~, x E G, for f , starting from x E fa,b for t = 0. Since the function f increases along this trajectory, such an instant exists and is unique and the map x ~ r(x) is continuous. Let,

x, x E f~, s t ( x ) = e

fa,

for all t, 0_< t _< 1. Then st, 0 <_ t <_ 1, is the family of continuous maps s t : f" ~ fa, continuously depending on the parameter t, which yields the required deformational retraction. The proposition is proved.

2750

Let Hi(~.) be the i-th singular homology group of a space :~ with coefficients in some field, and let bi(Y~) = rank Hi(Y) be the i-th Betti number of that space. According to Proposition 5.1, for Vi,i = 0 , 1 , 2 , . . . ,

Hi(f '~ , f b) = 0 r H i ( f a) = Hi ( f b)

provided a _< b and the interval [a, b] is free of critical values of f . By full analogy with [70, 72] one establishes P r o p o s i t i o n 5.2. Let f be a C I-function on a Hilbert submanifold ]) with corners of 7"[ bounded from

above that satisfies the generalized (C) condition in the sequential form. I f ~t(x), t > O, is a trajectory of the pseudogradient vector field X~, x E G, for that function f , starting from x ff G for t = 0 and [0, fl) is a maximal interval where it exists, fl = sup{t > Ol~t(x ) E G}, then the closure of the set

{~,(x);t ~ o}

contains a critical point of the function f . This proposition can be used to deduce in the standard way the existence theorem, which has been

stated in Sect. 2; see Proposition 2.2. P r o p o s i t i o n 5.3. Let f be a Cl-function on a HiIbert submanifold ]) C 7ff with corners which is

bounded from above and satisfies the generalized (C) condition in the sequential form. Then, for all x E G, there exists the limit

x0 = lira ~#t(x) t---* fl

which is a critical point of the function f provided that all critical points of f are isolated. Proo f . It is sufficient to consider only the case where/3 = +c~. Supposing the contrary, we have that there exists a neighborhood O of a limit critical point x0 of the

trajectory Tt(x), t > O, as t --~ cx~, that contains no other critical points besides x0, such that there exists a sequence t,~ --* 0 with r (x) • (9.

! II Let /d be a neighborhood of the point x0 wi th /~ C O. One can choose the sequence {[s,, s,]} of I t t nonintersecting intervals [s,, sn] such that

~,,(~)eO\u, ~" < t < " , ~ ,, ~ . , p ( ~ , , . ( ) , ~ , . . ( ~ ) ) >_p(oo, ou)=po > 0 ,

where p is the Riemannian metric. This is possible since x0 is a limit point (as t ~ +oo) of the trajectory ~t(x), t > 0: if ~t(x) C U, then there exists the smallest s" >_ t with ~,,,(x) E r \ / , / , and the greatest s' _< s" with ~ , ( x ) E/.~ \ /d. Since ~t(x) E O \ b /on every interval [s',,,s"], we have

o n Ur I II1 U t~., ~.J for some k _> 2. Thus, for t e [ J , ~"], n > l n > l

( v f ( v , ( x ) ) , x~ , ( . ) )~ , (~)>_ 6k

by the definition of a pseudogradient vector field. Further, by the inequality llX~,(=)il~,(,) _ 1, we have

I I I I # n 8 n

I I 8 n 8 n

I I 8rL

_> II@,(x)lG(=)dt _> o ( ~ , , . ( ) , ~ , . ( x ) ) _> p0 > 0 I

8 n

2751

in view of the choice of sn,' s n." There exists

B = l i m f ( ~ t ( x ) ) < o o t - * + o o

since f is bounded from above and t ~-* f(qvt(x)) monotonically increases. Hence

c r > l i m [ f ( q o t ( x ) ) - f ( x ) ] = t-.--*.+ oo

O0

0

I t 8 n

>~18f (vf(qa+(x))'X~(~))~(~)dv>-= OO

X ; " ( s " ' _ -- z__, , , , - > ,S po n = l n = l

1 = +oo.

The contradiction obtained proves the proposition. P r o p o s i t i o n 5.4. Let f be a C 1-function on a Hilbert submanifold V C 7-l with corners that satisfies

the generalized (C) condition in the sequential form and has only isolated critical points. Then

Hi(fa, fb) = 0 r Hi ( f a) = Hi ( f b)

provided that the half-interval [a, b) is free of critical values of that function. Proof . Let us consider the pseudogradient vector field X~, x E G, for the function f which is defined

in the domain G C ]) free of critical points of the function f . of the domain

= {x E f'+'blf(x )

Let X(X), x E V, be the characteristic function

< b } .

Then the vector field Y~ = X(x)X~, x E V, is well defined on f~,b and is smooth in the domain G f l ~. All trajectories et(x) , t > 0, of this field are defined for all t > 0 and coincide with the trajectories of the pseudogradient vector field in the domain G n f~; when x E f b they all are fixed points: Y~ = 0, x E fb. Thus, for the ordinary differential equation

= Y~, x E V,

the existence and uniqueness theorem for the solution starting from x E f,,b for t = 0 and defined for all t > 0 is valid. Moreover, the theorem on continuous dependence of solutions et(x) , t > 0, of this equation on the initial conditions over finite intervals of time is valid; this theorem can be reinforced to the following

assertion. L e m m a 5.1. Let all conditions of Proposition 5.4 be valid and let Y~, x E V, be the vector field

which is constructed above. Then, for any point xo E fa,b, there exists a unique trajectory et(xo), t >_ O, starting from xo for t = 0 and defined for all t >_ O. Moreover, for any point xo E fa,b and for an arbitrary neighborhood Lt C Y containing at most one critical point of f with et(x0) E U Vt > T > 0, there exists a neighborhood 0 with xo E 0 such that, for all y E O, the corresponding trajectory et (y) E U for all t >_ T.

P r o o f of L e m m a 5.1. The first assertion is obvious. Let us prove the second one. Assume the contrary, i.e., assume that there exists a neighborhood U containing at most one critical point, with et(xo) E U, t > T > 0, such that, for any neighborhood O, O 9 x0, there exists a point y E O and an instant of

2752

time T~ >__ T, with ~b% r H. In other words, suppose that there exist a sequence xk ~ Xo and a sequence Tk >_ T, such that

C n ( x k ) r C d x 0 ) e u , t>m, T,>T, k = 1 , 2 , . . . .

We can assume Tk --* +c~. Since the neighborhood H contains at most one critical point, we can find a neighborhood L/ l , /~ C H, such that Ct(xo) E H1, t >_ T, and/~ \/4~ is free of critical points of the function f; therefore

i nf my(z ) >_ 5ko-1. EU\bh

Since Tk --* +ec , there exists a sequence of closed intervals [T],,T],'] such that CTl(zk ) e ~-[1 k ~[1, eT~'(xk) e /) \ H, and Ct(xk) e /~ \/41, T~ < t < T~'. Moreover, T~ --, +oe as k --* co. Namely, ~r~, is the

smallest instant of time t :> 0 such that Ct(zk) E. lfl\Lt, and T~ is the greatest t _< T~' such that Ct(xk) E H~. From the definition of the vector field Y~, z C ~, it follows, that, for all z E f,,b, there exists

lim f(qat(x)) = b. t - - * + c ~

In fact, if f(qvt(x)) < b for all instants of time, then by Proposition 5.3 the trajectory Ct(x) converges to a critical point x0 of the function f as t --* +oe, and at that point xo, f (xo) = b, since on [a, b] there are no other critical values of f besides b.

If, for some finite t, f(r = b, then by the definition of the field Y

r = r t > i,

and hence, in all cases, in view of monotonic increasing of f along Ct(x), there exists the finite limit

b = lim f(r t----~+ c~

Thus, for Xk -~ xo as k --~ oe, we have

b = lira f ( r lira t - ~ + o o t - - * + ~

t

o

0

Hence, if TIc --4 +0% then for any e > 0

dT-- f (xk) .

f ( v f ( r Yr (=k))r dr T' k

< 6

provided k > k1(r Then

> f(vs(r Y+.<:,))+.(:~) <t~- _ f(vs(r Y+.r Tt Tt

dT >_

_> ~ko(:r,~' - T~) __. '~koP(r Cr:,(xk)) _>

>_ ~koP(OU, OHa ) = ~koPO.

2753

Whence, in view of the arbi t rary choice of e > 0 it follows tha t p0 = 0 or 5ko = 0, which is impossible. The contradict ion proves the lemma.

Let us cont inue the proof of Propos i t ion 5.4. We define, for all t, 0 < t < 1, the family of maps st : f~ --~ fa by the fo rmula

X, :r, E f b,

where Ct(x), t > 0, is the t ra jec tory of the vector field Y~, z E G, cons t ruc ted above. By Proposi t ion 5.3, this definit ion is correct since

l i m e , , ,=. l imr - - t---* l t --'Z'~t ix1

From the theorem on the cont inuous dependence of solutions of an ordinary differential equat ion on initial condit ions follows the cont inui ty of the map (t, x) ~ st(x) for t < 1. For t = 1 the cont inui ty of this map follows from L e m m a 5.1. In addit ion, so(x) ---- x, x E f~; st(x) E f a , 0 < t < 1, x E f~;

s t ( z ) - x , x E f b , O < t < l ; s l (x) E f ~, x E f f .

Hence the family of maps s t : f~ --* fa defines the deformat ion re t rac t ion f rom f~ onto fb and, thus, for all i > 0,

H i ( f " ) = Hi(f~) .

This proves the proposi t ion. Up to this point we have never used the Morse proper ty of the triple (13, l), f ) . Under this condit ion

we have the following P r o p o s i t i o n 5.5 . Let (1~, ~, f ) be a Morse triple. Then 1. All critical points of f on "P are isolated; 2. There exists at most a .finite number of critical points of the function f lying on every critical level

f ~ , b , a < b . P r o o f . For the proof of assertion 1 it is sufficient to prove tha t a critical point x0 is isolated on the

open face A/" with min imal codimension in 7-I containing x0 in its interior; fur ther reasons are completely analogous to those of [70, pp. 150-151], where the fact tha t x0 is isolated on Af has been proved by the Morse lemma.

Let (O, ~) be the chart at a point x0 E Af of the manifold .hf f rom the definit ion of a Morse triple (see II). Then u0(.) = 0 is the critical point of the funct ional F = - f o r : H --* 1~ on a Hilbert space H, and we have to prove tha t this point is isolated. Assuming the contrary, we have a sequence {un} of critical points of the funct ional F with un --* u0. Since un E H = H M 7-/, we have for every t, 0 < t < 1, tun + ( 1 - t)uo e [t,

1

(F'(u.) ' f - F ( u o ) , ~ . - ~o) = P ' ( t u .

o

+ ( 1 - t ) u o ) ( u , , - u o , u , , - ~ o ) a t = o

for all n = 1, 2, . . . . The quadrat ic form P ' ( t u n + (1 - t)uo) admits an extension to a cont inuous Legendre form F"(tu~ + (1 - t)u0); therefore

P ' ( tu~ + (1 - t )uo )( U~ - Uo , U. - Uo ) = ( V 2 r ( tu~ + (1 - t )uo )( U. - u o , u ~ - uo ) =

= ( P ( t u ~ + (1 - t ) u o ) ( U ~ - u o ) , U n - U o ) +

+ ( Q ( t u , , + (1 - t ) u o ) ( u , , - u o ) , u , , - u o ) ,

2754

where P(u) , Q(u) both are self-adjoint operators from the decomposition of the self-adjoint operator V2F(u) corresponding to the quadratic form F"(u) into a positive definite and a complete continuous part. Thus

_ 1

( F ' ( u . ) - F ' ( u o ) , u , , - uo) = Ilu~ u0JJ 2

1

= f(P(tu. 0

(,~.-,,o) where v~ - Jl~.-~011"

1

+(1-t)uo)v.,v.)dt +/(Q(tu. o

+(1- t )uo )vn , v , )d t ,

We can suppose that vn ~ v0 weakly in H. Then, by property II in the definition of a Morse triple, we have

1

(Q(tu,

0

Consequently, as n ---* c~,

which implies v0 # O. Since, for all h E H,

1

+ (1 - t)Uo)V,,v,)dt ~ / (Q(uo)vo ,vo)d t = (Q(uo)vo,vo), o

1

(P(tu~

0

+ (1 - t)uo)v=,v,)dt >_ c l l v . l l 2 = c > 0.

0 > c + ( Q(u0 )v0, v0 )

1 1

0 = / ( V ~ F ( t u n + ( 1 - t ) u o ) h , vn)dt--* / (V2F(uo)h, vo)dt,

o 0

when un --* uo, we have V2F(uo)vo = 0 and vo # 0; this contradicts the nondegeneracy of V2F(uo) and proves the first statdment.

The proof of the second assertion follows from the generalized (C) condition in the sequential form. In fact, if f,,b contains an infinite number of critical points of the function f , then we can choose from that infinite sequence of critical points {xn } a subsequence converging to a critical point xo; this contradicts the isolating property of critical points, which has been proved (see the first assertion of the proposition), The proposition is proved.

Let c~(f) be the number of critical points with coindex i of the function f on fa, a E ~. T h e o r e m 5.1 (Morse inequalities). Let (]),]), f ) be a Morse triple and f be a function bounded from

above. Then, for all rn = 0, 1, 2,.. . , the inequalities

m m

1 r n - i c a E ( - 1 ) m - ' b ' ( f f ) <- E ( - ) ' ( f ) , i = 0 i = 0

oo oo

i a E ( - 1 ) i b i ( f a) = E ( - 1 ) c i ( f ) i = 0 i = 0

hold, in particular, for all i, i = 0, 1 , . . . , b , ( f a) ___ c ~ ( f ) ,

and the last inequalities hold even in the case when a function f is not bounded from above. Proo f . Let Ka be a critical level of the function f : '12 ---, R,

Ko = {x ~ V I I ( x ) = a, m~,(x) = 0},

2755

and a, b be the neighbor pair of critical walues, i.e., a < b, and the interval (a, b) is free of critical points of this function.

By the exact sequence of a triple Z C Y C X,

�9 . . -~ Hi (Y ,Z) --* H i ( X , Z ) --* H i ( X , Y ) --* H i - I ( Y , Z ) - ~ ' . . ,

assuming X = fa, y = f~ \ Ka, Z = fb, we have

0 --* H i ( f ~ , f ~ \ g~) ~ H i ( f ~ , f b) --~ 0

since Hi( f ~ \ g ~ , f b) = 0 in view of Proposition 5.4, i.e.,

I-Ii(f'~,f b) = H i ( f a , f a \ Ks).

If O~ are sufficiently small neighborhoods of critical points z,~ corresponding to the critical value a, then, by the excision axiom,

H i ( f a , f " \ K , , )= Hi(U(o,~ M f~), U ( O ~ M f~ \ K ~ ) ) = ot ot

= O H i ( f a M O ~ , , f ~ ' M O ~ , \ K , ~ ) . or

Consequently, the problem of calculation of the critical group Hi ( f ~, fb) is reduced to the calculation of the critical group H~(f '~ N (9, fa N O \ {z0}), where O is a sufficiently small neighborhood of x0. We can suppose that f is a smooth function on the cone K: @ H, where H is a subspace in 7-/, 0 is the critical value, f(0) = 0, (9 is a sufficiently small neighborhood of the critical point 0, and K: is a polyhedral cone with vertex at the origin. As in [70, p. 162] one verifies that the pair

({z E Oi l (z) > 0), {x E (glf(z) > 0))

is homotopy equivalent to the pair

({x = (p,q) E Olp E H,q E K:,f(p, 0) > 0}, (x = (p,q) E OlP E H,q E IC, f(p,O) > 0}).

Let F(p) = - f ( p , 0); then we can assume that F is the function on H satisfying the conditions indicated in the definition of a Morse triple. Therefore, the problem is reduced to the calculation of the critical group

Hi( {u E U \ F(u) < 0}, {u E VlF(u) < 0))).

where U is the neighborhood of the critical point zero which is free of other critical points:

Hi(fa M O, fa M Ol{xo}) = Hi({u E UIF(u) < 0},{u �9 VIF(u ) <0}.

The calculation of this group is performed by the results of I. V. Skrypnik [173]. Since zero is an isolated critical point of F, 0 �9 H = H M 7~, the second differential ~11 of the function

P = FIHn~ admits an extension to a Legendre quadratic form F"(u). Let V2F(u) be a self-adjoint operator on H generating this form and H1 be a subspace of H corresponding to negative eigenvalues of ~72F(0). Denote by/-/2 the subspace of H which is orthogonal (in the sense of the quadratic form ~72F(0)) to the subspace H1 :

/-/2= { h E H l ( ~ Z 2 F ( 0 ) h , u ) = 0 V u � 9

L e m m a 5.2. For a sujficiently small neighborhood U of zero in H there exists a continuous functional : U --* ]~, w(u) ~ O, u ---* O, such that Vu �9 U = U M 7~,

(a) (V2F(u)h,h) > kllhl[ ~, h �9

2756

(b) [ ( V 2 F ( u ) h l , h 2 ) l < w(u)l]hi]lHh2[[ , hi E H1, h2 E / / 2 ; ( c ) ( V 2 F ( u ) h , h ) < -klihH 2, h e HI. P roo f . Assertion (b) follows from the finite dimensional property of HI C ~ Cl H and from the

definition of a Morse triple. Assertions (a) and (c) are proved in a similar way. Let us prove, e.g., assertion ( a )

Since V2F(0) is a Legendre form, there exists a k > 0 such that

(V2F(0)h, h} > kllhll 2, h e H >

Suppose the contrary, i.e., suppose that (a) does not hold. Then there exist u,, E U, un --~ 0, h,, E H2, ilh,~[[ = 1, such that

( V 2 F ( u n ) h n , hn) --* ~ < 0 as n ---+ oo.

We can assume that hn -+ h0 weakly and h0 # 0. In fact,

v ~ F ( u , ) = P ( u , ) + Q(u . ) ,

where P ( u n ) is a positive definite and Q(u, , ) is a completely continuous operator. Moreover, by the weak convergence hn --+ h0 and by the continuity of the mapping u ~ Q(u) in the strong operator topology, we have

( P ( u , , ) h , h ) >_ c[thlt, n = 1 , 2 , . . . , c -- const > 0;

lira (Q(u, , )h , , ,h , , ) = (Q(O)ho,ho). n --"+ O 0

Thus, 0 _>/3 = lim (V2F(u , , ) hn , hn} >_ lim c[Ih.Jl2+

n-- -+ ~ n - - - + ( ~

+ lim {Q(u,~)h, , ,hn) >_ c + (Q(O)ho,ho), n - - + O 0

which implies h0 r 0. Passing to the limit as n --+ eo in the relations

(V=F(u , , ) (hn - ho) , (h, , - h0)) = (P(u, , ) (h , , - ho),h,~ - ho)+

we obtain

+(Q(u , , ) (h , , - ho) ,h, , - ho) >_ (Q(u, , )(h, , - ho) ,h, , - ho),

lira ( V 2 F ( u n ) ( h , , - h0), h,, - h0) = lira ( V = F ( u , , ) h , , , h , ) - n - - * OO n ' - + OO

- 2 lim ( V 2 f ( u , , ) h o , h , , ) + lim (VaF(u , , )ho ,ho ) =

= ~ - (V2F(O)ho,ho} >_ O,

i.e., (V2F(0)h0, h0} </~ < 0, and h0 5r 0; this contradicts the Legendre property of V2F(0) and the choice of {h,} -+ h0, h0 E H2.

The lemma is proved. Let Pa and Pa be the orthogonal projection maps from H onto H1 and / /2 respectively.

B (1) = {h E H1]HhII _~ r}, B! 2) = {h e H2[Hh[[ _~ .s},

R = • < 2 )

L e m m a 5.3. There exist r, s, e, v > 0 such that (a) (VF(u),e=u) > u, u c R, IjP2ull = ~; (b) ( V F ( u ) , P l u ) <_ --u, u e R, r ( u ) = c - e; (c) F ( u ) < c - ~, u e R, IIPiu]l = r.

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Proof . Let us choose r, s such that R C U, where U is the neighborhood we dealt with in Lemma 5.2. Then, by this lemma, we obtain that, for all u �9 b" = U n ~ ,

<VF(u),P2u> k:llP ull - k= ,2(u)ilPxull ;

<VF(u) ,PIu) <-klllPlull 2 + k2 o2(u)llP2ul12;

k l l l P 2 u l t 2 - k211Plull F(u) < k211P2ull 2 - k , l l P l u l l 2

with some k~, k2 > 0. Since ~" is dense in U, we have in view of the continuity of the functional w(.) with respect to the norm.of H, that these inequalities also hold for all u �9 U, whence the 1emma follows. Now, by almost literally repeating the reasons given in [173, pp. 203-204], we have that the following statement holds true.

L e m m a 5.4. Let 0 be the only nondegenerate critical point of the function F, F(O) = O, on the neighborhood U. Then, for an r > 0 su~ciently small, the set {u �9 UlF(u ) < -~} has the homotopy type of the set {u �9 UtF(u ) <_ r with a k = dim H~-dimensional cell attached.

From Lemma 5.4 and via the exact sequence of a triple, it follows easily that

k, i = k, Hi({u�9149 0, i # k ,

where k is a coefficient field. Thus, for all i = O, 1 , . . . ,

b i ( f a , f b ) = b i ( f a , f a \ K ~ ) = E b i ( O ~ n r , O ~ f q f ~ \ K ~ ) ~,b = C i ,

ot

where c~ 'b is the number of critical points of the function f of the coindex i lying on f~,b. Consequently, if a0 = a < al < . . . < aN = sup f (x) are critical values of the function f numbered in

an increasing order, then, for all m = 0, 1 , . . . ,

E bk(fai' fa,+,) = E "~ka"ai+l ---- c ~ ( f ) , i i

m

E(--1)rn-kbk( f a) = k = 0

The theorem is proved.

m m

< i k = 0 k = 0

Bib l iograph ica l no tes

The list of references presented below includes papers concerning critical point theory, its applications, and related topics. The term "critical point theory" is not understood here in a wide sense, but the following basic directions are born in mind: problems of the calculus of variations in the large, in particular, geodesics theory, Lusternik-Schnirelman and Morse type theories, minimax principles of the mountain-pass type, dual variational methods of Clarke-Ekeland dual action principle types, topological aspects of optimal control problems and of mathematical programming, including some elements of local theory related with the problems of encoding of critical conditions and calculations of the corresponding topological invariants. The linkage of the themes is no mere chance in view of the synthetic character of the theory: it is also motivated by the results obtained in the basic part of this article. Indeed, the list of reference presented here is far from complete, but the author hopes that it reflects, at least, the diversity of applications of the theory under consideration, from topology to mathematical economy and control problems. A full review of the articles presented below is beyond the scope of this article, so we shall confine ourselves to mentioning the basic sources.

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The exposition of classical Morse theory (involving geodesics theory) can be found in the books [101], [134], [152], [153] and in the corresponding chapters of the monographs [99], [191], [196]. The local aspects of the theory (the index theory) are presented, as usual, in the books on Riemann geometry (see, e.g., [96], [108]). We point out the article [232] in this connection. The article [350], and the corresponding chapters of the book [852], are devoted to both classical and modern aspects of the Morse theory. The book [92] is devoted to the Morse theory for spaces with singularities. From the diversity of articles dealing with the further development of the theory we select the corresponding chapters of the books [107], [173] (see also [174], [199] and articles [175], [346], [353], [365], [382], [383]-[386], [404], [406], [432], [458], [459], [462], [552], [585], [587], [597], [615], [642], [675], [701], [7021, [704], [719], [724], [768], [772], [776], [780], [830], [839], [872], [899], [910], [931], [937], [952], [956], [963], [973], and others).

The original works by M. Morse are not mentioned here because they are not readily available. The exposition of the finite dimensional Lusternik-Schnirelman theory can be found in the books [99], [193], [691]. From the works concerning this theory we point out [100], [124], [125], the review [399] and also [415], [416], [430], [518], [627], [639], [690], [693], [708], [774], [831], [851], [895], [900], [954], [996], [998].

For minimax principles of the mountain-pass theorem type by A. Ambrosseti and P. H. Rabinovitz [249] one can consult the book [146]. The same principles and their generalizations are considered in many papers concerned with applications. The same remark refers to the dual action principle presented, e.g., in the book [105]. In connection with applications we mention the books and review papers [64], [66], [105], [111], [115], [137], [147], [235], [249], [319], [325], [399], [421], [718], [743], [757], [758], [803], [892].

For mathematical programming problems see the review [638] and the bibliography presented there. Finally, we present a number of papers on "the optimal control in the large" related to the problems studied in our paper. For comments concerning this subject see [72].

2759

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