SIAM J. CONTROL AND OPTIMIZATIONVol. 27, No. 4, pp. 776-785, July 1989

1989 Society for Industrial and Applied Mathematics006

A BOUNDARY VALUE PROBLEM FOR THE MINIMUM-TIME FUNCTION*

MARTINO BARDI?

Abstract. A natural boundary value problem for the dynamic programming partial differential equationassociated with the minimum time problem is proposed. The minimum time function is shown to be theunique viscosity solution of this boundary value problem.

Key words, nonlinear systems, time-optimal control, dynamic programming, Hamilton-Jacobi equations,viscosity solutions

AMS(MOS) subject classifications. 35F30, 49C20

1. Introduction. Given the control system

(1.1) y(s)+b(y(s),z(s))=O, s>=O, y, zZM,the minimum time function T(x) associates to a point x N the infimum of the timesthat the trajectories of (1.1) satisfying y(0)= x take to reach the origin. The problemof determining T and the optimal controls realizing the minimum is one of the mostextensively studied in the control-theoretic literature, especially in the linear case 1 ],[5], [6], [8], [15], [16], [20], [22]-[26].

It is well known that Bellman’s Dynamic Programming Principle implies that atpoints of differentiability T satisfies the following first-order fully nonlinear partialdifferential equation (PDE) of Hamilton-Jacobi type"

(1.2) sup b(x, z) DT(x)= 1.zZ

It is also known that in general T is not differentiable everywhere, but it satisfies(1,2) in some generalized sense [8], [24]. In the last five years the new concept ofviscosity solution for Hamilton-Jacobi equations has been introduced by Crandall andLions [11] and the theory of such solutions has developed quickly (see, e.g., [9]-[12],17], [21] and the references therein) and has been applied to many problems in control

theory and differential games (see, e.g., [3], [4], [7], [13], [21], the survey paper ofFleming [14], and its long list of references). Following Lions [21] it is not hard toshow that T satisfies (1.2) in the viscosity sense as soon as it is continuous. The goalof this paper is to complement (1.2) with a natural boundary condition and prove auniqueness result for viscosity solutions of such a boundary value problem (BVP).Ishii 17] has proved the uniqueness of viscosity solutions of the Dirichlet problem ina bounded open set for a class of equations that includes (1.2). However the Dirichletproblem does not seem to be the most natural one for the minimum time functionbecause in general we do not know a priori the value of T on the boundary of somegiven bounded set. Instead we consider (1.2) in the set \{0} where is the set ofpoints x such that there is a trajectory of (1.1) starting at x and reaching the origin infinite time, i.e., the largest set where T is defined (and finite); we propose the followingboundary condition:

(1.3) T(0) 0 and T(x) -> +o uniformly as x O.

In 2 we will prove that under quite general assumptions T satisfies (1.2) in \{0}in the viscosity sense and (1.3). In 3 we will show that for any open set Y having

* Received by the editors December 9, 1987; accepted for publication (in revised form) July 7, 1988.t Dipartimento di Matematica Pura e Applicata, Universitt di Padova, 1-35131 Padova, Italy.

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A BVP FOR THE MINIMUM-TIME FUNCTION 777

zero in its interior there is at most one viscosity solution of (1.2) in \{0} satisfying(1.3) and bounded below.

The main result is the uniqueness theorem in 3. it presents three difficulties withrespect to standard uniqueness results in the Hamilton-Jacobi theory: (i) the Hamil-tonian depends on the gradient of the unknown function but does not depend explicitlyon the unknown function itself, while it is usually required that it be strictly monotonein such a variable; (ii) the infinite boundary condition (1.3) if N; and (iii) thelack of regularity of the solutions to be compared and of the Hamiltonian if b is notglobally bounded and is unbounded.

To overcome the first difficulty we introduce a change of the unknown variablefirst used for this goal by Kruzkov [18]. It turns out that this transforms the infiniteboundary condition into a finite one, therefore automatically taking care of the seconddifficulty.

The third difficulty can be solved using the new approach to uniqueness presentedin the paper of Crandall, Ishii, and Lions [10]. Indeed our uniqueness theorem hasto be considered as a corollary of the methods developed in 10]. As one of the refereespointed out to us, this difficulty had already been overcome for different problems inan earlier paper by Ishii [28], whose methods could be applied effectively to ourproblem as well.

We remark that the proof of the uniqueness theorem does not make use of theconvexity of the Hamiltonian. Therefore the methods of this paper can be employedto study the minimum time problem in games of pursuit and evasion (see Bardi andSoravia [27]).

After the completion of this work we have learned that Kruzkov’s change ofvariables has been used recently by Lasry and Lions [19] to study the minimum timefunction of a differential game with state constraints in a bounded domain, and byBarles [2] for unbounded control problems. Moreover, one of the referees pointed outthat a uniqueness theorem for discontinuous solutions of (1.2), (1.3) can be provedby combining the Kruzkov transform and the results by Barles and Perthame [3],provided the target to be reached is a smooth set instead of a single point.

In the last decade the use of the theory of subanalytic sets has led to very strongresults on the regularity of the minimum time function and of feedback controls (seeBrunovsky [6] and Sussmann [26] and the references therein). However it is alsoknown that certain quite smooth systems exhibit very irregular behaviors that fail tofall within the theory of subanalyticity (see Lojasiewicz and Sussmann [22] or theclassical Fuller’s example in [23]). It is our hope that the PDE setting of the minimumtime problem proposed in this paper could be of some help in the study of theseproblems.

2. The BVP of the minimum time function. We begin listing the assumptions tobe used in the following.

(H1) b "vxZ-, where Z c__ 4, is continuous and there exist L, K suchthat Ib(x, z) b(y, z)[ =< L[x y[ and Ib(y, z)[ =< K(1 + [y[), for all x, y N,and for all z Z.

Let be the set of measurable functions z:[0, c)-, Z, and let y(s)= y(s; x, z)be the solution of

(2.0) y(s) x b(y( t), z( t)) dt

for s ==_ 0, x u, and z . Let 3-c_. RN be a given closed set, the terminal set (e.g.,

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778 M. BARDI

3-= {0}), and define

:= {x 6 [N: ::lz J//, >- 0 such that y( t; x, z) 3-}.

Clearly,_

3-. Now define the minimum time function

T: -> [0, oo), T(x) := inf { t: y( t; x, z) 3- for some

We will assume the following.

(H2) is open.

(H3) T is continuous in .(H4) For every Xo 0, limx_o T(x) +oo.

Conditions under which (H2)-(H4) are satisfied are well known in the literature,especially in the linear case. See, for instance, [1], [15], [20] for (H2), [25], [15], [1],[5], [8] for (H3), [8], [15] for (H4), and the references therein. Essentially (H2)-(H4)follow from some controllability around 3-.

Next we define a suitable boundary condition for functions u C(\3-):

(BC) u converges uniformly to 0 as x 03- and to +oo as x 0Y2, i.e., for every e,M > 0 there exists 6 > 0 such that dist (x, 03-) < implies ]u(x)] < e anddist (x, 0) < 6 implies u(x) > M.

Before proving that T satisfies (BC) we need a technical lemma.LEMMA 1. Assume (HI) is true. If y(t; x, z)=xl, then

1 (>_---- log 1 +K l+min (Ix[ IXl[)

Proof Hypothesis (H 1) implies

ly(s)-xl<- gs(1 +lxl)+ KIy(,)-xI d,

and then, using Gronwall’s inequality, we get

Ix1 xl-<- (1 + [x[)(e Kt 1),

which gives

( Ix,,-x.l t>=-log 1+l+lxI]"

The same calculation for 37(s):= y(t-s) leads to

1 ( + Ix, X,[t->log 1l+lXll]"

Remark 1. It follows easily from Lemma 1, choosing Xl 3-, that T(x) > 0 for allx \3- (since 3- is closed) and that 3- bounded implies limlxl_.o T(x)

LEMMA 2. Assume (H1)-(H4) and 3- bounded. Then T satisfies (BC).Proof Since T is continuous, null on 03-, and 3- is bounded, the first part of (BC)

is clearly satisfied.To prove the second part we fix M > 0. From Lemma 1 and Remark 1 the existence

of R > 0 such that T(x)> M for Ix > R follows. Now we use (H4) to get a coveringof the compact set 0 (q {x: Ixl <= R} made of a finite number of open balls B centeredon0 and having small radius so that T(x) > M for x f’) B. We conclude observingthat there exists > 0 such that Ix[ <- R and dist (x, 0) < 6 imply x Bi for some i.

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A BVP FOR THE MINIMUM-TIME FUNCTION 779

We recall that a continuous function u defined in an open set

_EN is defined

to be a viscosity solution of

H(x, u, Du)= 0 in ,if for every b C1() and Xo local maximum point of u-b we have

H(xo, U(Xo), D6(xo)) -< 0,

while for xo, local minimum point of u-b we have

H(xo, U(Xo), Dqb(xo)) >- O.

We will now prove that T is a viscosity solution of

(HJ) sup b(x, z). Du-1 =0 in \-.zZ

This fact is certainly known to experts, since it follows from arguments of Lions [21,Chap. 1]. We include a full proof for the sake of completeness.

Define for zeM, xeNu, tx(Z):=inf{t: y(t; x,z)e -} and denote by X{t<tx(z) thefunction defined on [0, oo) which is one if < tx(z) and zero if the opposite inequalityholds.

LEMMA 3 (Dynamic Programming Principle). Assume (HI). Then for all xand >= 0

T(x) inf {min (t, tx(z))+X{,<,x(z)iT(y(t" x, z))}.

Proof. Fix x and and let A be the right-hand side of the above equality. Toprove T(x)>= A we fix an arbitrary e > 0 and show that

(2.1) T(x)>=A-e.

Let zl M be such that

(2.2) T(x)>=t(Zl)-e.

If t>= t,(zl), then (2.1) holds. Now suppose t<t(zl) and let z(s)=zl(t+s). Then

tx(Zl) t+ ty(t;x,z,)(z2) t+ T(y(t; X, Zl))>=a,

and so by (2.2) we have (2.1).Now we want to prove

(2.3) T(x)<=A+e,

and for this we choose zl such that

(2.4) A+2-=>min (t, tx(gl))+X{t<t(Zl)}T(y(t; x, Z1) ).

If >_-t(zl) then (2.3) holds. If < t(zl) let z M be such that

(2.5) T(y(t; x, z1)) ty(t;X,Zl)(Z2)-_..z

Now define the control

ifs< t,z3(s):=

z2(s-t) if s>=t.

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780 M. BARDI

Clearly

tx(z3) t+

and therefore we get from (2.4) and (2.5)

E E E->= t+ T(y( t; xl, zl)) >- /x(Z3)-e Z(x)-’,A+2

which proves (2.3). By the arbitrariness of e the proof is complete.THEOREM 1. Assume (H1)-(H3). Then the minimum time function T is a positive

viscosity solution of (HJ). If moreover (H4) holds and the terminal set 3- is bounded,then T satisfies the boundary condition (BC) as well.

Proof The second statement is just Lemma 2. To prove the first statement webegin by considering Xo \ 3- and b Cl(y \ 3-) such that for all x sufficiently closeto Xo,

(2.6) T(xo)- 49(Xo) >= T(x)- oh(x).

Let Zl be any constant control, Zl(S) 6Z. By Lemma 3 we have for all O<s < T(xo)

r(xo)- r(y(s; Xo, z,)) <- s,

and so by (2.6) we have for sufficiently small positive s,

T(xo)- r(y(s" Xo, z,))5 1.

6(Xo) b(y(s; Xo, z,))=<S S

Now letting s % 0 and using (2.0), we get

Dch(Xo)" b(xo, e)=< 1,

and by the arbitrariness of ,sup b(xo, z) D4)(Xo)- 1 <= O.zZ

Let us now consider new xo and b as above but such that

T(xo)- qb(xo) <= T(x) qb(x)

for all x in a given neighborhood of xo. By Lemma 1 there exists s such that

b(xo)- b(y(s; xo, zl)) >= T(xo)- T(y(s; Xo, z)) Vs =< s, /z

Fix e > 0. By Lemma 3 for every s <= T(xo) there is z* such that

T(xo) >_ s + T(y(s; Xo, z*)) es,

and thus for 0 < s =< s2

(2.7)6(Xo)- 6(y(s; Xo, z*))

> 1 e.S

Using (2.0) and the expansion

oh(x) 6(Xo) + D6(xo) (x Xo) + m(x)]x xo]

we can write the left-hand side of (2.7) as follows:

_1 Dqb(xo) b(y( Xo z*), z*( t)) dt m(y(s; Xo z*)) _1S S

with lim m(x) 0,

z*) z*( t))b(y(t;Xo, dt

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A BVP FOR THE MINIMUM-TIME FUNCTION 781

The second term in this expression can be made smaller than e for small s by (H1)and Lemma 1; by using (H1) again, the first term can be written as

1Is o Dc(Xo) b(xo, z*(t)) dt plus a correction term smaller than e for small s. Thenby (2.7),

sup Dd(xo) b(Xo, z) >=- Dd(xo) b(xo, z*( t)) dt >= 1 3e,zeZ S

which provides the desired inequality by the arbitrariness of e.

3. Uniqueness. In this section we will prove the following theorem.THEOREM 2. Assume (H1), let be an open subset of R1, and let

_be a

closed set. Iful, u2 C(\) are viscosity solutions of (HJ) satisfying (BC) and boundedfrom below, then U u2.

The main tool of the proof is a lemma of Crandall, Ishii, and Lions [10, Lemma1], which we report below in a version simplified for the present purpose. We say thatH "Rnx n - It satisfies condition (H) if the following holds"

(H) H is continuous; there exist a Lipschitz continuous, everywhere differentiablefunction/x - [0, oo) and a continuous, nondecreasing function[0, oe) satisfying tr(0) 0, such that H(x, p) H(x, p + AD/x(x)) =< tr(A), forall x, p N", h [0, 1 and liml,l_ /x (x) /.

LEMMA 4. Let H satisfy condition (H) and let f be an open subset of Nn. Letz C() be a viscosity solution of z + H(x, Dz) 0 in f, and let w C1(() satisfy

w(x)+ H(x, Dw(x)) >-0 and IDw(x)l <= C Vx

Assume that

sup (z w) < sup (z w) < c.

Then z <-w in

Proof See 10] for the proof of this lemma.The other key tool in the proof of Theorem 2 is a change of the unknown variable

in (HJ). For this we need a slight extension of Corollary 1.8 in [11].LEMMA 5, Let u be a viscosity solution of H(x, u, Du)= 0 in , open subset of

let P CI(), dP’(r) >O for all r; and let ’() be the inversefunction of . Thenv u is a viscosity solution of

H(x, (v), ’(v)Dv)=0 in .Proof Let Xo 6 and ’ Cl(t) be such that v-ff has a local maximum in Xo.

Define sO(x)= ’(x)-r(Xo)+ V(Xo). We have

D:= Dsr, :(Xo) V(Xo)= (U(Xo)),

v(x) <= (x) in a neighborhood of Xo.

Since (E) is open, is defined in a neighborhood of Xo and we extend it tor/ C1(6). By the monotonicity of we have

U(Xo) q(Xo), u(x) <-_ rl(x) in a neighborhood ofxo.

Then

H(xo, U(Xo), Dq(Xo)) <-- O,

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782 M. BAlII

and so

H(xo, (V(Xo)), ’(V(Xo))O(xo)) <- O.

If Xo is a minimum point we get the desired inequality in the same way. [3

Proof of Theorem 2. Define P(t) := 1- e-’, Vl :--- (I) Ul V2:= (I) U2, if:= \3-. ByLemma 5, vl and v2 are viscosity solutions of

sup { b(x’ z) ( 1-vDV)}-l=0 in,

and since vl, v2 < 1, they are also viscosity solutions of

(3.0) v+sup{b(x, z). Dv}-I =0 in ,zGZ

as is easy to verify from the definition. They can be extended in a unique way to va,v2 C() by setting

vi=0 on03-, vi=l on0R, i=1,2.

Now define

H(x, p):= sup {b(x, z) p- 1}.zGZ

We claim that H satisfies condition (H). Fix e>O and choose zeZ such thatH(x, p) <- b(x, z) p- 1 + e. Then by (H1)

H(x, p) H(y, q) <- b(x, z1). p b(y, z) q + e

<= L[x Yl IP[ + K(1 + [Yl)IP ql + e,

and thus

(3.1) [H(x,p)-H(y,q)l<-Llx-YllPl+K(l+lyl)[p-ql,

which implies the continuity of H. Now let h cl(N) be such that h(0)= h’(0)=0,h(e)= 1, h’(e)= I/e, and define

h(lx[) iflxl<e,(x) :=

log (Ix I) if Ixl--> e.

Clearly, z C1(u) and it is Lipschitz continuous since Dt.(x)= (1/[xlZ)x for Ixl _-> e.Moreover, by (3.1),

H(x, p) H(x, p/ tD(x)) <-_ K a / lxl)tlO(x)l <-C -:

for > 0, where C := max (2K, K (1 + e) sup Ih’l), which proves the claim.Next we define, following Crandall, Ishii, and Lions [10],

/(x, y, p, q):= H(x, p)- H(y, -q),

z(x, y):= vl(x)- v2(y),

and note that z is a viscosity solution of

z + I21(x, y, Dxz, Dyz) 0 in x .Our goal is to prove that z(x, x)<-0, because by interchanging the roles of v and v2we get vl v_ and then u u2. To reach our goal we are going to apply Lemma 4 toz defined above, f := A f)(G x ) where

Z:= {(x, y) N-rq" [x-yl < 1},

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A BVP FOR THE ’MINIMUM-TIME FUNCTION 783

and w we for suitable e where

we(x, y):= (e4L+lx- yl2)’/zI/e.We have to show that there exists Co>0 such that for all 0< e <-Co, we satisfies thehypotheses of Lemma 4. Once this is done we have z(x, x)<= we(x, x)= e, and lettinge 0 we conclude.

Since

1D,w(x,y)=--(e4I+lx-y (x-y)=-Oyw,(x,y),

we is Lipschitz continuous in A and, moreover, by (3.1),

we(x, y)+ I2I(x, y, D.w, Dyw) >= we Llx yl2(e4+ Ix yl)/)-a/eLg (1 + lyl)lDxw + Dyw

>-- we we O.

Since ul and u2 are bounded from below, vl and v2 are bounded, and

(3.2) a:=sup(z-w)<-_A<c foralle>O.

If

lim inf at =< O,eO

we immediately obtain z(x, x)<-O. Thus it remains to prove that a>-_a>O and0 < e _-< eo imply

sup (z- we) < ce sup (z-

Suppose that (:, 37)O is such that

(3.3) z(, 37) w (if, 37)2

Fix 0<3<1 such that xO and Ix-yl<, implies Iv,(x)-v,(y)l<o/2, i- 1, 2. Thiscan be done because v and v take up their boundary values uniformly as a consequenceof (BC). Suppose first that I-1 < so that either or , say , belongs to 0. Thenz(, ) 0 and w 0 imply

z(, y)- w,(, y) v,()- v(y)- v()+ v() <-2’

a contradiction to (3.3). On the other hand, if I-1 , (3.2) and (3.3) imply

z(, y) w(, y) sup (z w) z(, ) w(x, ) A,

from which we obtain

v:() v(y) w(, ) A.

The right-hand side of the last inequality can be made arbitrarily large choosing e

small because

lim inf {w(x, y): x- y] 6} +,e0

and we get a contradiction because the left-hand side is bounded.

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784 M. BARDI

Remark 2. Under the stronger assumption that [b(y, z)[ =< K for all y \ -, z Z(which excludes the linear case if is unbounded), and strengthening the boundaryconditions, we can give a weaker uniqueness theorem with a much shorter proof basedon the earliest uniqueness result for viscosity solutions, that is, Theorem III.1 ofCrandall and Lions [11]. In fact, under such an assumption on b, (3.1) implies thatthe Hamiltonian H(x, p) is uniformly continuous in ENx {p" Ip]_-< R} for all R >0.The additional boundary condition is

lim u(x)= +o,(ABC)

which is satisfied by the minimum time function T if - is bounded (see Remark 1).The proof of the weaker theorem begins with the same change of variables as the proofof Theorem 2. Next, we note that if u, u2 satisfy (BC) and (ABC) then v, v26 BUC()and that (3.0) satisfies the hypotheses ofTheorem III.l(ii) in 11], which implies va v.

Remark 3. If is bounded, ua and u satisfy (ABC), they are zero on 0-, andthey tend to + as x -* Xo 6 0R, then the hypotheses of Theorem 2 are satisfied. In fact,ua and u2 are clearly bounded from below, and they satisfy (BC) by the proof ofLemma 2.

Remark 4. If we drop the assumption that Ul and u2 are bounded below, thenthe conclusion of the theorem is false. In fact, if we take b(x, z)= z, Z the unit ballin N, 0-= {0}, then we get the BVP

IDu[- 1 0 in

u(0) =0,

which has the classical solutions u(x)= Ix] and u(x)=

Acknowledgment. The author thanks Alberto Bressan for some interesting con-versations which stimulated this research.

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A BVP FOR THE MINIMUM-TIME FUNCTION 785

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fail to be subanalytic, SIAM J. Control Optim., 23 (1985), pp. 584-598.[23] C. MARCHAL, Chattering arcs and chattering controls, J. Optim. Theory Appl., 11 (1973), pp. 441-468.[24] F. MIGNANEGO AND G. PIERI, On a generalized Bellman equationfor the optimal-timeproblem, Systems

Control Lett., 3 (1983), pp. 235-241.[25] N. N. PETROV, The continuity of Bellman’s generalized function, Differential Equations, 6 (1970), pp.

290-292.[26] H. J. SUSSMANN, Analytic stratifications and control theory, in Proc. Internat. Congress of

Mathematicians, Helsinky, 1978.[27] M. BARDI AND P. SO:AVIA, A PDEframeworkfor games ofpursuit-evasion type, in Differential Games

and Applications, T. Basar and P. Bernhard, eds., Lecture Notes in Control and Information Sci.,Springer-Verlag, Berlin, New York, to appear.

[28] H. ISHII, Uniqueness ofunbounded viscosity solution ofHamilton-Jacobi equations, Indiana Univ. Math.J., 33 (1984), pp. 721-748.

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