IF1 - core.ac.uk · (ii) IF1 (u) < for all admissable u. (c) Fl(u), a lull a >0 p and F2(x). 0...

lnternat. J. Math. & Math. Sci.Vol. i0, No.3 (1987) 609-612 RESEARCH NOTES

AN EXISTENCE THEOREM FOR OPTIMAL CONTROL WITHNONSTANDARO COST FUNCTIONALS

609

R.N. MUKHERJEE and SHIV PRASAD

athematics SectionSchool of Applied Sciences

Institute of TechnologyBanaras Hindu UniversityVaranasi-221 005, INDIA

(Received September 21, 1983)

ABSTRACT. An existence theorem for optimal control is obtained for a general nonstan-

dard cost functional of fractional type in this work. As an application of our result

we can derive an existence theorem for optimal control given by M. B. Subrahamanyam

for a cost functional, which is a ratio of two given integral cost functionals.

KEY WORDS AND PHRASES. Nonstandard cost functional, optimal control.

1980 AMS SUBJECT CLASSIFICATION CODE. 49B27, 65KI0.

I. INTRODUCTION.

Consider the n-dimensional system

x A(t)x + B(t)u, x(to) 0 (I.I)

for t [t tl] t < ; where A(t) and B(t) are matrices of size n x n ando

n x r respectively. We lay some further restrictions on x(t) and u(t) later.

We minimize a functional of the following type

F (u)F(x,u) (1.2)

[F2 (x) ]a

where a > 0 and u is a measurable control. We make the following assumptions:

(a) A(t) and B(t) are continuous n x n and n x r matrix functions respectively.

(b) (i) F (.) is convex in u and F F2 both are continuous and

(ii) IF1 (u) < for all admissable u.

(c) Fl(u) , a lull a > 0 p > and F2(x) . 0 along any x(t) which is responseP

to some admissable u(t).

(d) For each K < such that if lull < K thenp

IF2(x) < (1.3)

for any admissable u whose trajectory obeys any constraints imposed.

(e) There exists k 0 such that for every c > 0

610 R.N. MUKHERJEE AND S. PRASAD

Fl(cu) Ck

F1(u)

F2(cx) ck/aF2(x) (1.4)

By (1.3) this assumption implies that for every c 0 F(cx, cu) F(x, u)

(g) There exists an admissable control, the trajectory of which satisfies the imposed

constraints and is such that

F2(x) > 0 (1.5)

i(h) If {u is a sequence of functions in L [t ,t I] going weakly to u inp o o

L [t ,tl thenp o

Fl(u) < lim inf Fl(uI) (1.6)

We call a constraint regular if the following two conditions hold:

(I) (x,u) satisfies the constraint => (cx,cu)

satisfies the constraint for every c > 0

i o(2) Let (xl,uI), (x2,u2), be admissable pairs such that u u weakly in

L (to

tl). Suppose (xn un) satisfies the constraint for each n >.. Then (x,u)

obeys the constraint.

REMARK. Observe that in the proof of theorem I. below it has been shown that u is

necessarily admissable.

First of all we prove the following lemma which is the modificaiton of Proposition

I.I of [I] in the general setting.

LEMMA I.I.

Consider all pairs that obey (1.1) and the constraints. Assume that all the con-

straints are regular, and let

F (u)I if F(x,u) inf -/----- (1.7)

(x,u) (x,u) IF2 (x) ]e

(% is well defined by assumption (b) part (ii) and (g)). Also let

inf F1(u) J subject to [Fl(x)] M o. (1.8)U

Then % J/M.

PROOF. One can easily see that J/M > % To prove the reverse inequality, let

be such that F(,) < + for some > o. Let [F2()] (< by assumption

(b) part (ii), (c) and (d)) and (M/) I/kThen ( ,B ) obeys all the con-

straints by regularity of the constraints and by assumption (e), [F2(u)]e M and

F(, B) < + e By (1.8), J/M < + since is arbitrary, the conclusion of

the lemma follows.

THEOREM 1.1.

Consider the system (I.I) and (1.2) along with assumptions (a) to (h). Also

assume that the constraints on x and u are regular. Then there exists a control

among all admissable controls that minimizes (1.2).

PROOF. By lemma i.i it suffices to exhibit a minimizing control over all admlssable

controls for which [F2(x)]e M > o and trajectories of which satisfy (I.I) and all

EXISTENCE THEORF#I FOR OPTIMAL CONTROL 611

the constraints. Let J inf Fl(u) subject to [F2(x)]a

M Choose {(xi,ui)}u

isuch that lira Fl(ul) J with [F2(xl)]a M for each i. By assumption (c) u

i+

form a bounded sequence in L (to

t and hence a subsequence still denoted by {ui}p

o oconverges weakly to some u in L (t

ot Let x be the response of (I I) to u

pi

xo

By assumption (a) and by the convergence, x (t) (t) for all t[to,tl (see ref.

x[23). By regularity of constraints, (t) obeys all the constraints. Assumption (b)

i oF2(xI) F (x) as i (Since x x in L (t

ot )) Sinceimplies p

for some K < By assumption (d),

By assumption (h)

[F2(x)]a lira [F2(xI)] Mi+

F (u) < lira inf F1(ui) J

From which it follows that u is the minimizing control.o

APPLICATION. If we specialize the functional F(x,u) as given by F(x,u)

otl l(x(t),t) dt

tl 2 (u(t), t)dt]e

Where I and 2 satisfy the condition as given in ([I], theorem I.I) we get the

existence theorem of [I] as a particular case of our thoerem.

DISCUSSIONS. The general method, what we have discussed, can be extended to cover

other variants of the functional given by Subrahmanyam [I]. For example, we can con-

sider functional of the form

F(x,u) . 1 (x(t),t) dt

l(U(t),t)]el [{o n(U(t)’t)dt]n

where I, {i}.=l all have to satisfy certain restrictions analogous to the restric-

tions given as in [I].

REFERENCES

i. SUBRAHMANYAM, M.B. On Applications of ContrQl Theory to Integral Inequalities II,Siam J. Con__t_r__o_an_d_._O_t.im_ization, 19(1981), 479-489.

2. LEE, E.B. and MARKUS, L. Foundations of Optimal Control Theory, John Wiley, NewYork, 1967.

3. SUBRAHMANYAM, M.B. Necessary Conditions for Minimum in Problems with NonstandardCost Functionals, J. Math. Anal. Appl. 60(1977), 601-616.

4. SUBRAHMANYAM, M.B. On Applications of Control Theory to Integral in Equalities,J. Math. Anal. Appl. 77(1980), 47-59.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

IF1 - core.ac.uk · (ii) IF1 (u) < for all admissable u. (c) Fl(u), a lull a >0 p and F2(x). 0...

Documents

Transcript of IF1 - core.ac.uk · (ii) IF1 (u) < for all admissable u. (c) Fl(u), a lull a >0 p and F2(x). 0...