if1 - core.ac.uk · (ii) if1 (u) < for all admissable u. (c) fl(u), a lull a >0 p and f2(x)....
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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.
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