MiNiMAX OPTIMAL CONTROL FOR ONE CLASS OF UNCERTAIN SYSTEMS

II J[u(.),xo' v()] == h [x(1 f }J+ f g{I,u(t }}it (2) '0 where X(I f) is the value of solution of Eqn, (l) at t == tf ' h: ~ n .~ \Ris convex, and g:{lo.lf Ix P ~ !Ris continuous ."ith respect to I and convex with respect to x, It is desired to select control function u minimizing the cost J Also it is given that the state vector x in the system given by Eqn.(l) is not observable, therefore, u(t)will be designed viJithout having feedback from it,

.Dept. of Electrical and Electroni.cEng.Osmangazi UniYersity Dademlik 26030 F,ski~hir, TURKEY.**D\.-pt, of Mathematics, Dumlupmar Uru"emty 43100 Kiltahya, TIJRKEY.Abstact A controllable system described by line..>rdifferential equatiOl1 with uncertainties in the initial condition and forcing function is COllsidered, We aim to find a cartrol which minimizes a cost function having terminal and integral parts, Using game theory and cmvex analysis, under some sufficient cond.'tioos, the optimal control is obtained, where ' 0 where  In this paper we present sufficient conditions fur the solution of problem given by ( 3), and we present illustrative examples.
If the initial position is known, i.e. S == {x o }, then the problem (1)-( 3) is a programmed minimax problem and can be investigated by methods of [1-3] .
Schmitendorf [4] gives sufficient conditions for nonlinear systems with unknown initial vector X o and unknown parameters, but the conditions obtained there are very complicated.
In I5] the problem (1)-( 3) without integral term in the cost functional was investigated.
. fundamental matrix for the homogeneous system X(I) == A (I )x( 1j, and let h' (x) where <: .Utilizing Cauchy formula to obtain the solution ofEqn.
(1), and from the property of interchangability of inner and outer maximums we can express the problem given by (3) in terms of the notations introduced above: Fortbe system (4) find t/ EU satisf)ing The initial condition and uncertainty in the problem given by ( 1)-( 3) have been transferred into the supremum.Due to the definition of functional J, compared to (1)-(3) the solution of( 4)-( 5) is sometimes casier.We, however, analyze the problem (1)-( 3 holds.In Eqn. ( 9) the outer minimum and outer maximum pair is a saddle point.The vector X o which ll".inimi7.es the lefthand side of(9) and the vector Yo which maximizes righthand side of (9) are called .mininl.!lX and maximin respectively.
Lemma 2: Let (xo,Yo) be a saddle point.Then the vector X o minimizes the function m(r) '" j(x,yo)' 0 If Yo is a maximin vector qualifying as a maximum for the righthalld side of (9) then the vector minimizing m(x) = j(x,yo) is not necessarily the minimax vector.
Consider the example below.Corollary 1: Let the fun,,1:ion g(t,u) be strictly convex with respect to u.Then uO(/) satisfying the condition ( 11) is the optimal minimax control.
Proof Since the function g(t,u) is strictly convex with respect to u, the function UO{t) minimizing the expression (II) is unique.According to Lemma 2 it is the optimalcontrol.0 Theorem 2: Let the function lP< i) be convex, L be the convex hull of {i l , ..., i k }.In the sequel it will be denoted by co{.e 1" .. ,!.k }.Then Proof Note that by the hypothesis of the theorexr..,the fjJnction J(u{.),f) is convex with respect to f. the foUowing equalities can be shown easily: • (.)eU pep Now we can restate the problem in the terms defined above: It is desired to find UO E U such that min sup J[u(),xo'v(.)]=sup J{u°(.),xo,v(.)]=:Jo(3) .(.)al xoe8.v();¥><,e8.v(.)<VThe function U O satisfYingEqn.(3) is called guarantt"€ldoptimal control.
X(I f) is the value of solution of Eqn, (l) at t == tf ' h: ~n .~\R is convex, and Assuming that the unknown inputs are measurable and taking values from the set Q we denote such set of unknown inputs by V.