Advances in Dynamic Games and Applications by Eitan Altman, Tamer Başar, Naira Hovakimyan (auth.), Jerzy

By Eitan Altman, Tamer Başar, Naira Hovakimyan (auth.), Jerzy A. Filar, Vladimir Gaitsgory, Koichi Mizukami (eds.)

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Tf]; W) into nonnegative reals which we denote by Fp.. The minimax (HOO-optimal) controller design problem with state feedback is one of finding a state-feedback controller {l that minimizes the index sup (4) p(J-L, w), WEL2([to,t I]; W) where p(J-L, w):= :F 1 1/2j 11:112 and IIwII2:= {l tl to IIw(t)lI~dt }\/2 (5) If this minimum value (whenever it exists) is denoted by y*, then a related problem is the following: Given a y > y *, find a state-feedback controller, {ly, such that inf sup p(J-L. w) < y.

Let Assumption Al hold. Then y* defined by (15) is finite, andfor all y > y*: (1) There exists a minimal nonnegative-definite solution, Mo, to (37) such that (39) holds. (2) There exists a minimal nonnegative-definite solution, P, to (46) such that (47) holds. (3) Condition (48) holds. (4) An Roo-controller that ensures the bound y2 in the index (12) is given by (56). 1), part (5)). For a given y > 0, if anyone ofthe properties (1)-(3) above does not hold, then y ::: y*. 1. Under A 1, the optimal controlproblem with the disturbance model M1 admits the solution: u';' = foo (X ~) = -B'M(I IA + BB'M)n, f3nln ( ~in) =: K ( ~in ) , f3nln f3nln where Mis the unique nonnegative-definite solution of M = N + A'M(I + BB'M)-IA (58) (59) in the class ofnonnegative-definite matrices, and ~nln = ~n+1 (I + PfI'(££')-1 fIrl (~n + PfI'(££')-l yn ) , = A~n + ASfI'(££')-1 (Yn - fI~n), pr (60) (61) l , and P is the unique nonnegative-definite where S := P[l + iI'(£ £')-1 fI solution of P = DiY + ASA' in the class ofnonnegative definite matrices.

Perros and Y. ). North-Holland, Amsterdam, pp. 121-140, 1994. [3] Altman, E. and T. Ba~ar. Optimal Rate Control for High Speed Telecommunication Networks. Proceedings ofthe 34th IEEE Conference on Decision and Control, pp. 1389-1394, New Orleans, Louisiana, December 13-15,1995. [4] Altman, E. and T. Ba~ar. Optimal Rate Control for High Speed Telecommunication Networks: The Case of Delayed Information. 2 Broadband Communication, pp. 115-122, Paris, France, December 6-8, 1995. [5] Altman, E. and G.

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