By Alain Bensoussan, Jacques Louis Lions

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At. -h investigating closed-loop closed-loop Considering system this fact. in t b e with f o r m e d by (12) that to the Riccatl Then. definite weighting k > 0. respect corresponding the system (1) (15c) as the solution triplet equation, and (16) system cost (12) equations triplet is stated iS taken poles following on t h e left is B ) = (exp k(a the and In o r d e r See feedback (16) have Appendix to s o l v e of shown. and and f u r t h e r assume above have a non-negative defined equations by (15) (3)-(5), is if the as (17a) Q,(B ) = {~I(~ ) ÷ 2k~t(B ))(exp Moreover, (16) the result stabilizable (13).

M. Biernacki, H. P. Bhattacharyya, "Robust stability with structured real parameter perturbations," 1EEE Trans. Automat. , voi. AC-32, no. 6, pp. 495-506, June 1987. S. Thesis, Department of Electrical Engineering, Texas A&M University, Nov. 1987. P. Bhattacharyya, "A Generalization of Kharitonov's Theorem for Control Systems," Proceedings of the 1988 Princeton Con/crence on In]ormation Sciences and Systems, Princeton, New Jersey, March 15-17, 1988. OPTIMAL REGULATOR FOR LINEAR SYSTEMS WITH DELAYS - SPECTRUM DECOMPOSITION K.

Deternfine (o) (o) Pz ,Pz ,P3 ,P~°) the radii of the largest stability hyperspheres around each of them. ,. mzntPz ,Pz ,P3 ,P4 ) and set p(z) = p ( O ) +p~). ~)~ + ( ~ + ,~p~'~),~ + {~3 + ,,~p(,')s ~ + . . ~), + (u~ + ~ p ~ ' ) ) ~ + (:~ - ~,-,(,~),~ + ' " 28 And thus for any ( in (0,1), the box BI '~ = [~o - ,~opP~,,so + , ~ o A ' ] × l,s, - ,~,pP~, ~, + ~,,pp~] × . . × [,~. p~'~] is all stable. With e going to 1 we then conclude that the open box B O~ = (~5o - ~ 0 p O ) , ~ 0 + ~0p [1)) x ......