1 Upper-Lower Solution Method for Differential Riccati Equations ...

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4 with parameters EßFßGßHßK ßV and W satisfying (4). Thus (1) becomes To indicate the parameters, we may say equation (1) with parameters EßFßGßHßKßVßWßC and RÞ w T € L Q aT b€ C aTb œ!ßTa> b œ R. (1) " LQ a† b with parameters EßFßGßHßK ßV and W, and say X We remark that Q aTb and LQ aTb may be well-defined even if V€HTH is singular. If V€HTH X is nonsingular, then Q aTb can be written as where eaTb, faT b and âTb are " X Q aTb œ faTbeaTb faTb œ âTb eaTbâTb, (6) X eaT b œV€HTH, (7) X X faTb œ F T€HTG€W, âTb œ eaTb faTb. Here eaT b and faTb can be considered as perturbations of V and W with respect to T. The term " âTb œ eaTb faTb appears frequently as the optimal feedback matrix for problem (3) when T is a solution to (1) with C œ! ; see Theorem 1 below. If eaT b is singular, then we define " € âTb œ eaTb faTb, (8) € € where eaTb is the pseudoinverse of eaTb. Recall that any matrix Q has a unique pseudoinverse Q with the following properties (see [18] and [1]). € € € € QQ QœQßQ QQ œQ . (9) 8 € 8 € € If Q ’ , then Q ’ , and QQ œQ Q. € Q ! if and only if Q !. € Note that âTb œ eaTb faTb always exists, but it may not satisfy faTb œ eaTbâTb. This leads to the following definition. Definition 1. If T P _ aMß’ 8 b satisfies f a T b œ e a T b ^ a T b , then T is said to be feasible . _ 5‚8 Denote ŠaTb œ eO P aMß‘ bß faTb œ eaTbOf. Obviously, if T is feasible, then âT b ŠaTb and so ŠaTb Ág. The converse is also true. We have Proposition 1. If ŠaTb Ág, then T is feasible and for each O ŠaTb, X X Q aTb œ âTb eaTbâTb œO eaTbO, (10)
5 where âTb œ eaTb faTb. € Proof. If OŠaT b Ág, then faTb œ eaTbO. Since eaTb œ eaTbeaTb eaTb, we have € € eaTbâTb œ eaTbeaTb faTb œ eaTbeaTb eaTbO œ eaTbO œ faTb. This shows that T is € € feasible. Furthermore, from that âTb œ eaTb faTb œ eaTb eaTbO and property (9), we have € € âTb X eaTbâTb œO X eaTbeaTb eaTbeaTb eaTbOœO X eaTbO. This shows (10). ¨ Proposition 1 shows that if ŠaTb Ág, then Q aTb is well-defined by (10). Each O s ŠaTb will "ß_ 8 be called a feedback matrix associated with TP aMß’ b. This term is motivated by the fact (see "ß_ 8 Theorem 1 below) if TP aMß’ b is a feasible solution to (1) with C œ! , then each O s ŠaTb is an optimal feedback matrix. € Definition 2. Suppose T P "ß_ aM, ’ 8 b is feasible. T is an upper solution to (1) if T is a lower solution to (1) if w T € LQ aTb€ CaTb Ÿ!à T a> b R. w T € LQaTb€ CaT b !à Ta> b Ÿ RÞ T is a solution if it is both an upper solution and a lower solution. An upper or lower solution is called strict if at least one of the inequalities in the definition is strict. " " All differential equations and differential inequalities in this paper are considered pointwise for almost every > in an interval indicated by context. For brevity, we will write, for example, "K !" or " K ! in M " to mean that " Kab > ! for almost every >M." basic results are still of interest if it is assumed that The variable > is often suppressed. The EßFßGßHßK ßV and W are all continuous (or piecewise continuous) and bounded in M, and it is assumed that T is (piecewise) continuously differentiable in M. The equations and inequalities will be then pointwise (except at finite points). Note that ! ’ 8 X is a lower solution to (1) if and only if K W V " W !, R ! and that ! is X " an upper solution to (1) if and only if K W V WŸ! , R Ÿ! . These are the common cases studied in the classical literature of Riccati equations; see [7] and [8] for remarks. Also, ! is a strict upper or lower solution if one of the inequalities is strict. See Proposition 6 for an equivalent description. §2. Interpretation of Solutions and Structure of LQaT b
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Page 3: 3 the solution can be approximated
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