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

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12 So T is an upper solution to (1). The proof of (ii) is similar; it also follows from Remark 3. ¨ Now we prove a general comparison theorem for equation (1). Theorem 9. Denote XaT b ´ T w € LQ aTb€ CaTb. Suppose ], ^ P "ß_ aMß’ 8 b are feasible and satisfy Xa] b Ÿ Xa^ bß ] a> " b ^> a " b, (36) and either ea^ b ! or ea] b Ÿ!. Then ] ^ . If one of (36) is strict, then ] ž ^. Proof. Let Tœ ] ^ . Then T satisfies T a> " b !. Suppose ea^ b !. Setting Oœâ] b in (29), we have w ! Xa] b Xa^ b œ T € LQa] b LQa^b€ CaTb w œT € _^ a a] bàT b€ CaT b€ aâ^b X â] bb ea^baâ^b â] bb w T € _^ a a] bàT b€ CaTbÞ (37) So aTß! b is a pair of upper-lower solutions to the equation T w € _^ a a] bàT b€CaTb œ! with T a> " b œ! . By Proposition 7 (iii), we have ] ^. If one of (36) is strict, then T will be a strict upper solution, which implies Tž! , or equivalently, ]ž^. If ea] b Ÿ! , then set Oœ â^ bin (29) to get w ! Xa] b Xa^ b œ T € LQa] b LQa^b€ CaTb w œT € _^ a a^ bàT b€ CaTb aâ^ b X â] bb ea] baâ^b â] bb w T € _^ a a^bàT b€ CaTbÞ The rest of the proof is the same as the case ea^ b !¨ . The conclusion that ] ^ implies that either ea] b ea^ b ! or ! ea] b ea^b. In other words, ] and ^ turn out to have the same definiteness. Therefore, ea^ b ! and ea] b Ÿ! usually do not occur at the same time. Theorem 9 immediately extends to two equations like (1): w X 3 aT b ´T € LQ 3 aTb€ aTb œ! , T a> b œT , (38) C 3 " "3 where the parameters EßFßGßHßK ßVßW , C and T satisfy (4) for 3œ"ß# , and T T . X Denote e 3 aT b ´ V 3 €HTH 3 , for 3œ"ß#Þ 3 3 3 3 3 3 3 3 "3 "" "# 3 Theorem 10. Suppose ], ^ P "ß_ aMß ’ 8 b are feasible for both equations, ] is an upper solution to (38)" while ^ is a lower solution to (38)#. If one of the following conditions holds, then ] ^ in M. (i) X a] b X a] b and either e a^ b ! or e a] b Ÿ! . " # # # (ii) X a^b X a^ b and either e a^ b ! or e a] b Ÿ! . " # " " 3
13 Proof. The proof is almost trivial. The assumptions imply that X have that X a] b Ÿ!Ÿ X a^b; that is, # # a] b Ÿ!Ÿ X a^b. Then in case (i), we " # a]ß^b is a pair of upper-lower solutions to (38)#. The conclusion follows from Theorem 9 applied to X# . Similarly, in case (ii), we have that X" a] b Ÿ ! Ÿ X" a^b. So a]ß^ b is a pair of upper-lower solutions to (38)" and the conclusion follows from Theorem 9 applied to X " .¨ Theorems 9 and 10 are very general comparison results. Comparison theorems have been proved (e.g., in [10], [11], [19] and [24]) for solutions to (1) with GœHœWœ! under the assumptions that " X E3 FV 3 3 F3 L " L # and C" C# , where L3 œ Œ X . These assumptions imply that K E X 3 3 aTb X aT b for all T’ 8 , which is stronger than the conditions (i) and (ii). In this special case, the " # conditions on e a] b and e a^b are not necessary; see [17]. 3 3 Theorem. From Theorem 9 and the local theory of ODE we prove the following upper-lower solution Theorem 11. Suppose that a] , ^b is a pair of upper-lower solutions to (1). (i) If either ea^ b ! or ea] b Ÿ! , then ] ^ . In addition, if one of ] and ^ is strict, then ] ž ^. (ii) If either ea^ b ž! or ea] b !, then equation (1) has a unique solution T with ] T ^. Proof. Part (i) follows immediately from Theorem 9. For part (ii), we first consider the case Mœc>ß ! > " d. The local existence theory of ODE implies that equation (1) has a solution that exists in a maximal interval Ð7ß> Ó§M . Part (i) implies that ] T ^ in Ð7ß> Ó. By using the equation, we see that " " T ´ lim T ab > exists and ] a7b T â7b. In particular, eaT b ea^ a7bb ž! . If > ž 7 ž> , 7 7 7 >Ä 7€ then the local existence theory of ODE again shows that T can be extended further left beyond 7. This would contradict the definition of 7. Therefore, 7 œ > ! and so T ab > exists in c>ß ! > " d. For the case MœÐ _ß> Ó, the same argument shows that the solution T exists on c>ß > d for every > > . So the " ! " ! " solution exists on Ð _ß> " Ó. ¨ " ! Theorem 11 (ii) implies that a necessary and sufficient condition for the existence of a solution T to (1) with either eaTb ž! or eaT b ! is the existence of a pair a] , ^b of upper-lower solutions to (1) satisfying either ea^ b ž! or ea] b !, respectively. Proposition 8 shows that, in fact, the existence of one of ] and ^ is sufficient for the existence of a solution. We have Theorem 12. (i) Equation (1) has a solution T with eaT b ž ! if and only if it has a lower solution ^ with ea^b ž !.
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