Page 69 - Textos de Matemática Vol. 36

2.5. Global existence results 59
Before passing to another application of Theorem 2.5.1 we present a pos- sible application of the previous results to optimal control problems. Since ap- plicability of Theorem 2.5.2 to such problems was already discussed in Corol- lary 2.5.3 we emphasize here the application of Theorem 2.5.6 and Corollary 2.5.7. We consider here Riccati matrix differential equation
K˙ = −A∗(t)K − KA(t) − Q(t) + KS(t)K, K(t0) = K0. (2.65)
This equation was denoted HRDE, i.e., Hermitian Riccati differential equation, if Q(t) = Q∗(t), S(t) = S∗(t) and K0 = K0∗. In classical Optimal Control Theory there holds moreover Q(t), S(t) ≥ 0. In [KnPo97] one dealt with the case where S(t) ≤ 0 and Q may be non-symmetric, while in H∞-control problems S(t) is in general indefinite and Q(t) ≥ 0. Another generalization is investigated in [ThVo75] where in general Q∗(t) ̸= Q(t) and therefore the solutions under consideration usually turn out to be non-symmetric. Complex valued matrix solutions have been considered in [Zeli92], [Zeli93], [Zeli98].
Example 2.5.8. In order to apply Theorem 2.5.6 to such generalized equations, we can choose the parameters C and D in an adequate way; let us choose for instance C = 0n and D = In. Then the matrix Θ in (2.53) has a simple form and we obtain
∗ −(Q+Q∗) 0n
Θ+Θ = 0n −(S+S∗) . (2.66)
From (2.54) and (2.55) we then infer global existence of solutions for t ≤ t0 if Q(t)+Q∗(t) ≥ 0 and S(t)+S∗(t) ≥ 0 for t ≤ t0 and if the terminal value fulfills
K 0 + K 0∗ > 0 .
Applying Corollary 2.5.7, this latter condition could be weakened to
and
K 0 + K 0∗ ≥ 0
K 0 + K 0∗
rank −(Q(t )+Q∗(t ))−K∗(S(t )+S∗(t ))K 000000
=n.
This yields appropriate global existence results in all cases mentioned above. Choosing other parameter matrices C, D one would have obtained different con- ditions for global solvability.
Remark 2.5.9. i) The approach using a Lyapunov-type function as presented in Theorem 2.5.6 and Corollary 2.5.7, respectively, yields global existence of solutions of RDE, but, contrary to the approach presented in Theorem 2.5.2, the solutions need not be bounded.
ii) Let Mij , 1 ≤ i, j ≤ 2, be constant matrices and let the reverse dichotomic solution of ARE exist. Assume there exist matrices W0,C,D, such that (2.54) and the following conditions hold: