Page 70 - Textos de Matemática Vol. 36

60
Chapter 2.
Global aspects
α) (2.55) or (2.59) and (2.60).
β) The terminal value W0 is such that
 .  −1 In
where . =V W andV isdefinedasin(2.11).
c 1
cm+n
0
c1 .
det  .  ̸= 0, cn
Then the solution W (t, W0) (with W (t0, W0) = W0), of RDE is uniformly bounded in (−∞,t0]. Indeed, there is no finite escape time phenomenon and from Remark 2.4.11 we infer convergence towards the reverse di- chotomic solution.
In the last part of this subsection we shall again apply Theorem 2.5.1. But, in contrast to the previously investigated Riccati differential equations RDE we shall study here classes of generalized or perturbed Riccati differential equations admitting Hermitian or symmetric solutions. In particular we deal with terminal value problems of the form
K˙ =−A∗K−KA−Q+KSK+g(K), K(tf)=Kf, (2.67)
where the coefficient matrices A = A(t), Q = Q(t), S = S(t) are piecewise continuous and locally integrable in (−∞,tf], Kf ∈ Cn×n, with Q∗ = Q, S∗ = S, Kf∗ = Kf. If furthermore t → g(t,K) is piecewise continuous and locally integrable on (−∞,tf], g(t,·) is a Lipschitz continuous map of Cn×n into itself and
(g(t, K∗))∗ = g(t, K), t ∈ (−∞, tf ], (2.68) then (2.67) admits a Hermitian solution
K(t) = K∗(t)
in a neighborhood of tf .
In the sequel we present conditions for these solutions to exist in (−∞, tf ]. Since we are interested in specific applications, we restrict our considerations to the perturbation terms g(K) which arise from these applications.