Page 18 - Textos de Matemática Vol. 36

8 Chapter 1. Introduction For details concerning linear equations see the Appendix A.1.
A particular case of a matrix Riccati differential equation is the symmetric or Hermitian matrix Riccati equation:
X˙ = −A∗(t)X − XA(t) − Q(t) + XS(t)X, where X(t), A(t), Q(t), S(t) ∈ Rn×n.
The associated matrix
A −S M = −Q −A∗
is Hamiltonian and the equation (HRDE) admits Hermitian (symmetric in case of real coefficients) solutions if the initial matrix is Hermitian (symmetric). This type of equations mostly appears in optimal control problems.
Another important class of Riccati equations is that of discrete time Riccati equations. With
MM M = 11 12
M21 M22
∈ C(m+n)×(m+n) (1.15)
we denote an arbitrary matrix where the sub-matrices M11, M12, M21, M22 are of dimensions n×n, n×m, m×n and m×m, respectively. If detM11 ̸= 0, then we associate with M the matrix
TT (T(M) =) T = 11 12 T21 T21
I 0M−1 0I−M n 11 n 12
−M21 Im 0 −M22 0 Im M−1 −M−1M
(1.16)
:=
21 11 22 21 11 12 if detM22 ̸= 0 we consider alternatively the matrix
=11 1112; −M M−1 −M +M M−1M
M −M M−1M T˜= 11 12 22 21
−M−1M 22 21
−M M−1 12 22 .
−M−1 22
If both detM11 and detM22 are non-zero, then we have T˜ = T−1. Since the cases detM11 ̸= 0 and detM22 ̸= 0 can be treated similarly, we assume in the sequel that detM11 ̸= 0.
Further we associate with T and T˜ the (discrete-time) algebraic Riccati equations
W = −M21 − M22W (I − M12W )−1M11 (DARE)
(HRDE)
(1.14)