Existence and comparison theorems for algebraic Riccati equations and Riccati differential and difference equations (Q1972729)

The generalized Riccati differential equations \[ \dot W= -A^*W- WA- Q+ WSW- \Pi(W) \] and the corresponding generalized algebraic Riccati equations \[ -A^* W- WA- Q+ WSW= \Pi(W) \] are studied. Here \(A,Q= Q^*\), \(S= S^*\) are \(n\times n\) complex matrices, and \(\Pi(W)\) in a monotone linear function of the variable Hermitian matrix \(W\). Generalized Riccati equations of this form appear for example in optimal control problems of linear systems with Markovian jumps. The authors prove a comparison theorem for the algebraic equation, under the additional hypotheses that \(S\) is positive semidefinite, the pair \((A,S)\) is stabilizable, and a scaling condition on \(\Pi(W)\). The theorem extends a well-known comparison theorem for algebraic Riccati equations. The result is used to establish intervals of existence of solutions to the differential equation. Comparison theorems are also obtained for the generalized Riccati difference equations \[ K(m+ 1)= A^*K(m) A- A^*K(m) B(I+ B^* K(m) B)^{-1} B^* K(m) A+ Q+ \Pi(K(m)), \] and for the corresponding generalized discrete algebraic Riccati equations.