Existence theorem for the difference Riccati equations (Q1109242)

The following matrix Riccati difference equation is considered: \[ P(k+1)=AP(k)B-AP(k)D[R+CP(k)D]^{-1}CP(k)B+Q, \] with initial conditions \(P(0)=P_ 0\), \(k\geq 0\). Here, P is a sequence of \(r\times n\) matrices, A, B, C, D, Q, R are constant matrices of corresponding dimensions. Investigating interconnections between the existence of solution of this initial value problem and the existence of equilibrium points of this equation, the author succeeds in generalizing known results to nonsymmetrical cases and in giving some new ideas in the proof of existence and uniqueness of solution of the initial value problem under investigation.