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Maximal and stabilizing hermitian solutions for discrete-time coupled algebraic Riccati equations. (English) Zbl 0928.93047
Discrete-time coupled algebraic Riccati equations (CARE) that arise in quadratic optimal control and ${H}_{\infty }$-control of Markovian jump linear systems are considered in this paper. In Section 3, the CARE that arise from the quadratic optimal control problem are considered. The matrix cost is only assumed to be Hermitian. A sufficient condition for the existence of a maximal solution is presented (Theorem 1). Theorem 2 establishes a link between the LMI optimization problem and a maximal solution. Section 4 deals with necessary and sufficient conditions for the existence of a stabilizing solution (Theorem 3 and 4). Section 5 presents a recursive procedure for obtaining a stabilizing solution of the CARE that arise in the ${H}_{\infty }$-problem, whenever it exists (Theorem 6).
##### MSC:
 93D15 Stabilization of systems by feedback 93E15 Stochastic stability 15A24 Matrix equations and identities 93C55 Discrete-time control systems 93B36 ${H}^{\infty }$-control 49N10 Linear-quadratic optimal control problems