General LQ Problem for Infinite Jump Linear Systems and the Minimal Solution of Algebraic Riccati Equations

The paper addresses the LQ control problem for systems with countable Markov jump parameters, and the associated coupled algebraic Riccati equations.The problem is considered in a general optimization setting in which the solution is not required to be stabilizing in any sense. We show that a necessary and sufficient condition for a solution tothe control problem to exist is that the Riccati equations have a nonempty set of solutions, which generalizes previous known results requiring stabilizability as a sufficient condition.We clarify the connection between the minimal solution of the Riccati equation and the control problem, showing that the minimal solution provides the synthesis of the optimal control. The derived results strengthen the relations of the theory of Markov jump systems with the one of linear deterministic systems. An illustrative example is included.