Sufficient conditions for Nash equilibrium point in the linear quadratic game for Markov jump positive systems

The infinite horizon linear quadratic differential games for positive linear systems with Markovian jumping is considered. The authors' goal is to propose a set of sufficient conditions that guarantee the existence of the stabilising solution of a system of game theoretic algebraic Riccati type equations associated to the considered differential game. To this end the authors' introduce a new type of solution of the Riccati equation namely the strong stabilising solution and they prove that this strong stabilising solution is just a stabilising solution of this kind of Riccati equation. The main contribution is the formulation of a set of sufficient conditions which guarantee the convergence of the proposed iterative procedure to the stabilising solution of game theoretic algebraic Riccati type equation. Finally, a numerical example shows the performance of the proposed algorithm.