A new algorithm for solving cross-coupled algebraic Riccati equations of singularly perturbed Nash games

In this paper, we study the linear quadratic Nash games for infinite horizon singularly perturbed systems. In order to solve the problem, we must solve a pair of cross-coupled algebraic Riccati equations with a small positive parameter ε. As a matter of fact, we propose a new algorithm, which combines Lyapunov iterations and the generalized Lyapunov equation direct method, to solve the cross-coupled algebraic Riccati equations. The new algorithm ensures that the solution of the cross-coupled algebraic Riccati equations converges to a positive semidefinite stabilizing solution. Furthermore, in order to solve the cross-coupled algebraic Riccati equations, we propose a new Riccati iterations method different from existing method. As another important feature of this paper, our method is applicable to both standard and nonstandard singularly perturbed systems