Iterative solutions of coupled discrete Markovian jump Lyapunov equations

In this paper, two iterative methods are given to solve coupled discrete Markovian jump Lyapunov equations arising from the stability analysis of discrete-time Markovian jump systems. When such equations have unique positive definite solutions, a sufficient condition for the convergence of iterates for one iterative method is derived. When such equations only have unique solutions, but not necessarily positive definite, the above iterative method may fail and an alterative iterative method is provided by using the solutions of discrete-time Lyapunov equations as its intermediate steps. A sufficient condition is also presented for the convergence of the iterates.