Explicit Iterative Algorithms for Continuous Coupled Lyapunov Matrix Equations

In this article, two iterative algorithms are proposed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. First, an explicit iterative algorithm is presented to solve the considered matrix equations via introducing a tuning parameter. A necessary and sufficient condition is established for the proposed algorithm to be convergent. Furthermore, the optimal value of the tuning parameter achieving the fastest convergence rate is given explicitly. Second, a modified version of the previous explicit algorithm is established by using a new information update scheme. Similarly, a necessary and sufficient condition is provided for the modified explicit iterative algorithm. Based on this convergence condition, some easily verifiable convergence results are further derived, and the optimal tuning parameter is given explicitly. Finally, two examples are presented to validate the obtained theoretical results.