Fast Iterative Algorithms For Solving Lyapunov Equations In It? Stochastic Systems With Markovian Jumps

Markovian jump systems consist of a series of subsystems which jump according to Markovian distribution. Such kind of systems have found wide practical applications. For example, they can be used to describe some dynamic behavior with sudden change which is influenced by some random factors such as external environmental changes and internal emergencies. Recently, a kind of more complicated Markovian jump systems, which include It? stochastic process, has been extensively investigated. In the analysis of this kind of systems, the coupled Lyapunov matrix equations play a vital role. In this dissertation, we will focus on some iterative algorithms to solve coupled Lyapunov matrix equations.For Lyapunov matrix equations of discrete-time It? stochastic systems with Markovian jumping parameters, we prove the monotonicity and boundedness of the explicit parallel iterative algorithm under zero initial conditons. Furthermore, the convergence of the algorithm is proved.According to the special structure of coupled Lyapunov matrix equations of discrete-time It? stochastic systems, a series of implicit fast parallel iterative algorithms with weighting factors are proposed in this dissertation. The convergence of these algorithms is proved and the influences of different weighing factors on convergence performance are investigated.For Lyapunov matrix equations of discrete-time It? stochastic systems, by using the method of latest estimate information, one explicit fast iterative algorithm and one implicit fast iterative algorithm with weighting factors are proposed and the convergences of two algorithms are proved respectively. Numerical experiments show that fast iterative algorithms converge faster than parallel iterative algorithms.Similarly, by using the method of latest estimate information, one implicit fast iterative algorithm is proposed for continuous-time It? stochastic systems. Numerical experiments imply that implicit fast iterative algorithm is better than implicit parallel iterative algorithm. In addition, the different constant factors are introduced in this algorithm. Numerical experiments show that the smaller the constant factor is, the faster the iterative sequences converge.