Optimal Parto control of stochastic system with multiplicative noises and Markovian jumping

In this paper, a Parto optimization problem of N players in stochastic system is copied with. An optimal Parto solution of the stochastic system with multiplicative white noises and Markovian jumping is presented in infinite time horizon. A complete analytical optimal controller is obtained by using the generalized Lyapunov equation approach and solving a generalized algebraic Riccati equation. It is proved that the controller is a stabilizable feedback control and the solution of the generalized algebraic Riccati equation corresponding to the feedback gain of the optimal control is minimal.