The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping

In this paper, the linear quadratic optimization problem for a class of linear stochastic systems subject both to multiplicative white noise and Markovian jumping is investigated. Two classes of admissible controls are considered. One of these classes contains controls with additional property that corresponding trajectories tend to zero (in mean square) when tends to (infinity), while concerning the controls contained in the second class of admissible controls there is not any stability assumption. In the optimization problem over the first class of admissible controls, the cost functional could have indefinite sign of weights matrices. An iterative procedure to compute the maximal solution of the systems of generalized Riccati equations is provided. A numerical example to illustrate the applicability of the iterative procedure is given.