Linear quadratic optimal regulation for multiplicative noise systems with special terminal penalty

Abstract

This paper focuses on the linear quadratic optimal control problem for stochastic systems with a special terminal penalty. The specialty lies in that the terminal penalty is a quadratic term over the conditional expectations of the terminal state. The essential difficulty lies in an additional measurability constraint on the input induced by the conditional expectation term, which is shared by the control problems for the stochastic system with delays, rational expectations problems, control problems involving conditional expectation terms, asymmetric information control and so on. A new idea, dynamic programming combined with orthogonal decomposition, is proposed to deal with this problem. The orthogonal decomposition eliminates the additional measurability constraint on the input. The solvability condition and the optimal controller are obtained by developing generalised Riccati equations. Numerical experiment is provided to support the achieved results.