Study On Several Problems Of Indefinite Stochastic Linear Quadratic Optimal Control

The mathematical model of the dynamics is given by linear equation, and the performanceindex is taken as quadratic functional of the state and the control. This kind of optimal controlproblem is the so-called linear-quadratic (LQ) regulator problem. The optimal solution of theLQ problem has a unified analytic expression, by which the optimal linear state feedbackcontrol can be represented easily. Due to the linear state feedback control is convenient torealize, the LQ theory has been one of the most perfect parts in the optimal control theory andapplication.This paper deals with three problems:1,Multi-objective indefinite stochastic LQ problem: in many practical applications, we usuallywant to seek a control that can achieve many objectives simultaneously. The main objective isselected as the cost function, while the less important objectives are treated as constraints.Moreover, the cost and constraints functions are all represented by quadratic integration. Thesufficient condition for the existence and the uniqueness of the optimal control is provided viaa group of generalized differential Riccati equations.2,Indefinite stochastic optimal modified regulator: in traditional theory of indefinite stochasticLQ problem for infinite time horizon, the objective is only to find an optimal control thatminimize the expression index under the assumption that the state is mean-variance stable. Theconvergence rate of the state corresponding to the optimal control is not in mind. However, inpractical engineering, the convergence of the state is of enough importance to consider. Thispaper proposed a necessary and sufficient condition for the well-posedness and attainability ofthe modified regulator problem. Furthermore, the state corresponding the optimal control has amean-square convergence rate faster than the given rate.3,Indefinite stochastic LQ problem with partial information: in engineering problems, the state of thestochastic system often can not be completely observed. Under the condition that we can only get partialinformation of the state, how to obtain the optimal control is of great importance for the practicalapplications of the indefinite stochastic LQ theory. This paper gives the Kalman filter of the state by means of the Girsanov transformation, upon which the suboptimal linear output feedback control of the indifmiteLQ problem is determined.