A Stochastic Linear-quadratic Optimal Control Problem With Lévy Process In Infinite Time Horizon

This paper is concerned with a stochastic linear-quadratic optimal control problem with Levy process in infinite time horizon as follows:minimizesubject toA generalized algebraic Riccati equation which involves a matrix pseudoinverseis introduced:whereUnder some stabilizing conditions, we establish a relation between the sovability of the generalized algebraic Riccati equation and the existence of the optimal controls of the LQ control problem. Based on this relation, the optimal controls can be constructed by the solution of the generalized algebraic Riccati equation if it exists. Finally, we studies the solvability of the general ized algebraic Riccati equation based on the semidefinite programming.This paper is organized as follows.In Section 1, we simply introduce the history of the study to stochastic linear-quadratic optimal control problems.In Section 2, we give some definitions and lemmas. Definition 2.1 introducesthe Lévy process. Definition 2.2 and Definition 2.3 introduce the mean-square stability of stochastic linear-quadratic control problems.Then we give a stochastic linear-quadratic optimal control problem with Levy process in an infinite time horizon (2.1)-(2.2) and the associated generalizedalgebraic Riccati equation(2.4). Lemma 2.1 gives the definition of Moore-Penrose pseudoinverse. In the following there are lemma 2.3.1 (Schur's Lemma) and lemma 2.3.2 (Extended Schur's Lemma).In order to study the sovability of the generalized algebraic Riccati equation,we give the primal and dual semidefinite programming and linear matrix inequalities. Finally, we give the existence and uniqueness of solutions for stochastic differential equation with Levy process.In Section 3, We get the necessary and sufficient conditions for the stabilityof the stochastic linear-quadratic optimal control problem (2.1)-(2.2). Then we give the relationship between the solvability of the linear quadratic optimalcontrol problems (2.1) - (2.2) and the corresponding generalized algebraic Riccati. equation (2.4).From Theorem 3.1. we know that if the generalized algebraic Riccati equation(2.4) exists a solution and the linear quadratic optimal control problems (2.1) - (2.2) satisfies certain conditions of stability, then the linear quadratic optimal control problem (2.1) - (2.2) exists optimal solution.In Section 4, we give the relationship between the sovability of the generalizedalgebraic Riccati equatio, semidefinite programming and the optimal stochastic linear-quadratic control problem. From theorem 4.1 we know that semidefinite programming exists a solutionwhich satisfies the generalized algebraic Riccati equation (2.4) under certain conditions.In Section 5, we give an example with an explicit solution.