Study on Stochastic Linear Quadratic Optimal Control with Quadratic and Mixed Terminal State Constraints

Yang Hongli

doi:10.1155/2013/674327

Abstract

This paper studies the indefinite stochastic LQ control problem with quadratic and mixed terminal state equality constraints, which can be transformed into a mathematical programming problem. By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result given in this paper is the necessary condition for indefinite stochastic LQ control with quadratic and mixed terminal equality constraints. The result shows that the different terminal state constraints will cause the endpoint condition of the differential Riccati equation to be changed. It coincides with the indefinite stochastic LQ problem with linear terminal state constraint, so the result given in this paper can be viewed as the extension of the indefinite stochastic LQ problem with the linear terminal state equality constraint. In order to guarantee the existence and the uniqueness of the linear feedback control, a sufficient condition is also presented in the paper. A numerical example is presented at the end of the paper.

Highlights

Linear quadratic (LQ) control is an extremely important class of control problems in both theory and application
By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result in this paper is the necessary condition for indefinite stochastic LQ control with quadratic terminal constraints and mixed terminal constraints
This paper studied a class of indefinite stochastic LQ control problems with quadratic terminal state constraints and mixed terminal state constraints

Summary

Introduction

Linear quadratic (LQ) control is an extremely important class of control problems in both theory and application. This paper studied the indefinite stochastic LQ control problem with quadratic terminal equality constraints and mixed constraints, which can be viewed as the extension of [8]. By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result in this paper is the necessary condition for indefinite stochastic LQ control with quadratic terminal constraints and mixed terminal constraints. The result showed that the difference of the terminal state constraints will cause the endpoint condition to be changed in the differential equations we obtained for the linear constraint control problem, which coincides with the reality. H1: Ml, Mq (the coefficient matrix for terminal linear and quadratic constraints respectively in Problem 1 to Problem 21) are full row rank and the set defined by the terminal state constraints is not empty; H2: L∞(0, T; X) := {f(t) : f(t) is an Fl-adapted, X-valued measurable process on [0, T], and E ∫0T ‖f(t)‖2Xdt < +∞}.

Problem Statement and Preliminaries

Main Results

Necessary Condition

Numerical Examples

Conclusion