THE LINEAR QUADRATIC REGULATOR PROBLEM FOR A CLASSOF CONTROLLED SYSTEMS MODELED BY SINGULARLY PERTURBED ITO DIFFERENTIAL EQUATIONS

摘要

This paper discusses an infinite-horizon linear quadratic (LQ) optimal control probleminvolving state- and control-dependent noise in singularly perturbed stochastic systems. First,an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation(ARE) are newly established. It is shown that the dominant part of this solution can be obtainedby solving a parameter-independent system of coupled Riccati-type equations. Moreover, ufficientconditions for the existence of the stabilizing solution to the problem are given. A new sequentialnumerical algorithm for solving the reduced-order AREs is also described. Based on the asymptoticbehavior of the ARE, a class of O(√ε) approximate controller that stabilizes the system isobtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthythat the resulting controller achieves an O(ε) approximation to the optimal cost of the original LQoptimal control problem. As a result, the proposed control methodology can be applied to practicalapplications even if the value of the small parameter ε is not precisely known.