On the Asymptotic Structure of the Stabilizing Solution of the Riccati Equation Arising in Connection with the Linear Quadratic Regulator Problem for a System Described by Itȏ Differential Equations with Two Fast Time Scales

Abstract

We consider a stochastic optimal control problem described by a quadratic performance criterion and a linear controlled system modeled by a system of singularly perturbed Itȏ differential equations with two fast time scales. Our goal is to analyse the asymptotic structure with respect to the small parameters εj > 0, j = 1, 2 associated to the two fast time scales of the stabilizing solution of the matrix Riccati equation associated to the optimal control problem under consideration. The results derived in this stochastic framework cannot be obtained mutatis-mutandis from the already existing ones in the deterministic case, as those from.