Research Article

Separation principle in the fractional Gaussian linear-quadratic regulator problem with partial observation

Kleptsyna, Marina L.^a1, Le Breton, Alain^a2 and Viot, Michel^a2

^a1 Laboratoire de Statistique et Processus, Université du Maine, av. Olivier Messiaen, 72085 Le Mans cedex 9, France; Marina.Kleptsyna@univ-lemans.fr

^a2 Laboratoire de Modélisation et Calcul, Université J. Fourier, BP 53, 38041 Grenoble cedex 9, France; Alain.Le-Breton@imag.fr

Abstract

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time with partial observation. For a controlled linear system where both the state and observation processes are driven by fractional Brownian motions, we describe explicitly the optimal control policy which minimizes a quadratic performance criterion. Actually, we show that a separation principle holds, i.e., the optimal control separates into two stages based on optimal filtering of the unobservable state and optimal control of the filtered state. Both finite and infinite time horizon problems are investigated.

(Received July 21 2006)

(Revised January 8 2007)

(Online publication January 23 2008)

Key Words:

• Fractional Brownian motion;
• linear system;
• optimal control;
• optimal filtering;
• quadratic payoff;
• separation principle

Mathematics Subject Classification:

• 93E11;
• 93E20. 60G15;
• 60G44