The Stochastic LQR Optimal Control with Fractional Brownian Motion

Abstract

We consider the stochastic linear quadratic optimal control problem where the state equation is given by a stochastic differential equation of the Itô–Skorokhod type with respect to fractional Brownian motion. The dynamics are driven by strongly continuous semigroups and the cost functional is quadratic. We use the fractional isometry mapping defined between the space of square integrable stochastic processes with respect to fractional Gaussian white noise measure and the space of integrable stochastic processes with respect to the classical Gaussian white noise measure. By this mapping we transform the fractional state equation to a state equation with Brownian motion. Applying the chaos expansion approach, we can solve the optimal control problem with respect to a state equation with the standard Brownian motion. We recover the solution of the original problem by the inverse of the fractional isometry mapping. Finally, we consider a general form of the state equation related to the Gaussian colored noise, we study the control problem, a system with an algebraic constraint and a particular example involving generalized operators from the Malliavin calculus.