A new method of solving the optimal control problem for a partially observable stochastic Volterra process

Abstract

A new method is proposed for solving the linear-quadratic problem of optimal control for a partially observable stochastic Volterra process. The method relies on the representation and optimal estimation of optimal control in the form of integrals over the observable process. The integrands are non-stochastic and are defined by some system of integral equations, which may be solved numerically in advance. The optimal control is constructed directly from observations. An example demonstrating the implementation of the method by computer is given. Integral Volterra equations first arose in creep theory and they are the foundation of this theory /1, 2/. They include a fairly large class of equations with a memory /3–5/, which play a central role in control theory and in various applications. The theory of optimal control of Volterra equations is a natural outgrowth of the theory of controllable differentiable equations. Filtering and optimal control theory for stochastic integral equations is rapidly developing. The classical solution of the problem by the “separation principle” /6, 7/ reduces to solving the optimal filtering problem and the optimal control problem under complete information for some subsidiary controllable system. The optimal control of the original problem is obtained /8/ as a linear functional of the (mean-square) optimal estimate of the optimal trajectory of motion, which in its turn is the solution of a system of stochastic integral equations. Thus, in order to construct the optimal control at each instant of time t we need to solve a system of stochastic integral equations in the interval [0, t].