Abstract
This paper deals with the optimal control problem (regulator problem) of stochastic systems described by stochastic functional differential equations.
This problem has not been solved completely even in the deterministic case if the system is linear but includes delay terms. A stochastic version of this problem is very complicated because the solution of stochastic functional differential equations is not a Markov process with values in Euclidean space. This paper consists of the following parts, namely, the formulation of the problem, the functional equation to be satisfied by optimal control, the optimal control problem of linear systems with delay, and the construction of an approximately optimal control for the linear systems with a small delay term.
When the system is linear, a sufficient condition for the existence of an admissible control is also given. Specially, for the optimal control problem of the systems described by linear stochastic delay-differential equations with the quadratic performance criterion, we get a set of differential equations which the gains of the optimal control have to satisfy. We show that these equations can be reduced to a set of a nonlinear integro-differential equation of Volterra type and an algebraic matrix equation.
Moreover, it is shown that if the delay term of the systems is sufficiently small, we can successively integrate the set of differential equations.
