A Minimax Stochastic Optimal Control for Bounded-uncertain Systems

Abstract

A minimax stochastic optimal control strategy for bounded-uncertain stochastic systems is proposed. The minimax dynamical programming equation for an uncertain stochastic control system is firstly derived based on the optimality principle and Itô differential rule. A new type of bang-bang constraint on the bounded uncertain disturbance is proposed to form a class of minimax stochastic optimal control problems. Then the worst disturbance and minimax optimal control are obtained for the bang-bang-type uncertain system under stochastic excitations. According to this method, the quasi linear control law is obtained for linear stochastic systems with bounded uncertainty and the state-dependent quasi Riccati equation is derived from the minimax dynamical programming equation. Furthermore, a minimax stochastic optimal control strategy for uncertain stochastic quasi Hamiltonian systems is developed based on the stochastic averaging method and minimax dynamical programming equation. The worst disturbance and minimax optimal control for the stochastically averaged system are obtained by the similar procedure. The proposed and developed minimax stochastic optimal control strategies are illustrated with an example of a single-degree-of-freedom uncertain stochastic control system.