On Successive Approximation of Optimal Control of Stochastic Dynamic Systems
An approximation theory of optimal control for nonlinear stochastic dynamic systems has been established. Based on the generalized Hamilton-Jacobi-Bellman equation of the cost function of nonlinear stochastic systems, general iterative procedures for approximating the optimal control are developed by successively improving the performance of a feedback control law until a satisfactory suboptimal solution is achieved. A successive design scheme using upper and lower bounds of the exact cost function has been developed for the infinite-time stochastic regulator problem. The determination of the upper and lower bounds requires the solution of a partial differential inequality instead of equality. Therefore it provides a degree of flexibility in the design method over the exact design method. Stability of the infinite-time sub-optimal control problem was established under not very restrictive conditions, and stable sequences of controllers can be generated. Several examples are used to illustrate the applicati on of the proposed approximation theory to stochastic control. It has been shown that in the case of linear quadratic Gaussian problems, the approximation theory leads to the exact solution of optimal control.
KeywordsHamilton-Jacobi-Bellman equation optimal control nonlinear stochastic systems
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