Cost Moment Control and Verification Theorem for Nonlinear Stochastic Systems

In this paper we consider a system which is nonlinear in the state and linear in control action. Then we optimize the distribution of the cost function using the statistical control method. In statistical control, the optimal controller is found by optimizing any cost moments or cumulants. The optimal controller is found via the Hamilton-Jacobi-Bellman equation. As a part of statistical control, we investigate n-th moment optimal control in this paper. The Hamilton-Jacobi-Bellman equation for the n-th cost moment case is presented as a necessary condition for optimality. Then the verification theorem for n-th moment control is proved. We solve the optimizing controller for the first cost moment utilizing the derived HJB equation and the verification theorem, we verify that the new theory reduces to classical LQG result for a linear system with quadratic cost. Furthermore, we solve time-invariant nonlinear system by transforming the HJB equation to a first order partial differential equation using the pseudo-inversion method. Even though that the necessary and sufficient conditions are more easily derived, we conclude that n-th moment control generates more complicated controller then n-th cost cumulant case