Nonlinear n-th Cost Cumulant Control and Hamilton-Jacobi-Bellman Equations for Markov Diffusion Process

A general nonlinear stochastic system with non-quadratic cost function is considered for cost cumulant control of a Markov diffusion problem. The Hamilton-Jacobi-Bellman equation for the n-th cost moment case is derived as a necessary condition for optimality. The n-th cost cumulant Hamilton-Jacobi-Bellman equation derivation procedure is given. Second, third, and fourth cost cumulant Hamilton-Jacobi-Bellman equations are derived using the proposed procedure. The solutions of the nonlinear cost cumulant control problem is discussed using the state dependent Riccati equation method.