Nonlinear second cost cumulant control using Hamilton-Jacobi-Bellman equation and neural network approximation

Cost cumulant control is an optimal control method applied to the stochastic systems. Unlike the deterministic systems, the cost function in a stochastic system is a random variable. We use statistical control to optimize the distribution of the cost function via cost cumulants. We analyze the first and second cost cumulant optimization problems for nonlinear stochastic systems. The Hamilton-Jacobi-Bellman (HJB) equations are derived with respect to each cumulant minimization case. Then a neural network approximation method is used to numerically solve the HJB equations, which in turn is used to find the optimal controller. Two examples including a nonlinear system and a linearized satellite attitude control system are presented. The first and second cumulant optimal controllers are found. Then the performance of the optimal controllers for the first and the second cumulants are compared through the examples.