Effects of the control bounds in the stochastic optimal control problem

This paper proposes a method to solve the stochastic optimal control problem with fixed state terminal conditions and control bounds. Being applicable to strongly nonlinear stochastic systems under parametric and external excitations, the method makes use of Bellman´s principle of optimality and the short-time Gaussian approximation. The solution found is global in the sense that optimal controls for all the initial conditions in a bounded region of the state space are obtained. A Markov chain with a control dependent transition probability matrix is built using the generalized cell mapping method. This allow us to study the transient and stationary effects of the control on the system response in the probability sense. The method is applied to a few systems resulting in excellent control performances. The dependence of the response on the control bounds is studied for a 1D-state nonlinear system and the Van der Pol oscillator.