Optimal Nonlinear Dynamic Problem with Stochastic Impulse and Regular Control Laws

By combining a regular control with an impulse control into the dynamic process of a non-linear system, we formulate an optimal hybrid control problem. In order to obtain the maximal value of the objective function and associated optimal impulse and regular control laws, relying on both stochastic calculus and the classical hybrid control theory, we derive its quasi-variational inequality solution. Moreover, under some limitations, we also derive its closed forms of optimal hybrid control laws and objective function. Based on some explicit parameter values, we give some simulations and numerical examples.