Optimal containment control for a class of stochastic systems perturbed by poisson and wiener processes

In this note, we consider a class of nonlinear stochastic systems driven by Wiener and Poisson processes. The Wiener process input enters into the equations additively to the dynamics while the Poisson process input enters into the equations multiplicatively to the control input. Examples of applied problems that may lead to system models of this kind are discussed in the note. The optimal containment control problem is then formulated for these systems. It involves either maximizing the time of stay within an admissible set or a closely related performance measure. The optimal control and the optimal value function are characterized on the basis of Bellman's dynamic programming principle in the general case so that the optimal value function is a solution of a boundary value problem for a partial differential equation (PDE). For a special case defined by more restrictive assumptions the method of successive approximations is used to show the existence of solution to this boundary value problem and to set up an iterative solution procedure. An example is reported that illustrates the results.