Maximum principle for optimal control problems of extended mean-field forward–backward regime-switching systems with general singular controls

We study the optimal control problems for a class of extended mean-field forward–backward regime-switching systems with general singular controls, where the coefficients depend, nonlinearly, on the state and the control process as well as on their law. In particular, we assume that the singular control is a general finite variation process that is not necessarily non-negative non-decreasing. By virtue of separation and variational methods, we establish the related stochastic maximum principle, which consists of two components: the absolutely continuous part and the singular part. It is essentially different from that in the classical situation. Additionally, we obtain the verification theorem for optimal control problems under certain convexity assumptions. Since this paper extends singular controls to general finite variation processes, our conclusions can be applied to optimal impulse control problems. Finally, we also provide the feedback representation for the optimal singular control of a class of linear quadratic problems.