General Nonlinear Impulsive Control Problems

In this concluding chapter, an extension of the classical control problem is given in the most general nonlinear case. The essential matter is that now the control variable is not split into conventional and impulsive types, while the dependence on this unified control variable is not necessarily affine. By combining the two approaches, the one based on the Lebesgue discontinuous time variable change, and the other based on the convexification of the problem by virtue of the generalized controls proposed by Gamkrelidze, a fairly general extension of the optimal control problem is constructed founded on the concept of generalized impulsive control. A generalized Filippov-like existence theorem for a solution is proved. The Pontryagin maximum principle for the generalized impulsive control problem with state constraints is presented. Within the framework of the proposed approach, a number of classic examples of essentially nonlinear problems of calculus of variations which allow for discontinuous optimal arcs are also examined. The chapter ends with seven exercises.