Pontryagin's Direct Method for Optimization Problems with Differential Inclusion

We develop Pontryagin's direct variational method, which allows us to obtain necessary conditions in the Mayer extremal problem on a fixed interval under constraints on the trajectories given by a differential inclusion with generally unbounded right-hand side. The established necessary optimality conditions contain the Euler–Lagrange differential inclusion. The results are proved under maximally weak conditions, and very strong statements compared with the known ones are obtained; moreover, admissible velocity sets may be unbounded and nonconvex under a general hypothesis that the right-hand side of the differential inclusion is pseudo-Lipschitz. In the statements, we refine conditions on the Euler–Lagrange differential inclusion, in which neither the Clarke normal cone nor the limiting normal cone is used, as is common in the works of other authors. We also give an example demonstrating the efficiency of the results obtained.