Methods for improving the efficiency of the positional minimum principle in optimal control problems

The positional minimum principle is a necessary condition of global optimality, which strengthen the Pontryagin maximum principle and various extremal conditions for smooth and nonsmooth problems. It is based on iterations of the positional descent over the functional related to extremal strategies with respect to a solution of the corresponding Hamilton–Jacobi inequality. We discuss the main methods that allow one to increase the efficiency of positional descent iterations for uncertain extreme strategies and «stuck» on clearly nonoptimal processes. The positional descent from the sliding mode was examined in detail, i.e., from an admissible process of the convex problem with generalized controls, which are regular probability measures. Based on these ideas, we obtain the positional minimum principle for sliding modes.