Variational Optimality Conditions with Feedback Descent Controls that Strengthen the Maximum Principle

We derive nonlocal necessary optimality conditions that strengthen both classical and nonsmooth Maximum Principles for nonlinear optimal control problems with free right-hand end of trajectories. The strengthening is due to employment of feedback controls, which are assumed to ensure a descent of a value of the cost functional, and are extremal with respect to certain solutions of a Hamilton–Jacobi inequality for weakly monotone functions. The main results are Feedback Minimum Principles for smooth and nonsmooth problems, that are formulated through accessory dynamic optimization problems. Effectiveness of these necessary optimality conditions are illustrated by examples.