Feedback minimum principle for optimal control problems with terminal conditions and its extensions

The nonlocal necessary optimality condition, the so-called feedback minimum principle (F-PM) obtained in previous publications of the author for free endpoint problems, is generalized for problems with terminal constraints. The proof of the new necessary condition is based on abstract methods of support majorants and modified Lagrange functions (MLF) with a quadratic penalty. But the corresponding unconstrained problem does not necessarily have to be solved. If the reference process is optimal, then there is no descent for the MLF from it using F-PM. If this necessary optimality is violated, then we obtain an improved admissible process. The constructive basis of the feedback minimum principle is the descent method with feedback strategies. However, it is natural to use this descent method for minimizing the modified Lagrangian in the well-known Krotov and Pontryagin optimality conditions. As a result of such an extension of the F-PM descent method, we obtain feedback versions of the Krotov and Pontryagin methods, which are significantly more efficient than the traditional methods.