Feedback minimum principle for quasi-optimal processes of terminally-constrained control problems

In a series of previous works, we derived nonlocal necessary optimality conditions for free-endpoint problems. These conditions strengthen the Maximum Principle and are unified under the name “feedback minimum principle”. The present paper is aimed at extending these optimality conditions to terminally constrained problems. We propose a scheme of the proof of this generalization based on a “lift” of the constraints by means of a modified Lagrange function with a quadratic penalty. Implementation of this scheme employs necessary optimality conditions for quasi-optimal processes in approximating optimal control problems. In view of this, in the first part of the work, the feedback minimum principle is extended to quasi-optimal free-endpoint processes (i.e. strengthen the so-called $\varepsilon$-perturbed Maximum Principle). In the second part, this result is used to derive the approximate feedback minimum principle for a smooth terminally constrained problem. In an extended interpretation, the final assertion looks rather natural: If the constraints of the original problem are lifted by a sequence of relaxed approximating problems with the property of global convergence, then the global minimum at a feasible point of the original problem is admitted if and only if, for all $\varepsilon>0$, this point is $\varepsilon$-optimal, for all approximating problems of a sufficiently large index. In respect of optimal control with terminal constraints, the feedback $\varepsilon$-principle serves exactly for realization of the formulated assertion.