The accessory minimum problem and its importance for the numerical computation of closed-loop controls

Perturbations in constrained optimal control problems are discussed. Such problems as those with state and/or control constraints and discontinuities are considered. It is shown that an optimal solution of the disturbed control problem depending continuously differentially upon the perturbations exist in a neighborhood of an optimal solution of the undisturbed problem under an assumption which has to be intrinsically fulfilled for virtually all the important numerical methods for the computation of boundary value problems. The proof needs the existence of an optimal solution of the accessory minimum problem, and, moreover, the necessary conditions of this auxiliary control problem provide a linear multipoint boundary value problem accompanied by a system of linear equations and linear jump conditions which plays an important role for the real-time approximation of closed-loop controls