Supplementary Optimality Properties of the Family of Gradient-Restoration Algorithms for Optimal Control Problems

Abstract

The problem of minimizing a functional subject to differential constraints, nondifferential constraints, initial constraints, and final constraints is considered within the frame of the family of gradient-restoration algorithms for optimal control problems. This family includes sequential gradient-restoration algorithms (SGRA) and combined gradient-restoration algorithms (CGRA).The system of Lagrange multipliers associated with (i) the gradient phase of SGRA, (ii) the restoration phase of SGRA, and (iii) the combined gradient-restoration phase of CGRA is examined. It is shown that, in each case, the Lagrange multipliers are endowed with a suppiementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.These supplementary optimality properties have considerable computational implications: they allow one to reduce the study of an iteration of (i), (ii), (iii) to a mathematical programming problem involving a finite number of parameters as unknowns.