Dual properties of sequential 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 in connection with sequential gradient-restoration algorithms (SGRA) for optimal control problems. The system of Lagrange multipliers associated with (i) the gradient phase of SGRA and (ii) the restoration phase of SGRA is examined. For each phase, it is shown that the Lagrange multipliers are endowed with a duality 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 duality properties have considerable computational implications; they allow one to reduce the auxiliary optimal control problems associated with (i) and (ii) to mathematical programming problems involving a finite number of parameters as unknows.