Sequential gradient-restoration algorithm for optimal control problems with general boundary conditions

Abstract

This thesis considers the numerical solution of two classes of optimal control problems, called Problem P1 and Problem P2 for easy identification. Problem P1 involves a functional I subject to differential constraints and general boundary conditions. It consists of finding the state, the control, and the parameter so that the functional I is minimized while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include nondifferential constraints to be satisfied along the interval of integration. Algorithms are developed for both Problem P1 and Problem P2. The approach taken is a sequence of two-phase cycles, composed of a gradient phase and a restoration phase. The gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations and is designed to force constraint satisfaction to a predetermined accuracy while the norm squared of the variations of the control and the parameter is minimized. The principal property of both algorithms is that they produce a sequence of feasible suboptimal solutions: the functions obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the values of the functional I corresponding to any two elements of the sequence are comparable. The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functional J, while the stepsize of the restoration phase is obtained by a one-dimensional search on the constraint error P. The gradient stepsize and the restoration stepsize are chosen so that the restoration phase preserves the descent property of the gradient phase. Therefore, the value of the functional I at the end of any complete gradient-restoration cycle is smaller than the value of the same functional at the beginning of that cycle. The algorithms presented in this thesis differ from those of Refs. 1 and 2, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with Refs. 1 and 2, the present algorithms are capable of handling general final conditions; therefore, they are suited for the solution of optimal control problems with general boundary conditions. Their importance lies in the fact that many optimal control problems involve initial conditions of the type considered here. Numerical examples are presented to illustrate the performance of the algorithms associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of these algorithms.