Modifications and extensions of the sequential gradient-restoration algorithm for optimal control theory

Abstract

In Ref. 1, the sequential gradient-restoration algorithm for minimizing a functional subject to certain differential equations and boundary conditions was developed. This algorithm, here called Algorithm (α), includes the alternate succession of gradient phases and restoration phases. In the gradient phase, the first-order change of the functional is minimized subject to the linearized differential equations, the linearized boundary conditions and a quadratic constraint on the variations of the control and the parameter. In the restoration phase, the differential equations and the boundary conditions are restored to a predetermined degree of accuracy, subject to the least-square change of the control and the parameter. In this paper, several modifications and extensions of Algorithm (α) are studied. In parameter is modified by the inclusion of two positive-definite weighting matrices W 1(t) and W 2. In Algorithm (γ), an intermediate phase is interposed between the gradient phase and the restoration phase of Algorithm (α); in this intermediate phase, the control is reset at that value which minimizes the Hamiltonian. In Algorithm (δ), an intermediate phase is interposed between the gradient phase and the restoration phase of Algorithm (β); in this intermediate phase, the control is reset at that value which minimizes the Hamiltonian. Several numerical examples are developed, and it is shown that, if the matrix W 1(t) is the matrix of the second derivatives of the Hamiltonian with respect to the control, Algorithm (β) exhibits better convergence characteristics than Algorithm (α). On the other hand, Algorithms (γ) and (δ) are erratic in their behaviour: their convergence characteristics are found to be better or worse than those of Algorithm (α), depending on the particular problem.