Sequential gradient-restoration algorithm for optimal control problems

Abstract

This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. Asequential algorithm composed of the alternate succession of gradient phases and restoration phases is presented. This sequential algorithm is contructed in such a way that the differential equations and boundary conditions are satisfied at the end of each iteration, that is, at the end of a complete gradient-restoration phase; hence, the value of the functional at the end of one iteration is comparable with the value of the functional at the end of any other iteration.

In thegradient phase, nominal functionsx(t),u(t), π satisfying all the differential equations and boundary conditions are assumed. Variations Δx(t), Δu(t), Δπ leading to varied functions$\tilde x$(t),ũ(t),$\tilde \pi$ are determined so that the value of the functional is decreased. These variations are obtained by minimizing the first-order change of the functional subject to the linearized differential equations, the linearized boundary conditions, and a quadratic constraint on the variations of the control and the parameter.

Since the constraints are satisfied only to first order during the gradient phase, the functions$\tilde x$(t),ũ(t),$\tilde \pi$ may violate the differential equations and/or the boundary conditions. This being the case, a restoration phase is needed prior to starting the next gradient phase. In thisrestoration phase, the functions$\tilde x$(t),ũ(t),$\tilde \pi$ are assumed to be the nominal functions. Variations Δ$\tilde x$(t), Δũ(t), Δ$\tilde \pi$ leading to varied functions$\hat x$(t),û(t),$\hat \pi$ consistent with all the differential equations and boundary conditions are determined. These variations are obtained by requiring the least-square change of the control and the parameter subject to the linearized differential equations and the linearized boundary conditions. Of course, the restoration phase must be performed iteratively until the cumulative error in the differential equations and boundary conditions becomes smaller than some preselected value.

If the gradient stepsize is α, an order-of-magnitude analysis shows that the gradient corrections are Δx=O(α), Δu=O(α), Δπ=O(α), while the restoration corrections are$\Delta \tilde x = O(\alpha ^2 ), \Delta \tilde u = O(\alpha ^2 ), \Delta \hat \pi = O(\alpha ^2 )$. Hence, for α sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalI decreases between any two successive iterations.

Methods to determine the gradient stepsize in an optimal fashion are discussed. Examples are presented for both the fixed-final-time case and the free-final-time case. The numerical results show the rapid convergence characteristics of the sequential gradient-restoration algorithm.