Sequential conjugate-gradient-restoration algorithm for optimal control problems. Part 1. Theory

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 state and the parameter 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. First, the case of a quadratic functional subject to linear constraints is considered, and a conjugate-gradient algorithm is derived. Nominal functionsx(t),u(t), π satisfying all the differential equations and boundary conditions are assumed. Variations Δx(t), δu(t), Δπ 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 differential equations, the boundary conditions, and a quadratic constraint on the variations of the control and the parameter. Next, the more general case of a nonquadratic functional subject to nonlinear constraints is considered. The algorithm derived for the linear-quadratic case is employed with one modification: a restoration phase is inserted between any two successive conjugate-gradient phases. In the restoration phase, variations Δx(t), Δu(t), Δπ are determined by requiring the least-square change of the control and the parameter subject to the linearized differential equations and the linearized boundary conditions. Thus, a sequential conjugate-gradient-restoration algorithm is constructed in such a way that the differential equations and the boundary conditions are satisfied at the end of each complete conjugate-gradient-restoration cycle. Several numerical examples illustrating the theory of this paper are given in Part 2 (see Ref. 1). These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper. This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Professor A. Miele for stimulating discussions. Formerly, Graduate Studient in Aero-Astronautics, Department of Mechanical and Aerospace Engineering and Materials Science, Rice University, Houston, Texas.