Sequential conjugate gradient-restoration algorithm for optimal control problems with non-differential constraints and general boundary conditions, part I

In this paper, a new member of the family of sequential gradient-restoration algorithms for the solution of optimal control problems is presented. This is an algorithm of the conjugate gradient type, which is designed to solve two classes of optimal control problems, called Problem P1 and Problem P2 for easy indentification. Problem P1 involves minimizing a functional I subject to differential constraints and general boundary conditions. It consists of finding the state x(t), the control u(t), and the parameter pi 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 non-differential constraints to be satisfied everywhere along the interval of integration. The approach taken is a sequence of two-phase cycles, composed of a conjugate gradient phase and a restoration phase. The conjugate 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; each restorative iteration is designed to force constraint satisfaction to first order, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized. The resulting algorithm has several properties: (i) it produces a sequence of feasible solutions; (ii) each feasible solution is characterized by a value of the functional I which is smaller than that associated with any previous feasible solution; and (iii) for the special case of a quadratic functional subject to linear constraints, the variations of the state, control, and parameter produced by the sequence of conjugate gradient phases satisfy various orthogonality and conjugacy conditions. The algorithm presented here differs from those of References 1-4, 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 References 1-4, the present algorithm is capable of handling general final conditions; therefore, it is suitable for the solution of optimal control problems with general boundary conditions. The importance of the present algorithm lies in that many optimal control problems either arise naturally in the present format or can be brought to such a format by means of suitable transformations.5 Therefore, a great variety of optimal control problems can be handled. This includes: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with state-derivative equality constraints, (iv) problems with-control inequality constraints, (v) problems with state inequality constraints, (vi) problems with state-derivative inequality constraints, and (vii) Chebyshev minimax problems. Several numerical examples are presented in Part 2 (Reference 6) in order to illustrate the performance of the algorithm associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of the present algorithm.