DUALITY PROPERTIES AND SEQUENTIAL GRADIENT-RESTORATION ALGORITHMS FOR OPTIMAL CONTROL PROBLEMS (NUMERICAL METHOD)

This thesis considers duality properties and their application to the sequential gradient-restoration algorithms (SGRA) for optimal control problems. Two problems are studied: (P1) the basic problem and (P2) the general problem. In Problem (P1), the minimization of a functional is considered, subject to differential constraints and final constraints, the initial state being given; in Problem (P2), the minimization of a functional is considered, subject to differential constraints, nondifferential constraints, initial constraints, and final constraints. Depending on whether the primal formulation is used or the dual formulation is used, one obtains a primal sequential gradient-restoration algorithm (PSGRA) and a dual sequential gradient-restoration algorithm (DSGRA). With particular reference to Problem (P2), it is found convenient to split the control vector into an independent control vector and a dependent control vector, the latter having the same dimension as the nondifferential constraint vector. This modification enhances the computational efficiency of both the primal formulation and the dual formulation. The basic property of the dual formulation is that the Lagrange multipliers associated with the gradient phase and the restoration phase of SGRA minimize a special functional, quadratic in multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter. This duality property yields considerable computational benefits in that the auxiliary optimal control problems associated with the gradient phase and the restoration phase of SGRA can be reduced to mathematical programming problems involving a finite number of parameters as unknowns. Several numerical examples are solved using both the primal formulation and the dual formulation