Combined gradient-restoration algorithm for optimal control problems
Basapur, Venkatesh K.
Master of Science
The problem of minimizing a functional I subject to differential constraints, nondifferential constraints, and general boundary conditions is considered in this thesis. It consists of finding the state x(t), the control u(t), and the parameter-rr so that the functional I is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. A combined gradient-restoration algorithm is developed. This is an iterative algorithm characterized by variations Ax(t), Au(t), Air leading toward satisfaction of the optimality conditions, while simultaneously leading toward constraint satisfaction. The variations Ax(t), Au(t), Air are generated by requiring the first variations of the augmented functional J and the constraint error P to be negative. The procedure leads to a linear, two-point boundary-value problem, which is solved via the method of particular solutions. The descent properties of the algorithm are studied, and schemes to determine the optimum stepsize are discussed. In order to improve the convergence characteristics, the inclusion of a restoration phase is studied. In this connection, three versions of the algorithm are studied: the combined gradient-restoration algorithm (CGRA); the combined gradient-restoration algorithm with alternate restoration (CGRA-AR); and the combined gradient-restoration algorithm with complete restoration (CGRA-CR). A comparison of these versions with the sequential gradient-restoration algorithm (SGRA) is also made. Three numerical examples are presented to illustrate the different approaches. Key Words. Numerical methods, optimal control, gradient methods, combined gradient-restoration algorithm, differential constraints, nondifferential constraints, general boundary conditions.