Restoration algorithm for solving optimal control problems with nondifferential constraints and general boundary conditions
Huang, G. T. C.
Master of Arts
This thesis considers the numerical solution of the problem of minimizing a functional I, subject to differential constraints, non-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 is minimized, while the constraints are satisfied to a predetermined accuracy. First, a new version of the restoration algorithm is developed, in order to solve the following sub-problem: find a feasible solution, starting from a non-feasible solution. This task is accomplished in a cycle, composed of several restorative iterations. In each restorative iteration, variations of the state, the control, and the parameter are produced so as to achieve first-order constraint satisfaction, while minimizing the norm squared of the variations of the control and the parameter. Next, a transformation technique is employed. By proper augmentation of the state vector and the parameter vector, and by proper redefinition of the constraining relations, a transformed system is obtained. In this transformed system, the value of the functional I becomes a component of the augmented parameter. Then, the original minimization problem is replaced by the problem of finding the smallest value of the parameter I for which the transformed system admits a feasible solution. In this connection, ways and means are explored for approaching the minimum of the parameter I by cyclical application of the restoration algorithm. As a whole, the minimization algorithm is composed of a sequence of restorative cycles. Two consecutive elements of the sequence are such that the value of the parameter I at the end of any cycle is smaller than the value of the parameter I at the end of the previous cycle. The driving force which enables the restoration algorithm to continue is the lowering of the value of parameter I after a feasible solution has been obtained. This supplies the disturbance necessary for the restoration algorithm to continue. Depending on the strategy employed for the driving parameter and the strategy employed for the error in the feasibility equations, different versions of the minimization algorithm are developed: Algorithms A1, A2, A3 and Algorithms B1, B2, B3. These versions are tested through four numerical examples, and it is found that they perform in a satisfactory way. Thus, the numerical results show the feasibility as well as the convergence characteristics of the present algorithm.