SEQUENTIAL GRADIENT-RESTORATION ALGORITHM FOR OPTIMAL CONTROL PROBLEMS WITH CONTROL INEQUALITY CONSTRAINTS AND GENERAL BOUNDARY CONDITIONS
COKER, ESTELLE MATHILDA
Doctor of Philosophy
The problem of minimizing a functional I subject to differential constraints, control inequality constraints, and terminal constraints is considered in this thesis. 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 are satisfied to a predetermined accuracy. A sequential gradient-restoration algorithm is developed. It involves a sequence of two-phase cycles, the gradient phase and the restoration phase. In the gradient phase, the value of the functional is decreased, while avoiding excessive constraint violation; in the restoration phase, the constraint error is decreased, while avoiding excessive change in the value of the functional. The variations (DELTA)x(t), (DELTA)u(t), (DELTA)(pi) are generated by requiring the first variation of the augmented functional J to be negative during the gradient phase; and by requiring the first variation of the constraint error P to be negative, while imposing a least-square criterion on the variations of the control, the parameter, and the initial state during the restoration phase. This leads to a linear, two-point boundary-value problem, which is solved via the method of particular solutions. Various transformation schemes are employed, so as to convert control inequality constraints into control equality constraints. Considerable simplifications are possible if the control inequality constraints have a special form. In this connection, the following cases are studied: (P1) lower bounds on u(t); (P2) upper bounds on u(t); and (P3) upper and lower bounds on u(t). The algorithmic work per iteration is reduced due to (i) the special structure of problems (P1) through (P3); (ii) the fact that the multiplier (rho) associated with the auxiliary nondifferential constraint can be computed explicitly, bypassing the need of matrix inversion; and (iii) the fact that the auxiliary nondifferential constraint involves only the augmented control and not the state and/or the parameter. Because of this special situation, the number of integrations required to solve the linear, two-point boundary-value problem at each iteration can be reduced from n + p + 1. Here, n, p and b are the dimensions of the state vector, the parameter vector, and the vector of final conditions respectively.