MODIFIED QUASILINEARIZATION ALGORITHM FOR OPTIMAL CONTROL PROBLEMS WITH NONDIFFERENTIAL CONSTRAINTS AND GENERAL BOUNDARY CONDITIONS
Doctor of Philosophy
In this thesis, we consider two classes of optimal control problems. Problem (P1) involves 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 nondifferential constraints to be satisfied everywhere along the interval of integration. A modified quasilinearization algorithm (MQA) is developed. The main property of the algorithm is the descent property in the performance index R. R denotes the cumulative error in the constraints and the optimality conditions. Modified quasilinearization differs from ordinary quasilinearization because of the scaling factor (or stepsize) (alpha) present in the system of variations. The stepsize (alpha) is determined by a one-dimensional search on the performance index R. Since the first variation (delta)R is negative, the decrease in R is guaranteed if (alpha) is sufficiently small. Convergence to the solution is achieved when R becomes smaller than some predetermined value. To start the algorithm, we have to choose some nominal functions x(t), u(t), (pi) and some nominal multipliers (lamda)(t), (rho)(t), (sigma), (mu). In a real problem, we choose the nominal functions based on physical considerations. For the nominal multipliers, no useful guidelines have been available thus far. In this thesis, an auxiliary minimization algorithm (AMA) for the selection of the optimal multipliers is presented. In this auxiliary minimization algorithm, the performance index R is minimized with respect to (lamda)(t), (rho)(t), (sigma), (mu). Since R is quadratically dependent on the multipliers, the resulting variational problem is governed by optimality conditions which are linear; therefore, it can be solved without difficulty. To facilitate the numerical implementation, the interval of integration is normalized to the unit length. Several numerical examples are presented. The numerical results show the feasibility as well as the convergence characteristics of the present algorithms.