A Time-Decomposition Algorithm for the Solution of Multiple-Target Problems

Abstract

An algorithm for solving the nonlinear optimal control problem whose system equation contains discontinuities is proposed. The boundary conditions are specified at several corner times as well as at the initial and the final times. Assuming that the corner times are known, by using the variational principle, the problem is reduced to a nonlinear multipoint boundary-value problem (MPBVP), which is further reduced to a linear one by use of an interaction-coordination algorithm. The linear MPBVP is solved by a discontinuous version of a time-decomposition algorithm, which decomposes the problem into a number of subinterval TPBVP's. The exact boundary conditions of these TPBVP's are determined by an algebraic method which utilizes the solution obtained with arbitrarily chosen boundary conditions. After solving the nonlinear MPBVP, the assumed corner times are corrected by a gradient method. Correction is iterated until the optimum is attained. The solution in each iteration satisfies the boundary conditions exactly. In order to verify the effectiveness of the present algorithm, a linear and a non-linear problem are solved numerically and the solution to the linear problem is compared with the analytical one.