Gradient method in function space for solving a minimum path problem

An algorithm of searching for the optimal trajectory with the minimal cost W(x) of reaching the final state x_fin from the initial state x is presented. A system of ordinary differential equations is suggested to determine an optimal trajectory x(t) and an optimal control u(t). The algorithm represents the gradient method in function space. This work is a continuation of the author´s previous work (1998). The close-to-optimal trajectory x(t) and the control u(t) are determined by successive approximations. Learning consists in estimation of the unknown a priori minimal cost W(x) and ∂W(x)/∂x on the basis of analysis of the trial trajectories x(f) obtained earlier