Searching for optimal trajectory with learning

An algorithm of searching for the optimal trajectory with the minimal cost W(x) of reaching the final state x_{ft n} from the initial state x is presented. A system of ODEs is suggested to determine an optimal trajectory x*(t) and an optimal control u*(t). The trajectory x(t) and the control u(t) close to optimal ones are determined by successive approximations. The algorithm represents a development of a gradient method in the function space. Learning consists in estimation of an unknown a priori minimal cost W(x) and ∂W(x)/∂x on the basis of analysis of the trial trajectories x(t) obtained earlier