The time-optimal transfer of a perturbed dynamical object to a given position

Abstract

The problem of transferring a dynamical object (a point mass) of arbitrary dimensional to a required position in a time-optimal manner by means of a force of limited modulus is solved. The velocity of the object at the final instant is not specified. It is assumed that an arbitrary known perturbation, with a magnitude strictly less than that of the control, acts on the controlled system. For clarity when analysing the optimal controlled motion, considerable attention is paid to the case of a steady perturbation. A constructive procedure for finding the optimal response time and the control is developed for arbitrary permissible values of the governing parameters. The Bellman function and the feedback control are constructed over the whole of the phase space. The structural properties of the solution are established and an asymptotic analysis is carried out by small-parameter methods. The extremal directions of the perturbation vector and the corresponding response time and optimal control are found. A modification of the time-optimal problem to the case of a non-stationary perturbation is presented and the basic properties of the optimal solution are investigated.