The quickest transfer of a multidimensional dynamic object onto the surface of an ellipsoid

The time-optimal control of motion of a multidimensional dynamic object is investigated. Both open-loop and feedback controls are considered. It is assumed that the force vector is constrained by a non-degenerate ellipsoid. The terminal set is described by the surface of an ellipsoid that allows degeneracy, and the terminal velocity is not fixed. The initial position of the object can be either outside or inside the ellipsoid. Using the maximum principle, the necessary and sufficient conditions for the optimality of the control are established in the form of a system of polynomial equations, the orders of which depend on the dimension of the problem and the degree of degeneracy of the terminal ellipsoid. Iterative procedures and a technique for continuation with respect to the parameters of the approximating ellipsoid in the admissable region of change of the phase vector are proposed. Control problems for the limiting shapes of the terminal ellipsoid are investigated. The qualitative effect of the discontinuity of the functional and the control defined as functions of the phase vector is established.