Sufficient conditions in the collision avoidance problem under geometrical and integral limitations on control

Abstract

The paper considers the collision avoidance problem in a system, described by a controlled equation in partial derivatives, containing the second derivative with respect to time and elliptic operator. New spaces, depending on nonnegative parameter, are formed with the help of generalized eigenvalues and eigenfunctions. It is proved here that in the whole scale of these spaces there is unique solution of this hyperbolic equation with the elliptic operator. At that, the solution and its derivative are continuous in time with respect to the related norm. Sufficient conditions for the collision avoidance in the problems obtained under geometrical and integral limitations on the control parameters were gained.