Turnpike motions of control systems (I)

This paper is intended primarily for graduate students specializing in differential equations. It covers the applications to control systems of the well-developed theory of classical dynamic systems, methods of differential geometry and the theory of differential inclusions, mainly developed by A. F Filippov. The main content of the paper is the study of the so-called standard control system. The phase space of such a system is a finite-dimensional smooth manifold. This assumption is very important from the point of view of applications. In addition, it is assumed that the vector field of the system is locally Lipschitz, and the geometric constraints on the controlled parameters are compact. Admissible control functions can be program and / or positional. In the first case, we arrive at the so-called systems of Caratheodory equations. In the second case, if the vector field is discontinuous with respect to phase variables, we arrive at Filippov's differential inclusions. Serious attention is given to the study of the conditions under which specified properties of the control system continue to hold after closing the set of shifts (in the topology of uniform convergence on compact sets) of the initial standard control system.