On Diferential Inclusions Arising from Some Discontinuous Systems

Abstract

The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding modes. If one uses one of the well-known definitions of the solution of discontinuous systems, then the motion of an object while being on some surface can be described in terms of differential inclusions. With the help of the previously developed apparatus for solving differential inclusions, a method is constructed for finding the trajectories of a system moving in a such a mode. Since some of frequently used discontinuous controls contain nonsmooth functions of phase variables, the paper pays special attention to study the differential properties of such systems. At the end of the paper controls of a slightly different, in contrast to the classical, type are considered which have useful differential properties, and a method is constructed for solving systems with such controls considered both before hitting the required surface and moving in its vicinity.

Keywords:

• Differential inclusion
• discontinuous system
• Gâteaux gradient
• sliding mode
• support function

Additional information

Funding

The main results of this paper (Sections 5–7) were obtained in IPME RAS and supported by Russian Science Foundation (project no. 20-71-10032).