Tubes of discontinuous solutions of dynamical systems and their stability

Abstract

The article is devoted to investigation of nonlinear dynamical systems which applies the generalized effect. The generalized effect (or impulses) is the result of the presence on the right part of the system a generalized derivative of the function of bounded variation. Such system contains incorrect operation of multiplication of discontinuous function on the generalized function from the point of view of the theory of distributions. This incorrectness is overcome by the approximation of the generalized functions in the right part of system by the sequence of smooth approximations of the generalized influences by analogy with sequential approach of the theory of the generalized functions. This sequence generates a sequence of smooth solutions. Then limit of a sequence of smooth solutions is considered. If such limit exists it is offered to be used as a solution. The solution is understood as partial pointwise limit of such sequence if a sequence of smooth solutions does not converge. Such partial limits constitute a tube of solutions. The sufficient conditions are received for stability of tubes of discontinuous solutions. © 2017 Author(s).