Solution continuation on a discontinuity set

The movement of an object characterized by ordinary differential equations (ODE) with discontinuous right-hand sides along the surface of a gap is called a sliding mode. It is required to find the connection of the right-slip characteristics of the system (the system to continue the solution on the surface of the gap). The article prompted a sequel based on the solution of the averaged optimization. It is shown that for the known examples of methods for solving optimization averaged lead to results coinciding with the method of A. F. Filippov and allow to extend these techniques to a wide class of multidimensional problems. Optimality conditions set forth averaged nonlinear programming and examples of their use in the case of ordinary and degenerate solutions.