A regularization for discontinuous differential equations with application to state-dependent delay differential equations

We consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so-called “breaking points”, where the solution derivative is again discontinuous. Consequently, the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a Cauchy problem for an ode with a discontinuous right-hand side (see e.g. Bellen et al., 2009 [4]). Therefore a classical solution may cease to exist. The regularization is based on the replacement of the vector-field with its time average over an interval of length ε > 0. The regularized solution converges as ε→0+ to the classical Filippov solution (Filippov, 1964, 1988 [13,14]). Several properties of the solutions corresponding to small ε > 0 are presented.