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

作者：Fusco Giorgio; Guglielmi Nicola^*

来源：Journal of Differential Equations, 2011, 250(7): 3230-3279.

DOI：10.1016/j.jde.2010.12.013

• Neutral delay differential equations
• Breaking points
• Existence and uniqueness conditions
• Regularization
• Singular perturbations
• Generalized solutions
• Filippov systems

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 epsilon > 0. The regularized solution converges as epsilon --> 0(+) to the classical Filippov solution (Filippov, 1964, 1988 [13,14]). Several properties of the solutions corresponding to small epsilon > 0 are presented.

• 出版日期2011-4-1