From Lipschitzian to non-Lipschitzian characteristics: continuity of behaviors

Linear complementarity systems are used to model discontinuous dynamical systems such as networks with ideal diodes and mechanical systems with unilateral constraints. In these systems mode changes are modeled by a relation between nonnegative, complementarity variables. We consider approximating systems obtained by replacing this non-Lipschitzian relation with a Lipschitzian function and investigate the convergence of the solutions of the approximating system to those of the ideal system as the Lipschitzian characteristic approaches to the (non-Lipschitzian) complementarity relation. It is shown that this kind of convergence holds for linear passive complementarity systems for which solutions are known to exist and to be unique. Moreover, this result is extended to systems that can be made passive by pole shifting.