Singular limits for impulsive lagrangian systems with dissipative sources

Abstract

Consider a mechanical system, described by finitely many Lagrangian coordinates. Assume that an external controller can influence the evolution of the system by directly assigning the values of some of the coordinates. If these assignments are implemented by means of frictionless constraints, one obtains a set of ordinary differential equations where the right hand side depends also on the time derivatives of the control functions. Some basic aspects of the mathematical theory for these equations are reviewed here. We then consider a system with an additional dissipative term, which vanishes on a stable submanifold N. As the coefficient of the source term approaches infinity, we show that the limiting impulsive dynamics on the reduced state space N can be modelled by two different systems, depending on the order in which two singular limits are taken. These results are motivated by the analysis of impulsive systems with non-holonomic constraints.