Impulsive Dynamics and Measure Differential Equations
This chapter is devoted to introducing the mathematical basis on which various evolution problems involving impulsive terms rely. Impulsive forces in mechanics are first presented disregarding what they may be produced by. It is shown on simple examples why impulsive mechanics involves only measures (Dirac “functions”), and no distribution of higher degree (derivatives of the Dirac “function”). Various classes of measure differential equations (MDEs), or impulsive systems, are introduced. Then unilaterally constrained dynamical systems are presented, and the differences with the foregoing MDEs are discussed. Variable changes that allow one to transform MDEs into Carathéodory ordinary differential equations (ODEs) or unilaterally constrained mechanical systems into Filippov’s differential inclusions, are described in the last section.