Stability of nonsmooth dynamical systems
In chapter 1 we have discussed about the nature of solutions of dynamical problems involving impulsive impacts, and some of their properties like existence, uniqueness, continuous dependence on initial conditions and parameters. Roughly, we have seen that those dynamical systems are represented by measure differential equations (MDEs). We have also discussed the differences between dynamical equations of systems with unilateral constraints and MDEs. In view of some important applications like for instance control of manipulators subject to impacts, or simply the study of general impacting mechanical systems, it is important to study stability of solutions of those differential equations. In this chapter we start by presenting stability concepts which are the extension of Lyapunov stability to MDEs (as in section 1.2). Then we focus on the stability of impact Poincaré maps. In chapter 2 we have seen that under reasonable assumptions the solutions of compliant approximating problems converge to those of the limit rigid problem. This motivates us to study the relationship between stability of solutions of compliant approximating problems and stability of solutions of the rigid limit problem: in other words, we study how a particular stability property evolves when the stiffness becomes infinite.
KeywordsFeedback Gain Lyapunov Stability Periodic Trajectory Positive Definite Function Unilateral Constraint
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