A stability theory for second-order nonsmooth dynamical systems with application to friction problems

Abstract

A LaSalle's Invariance Theory for a class of first-order evolution variational inequalities is developed. Using this approach, stability and asymptotic properties of important classes of secondorder dynamic systems are studied. The theoretical results of the paper are supported by examples in nonsmooth Mechanics and some numerical simulations. Résumé Dans cet article, la théorie d'invariance de LaSalle est généralisée pour une classe d'inéquations variationnelles d'évolution du premier ordre. Des résultats de stabilité (au sens de Lyapunov) et d'attractivité sont ensuite obtenus pour des systèmes dynamiques du second ordre non réguliers. Une extension du théorème de Lagrange aux systèmes conservatifs non réguliers est également proposée. Enfin, quelques exemples et simulations numériques illustrent les principaux résultats théoriques.

Linear passive systems and maximal monotone mappings

“…Well-posedness of systems of the form (11) and their variants has been addressed in several papers [30,31,[39][40][41] for linear passive (or passive-like) systems and maximal monotone mappings. However, the relevant results appeared in these papers require extra conditions on the linear system and/or the maximal monotone mapping.…”

Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

“…In the absolute stability framework, strict passivity properties of a linear part of the system are required for proving the asymptotic stability of an isolated equilibrium point, which may be rather restrictive for mechanical systems in general. Adly et al [1,21] study stability properties of equilibrium sets of differential inclusions describing mechanical systems with friction. It is assumed that the non-smoothness is stemming from a maximal monotone operator (e.g.…”

“…Existence and uniqueness of solutions is therefore always fulfilled. A basic Lyapunov theorem for stability and attractivity is given in [1,21] for first-order differential inclusions with maximal monotone operators. The results are applied to linear mechanical systems with friction.…”

“…The results are applied to linear mechanical systems with friction. It is assumed in [1] that the relative sliding velocity of the frictional contacts depends linearly on the generalised velocities. Conditions for the attractivity of an equilibrium set are given.…”

“…Conditions for the attractivity of an equilibrium set are given. The results are generalised in [1] to conservative systems with an arbitrary potential energy function. In a previous publication [45], we provided conditions under which the equilibrium set of multi-degree-offreedom linear mechanical systems with an arbitrary number of Coulomb friction elements is attractive using Lyapunov-type stability analysis and a generalisation of LaSalle's invariance principle for non-smooth systems.…”