Asymptotic stabilization of coupled oscillators with dry friction by feedback control

In this paper we design a feedback control u=u(x,ẋ) so that each solution x(⋅)x(⋅) of the closed-loop system ẍ(t)+∂Φ(ẋ)+∇f(x)+u(x,ẋ)∋0 approaches the set of critical points of f(x)f(x) with |ẋ(t)|→0 as t→+∞t→+∞. The robustness of the control is also discussed in the case where f(x)f(x) has only a finite number of critical values. The approach is mainly based on LaSalle’s invariance principles and Morse decomposition theory of attractors for differential inclusions.