On the Response of Autonomous Sweeping Processes to Periodic Perturbations

If x ₀ is an equilibrium of an autonomous differential equation (dot x=f(x)) and det∥f ^′(x ₀)∥≠0, then x ₀ persists under autonomous perturbations and x ₀ transforms into a T-periodic solution under non-autonomous T-periodic perturbations. In this paper we discover a similar structural stability for Moreau sweeping processes of the form (-dot uin N_{B}(u)+f_{0}(u)), (uin mathbb {R}^{2}, )i. e. we consider the simplest case where the derivative is taken with respect to the Lebesgue measure and where the convex set B of the reduced system is a non-moving unit ball of (mathbb {R}^{2}). We show that an equilibrium ∥u ₀∥=1 persists under periodic perturbations, if the projection (overline {f}:partial Bto mathbb {R}^{2}) of f ₀ on the tangent to the boundary ? B is nonsingular at u ₀.