On the response of autonomous sweeping processes to periodic perturbations

Abstract

If $x_0$ is an equilibrium of an autonomous differential equation $\dot x=f(x)$ and $\det \|f'(x_0)\|\not=0$, 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 $-\dot u\in N_B(u)+f_0(u),$ $u\in\mathbb{R}^2,$ 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 $\overline{f}:\partial B\to\mathbb{R}^2$ of $f_0$ on the tangent to the boundary $\partial B$ is nonsingular at $u_0$.