Sweeping by a continuous prox-regular set
The authors consider the ``sweeping process' in a Hilbert space \(H\), specifically, the problem \[ -u'(t)\in N_{C(t)}(u(t)),\quad u(0)= u_0\in C(0),\quad u(t)\in C(t),\quad t\in [0,T], \] where \(C: [0,T]\to 2^H\) and \(N_K(x)\) represents the Clarke normal cone to \(K\) at \(x\). The goal of the paper is to extend results in the literature in which \(C\) is convex-valued (see [\textit{M. D. P. Monteiro Marques}, Differential inclusions in nonsmooth mechanical problems. Shocks and dry friction. Progress in Nonlinear Differential Equations and their Applications. 9. Basel: Birkhäuser (1993; Zbl 0802.73003)]) to the noncovnex case. Instead, \(C\) is required to be prox-regular-valued. (A closed set \(K\subseteq H\) is prox-regular if there exists a continuous function \(\varphi: K\to [0,\infty)\) such that for all \(x,y\in K\) and \(v\in N_K(x)\) we have \(v\cdot (y- x)\leq\varphi(x)|v||y-x|^2\). Note that a prox-regular set is convex if and only if \(\varphi\equiv 0\). Also, prox-regular sets are known under different names in the literature.) According to the authors, prox-regularity is a sort of ``a local version of convexity'' and ``the lack of monotonicity of the normal cone is controlled by \(\varphi\)''. Theorems are proven which guarantee local and global existence and uniqueness of continuous solutions with bounded variation, and a continuous dependence result is also proven. A number of examples is given, including examples that (a) give an application to Brownian motion, (b) illustrate that solutions need not be absolutely continuous, (c) demonstrate that infinitely many solutions might exist if assumptions are not satisfied, and (d) illustrate that solutions need not have bounded variation if assumptions are not satisfied.
scientific article; zbMATH DE number 1884214
scientific article