Sweeping processes and rate independence (Q2821978)

Let \(\mathcal{H}\) be a Banach space; \(\mathcal{C}(t),\) \(t \geq 0\), a nonempty closed comvex subset of \(\mathcal{H}\). The following sweeping process is considered: under the assumption that \(t \multimap \mathcal{C}(t)\) is a right continuous, locally bounded variation multifunction with respect to the Hausdorff distance one has to find a right continuous, locally bounded variation function \(y: [0, \infty) \to \mathcal{H}\) such that there exist a positive measure \(\mu\) and a \(\mu\)-locally integrable function \(\omega: [0, \infty) \to \mathcal{H}\) satisfying NEWLINE\[NEWLINE Dy = \omega \mu, NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(t) \in \mathcal{C}(t) \quad \forall t \in [0, \infty), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \omega (t) + N_{\mathcal{C}(t)}(y(t)) \ni 0 \text{ for } \mu-a.e.\,\,t \in [0,\infty), NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(0) = \text{Proj}_{\mathcal{C}(0)}(y_0), NEWLINE\]NEWLINE where \(Dy\) denotes the distributional derivative of \(y\), \(N_{\mathcal{C}(t)}(y(t))\) is the exterior normal cone to \(\mathcal{C}(t)\) at \(y(t)\) and \(y_0 \in \mathcal{H}\) is a given point.NEWLINENEWLINEThe author suggests a new reparametrization technique allowing to reduce this sweeping process to a locally Lipshitz case. This allows to prove for the initial value problem existence and continuous dependence results and to justify the convergence of the catching-up algorithm.