Polyhedral sweeping processes with oblique reflection in the space of regulated functions
The paper concerns existence, uniqueness, and dependence upon parameters of the solutions \(\xi:[0,L]\to{X}\) to the functional-difference inclusion \(\xi(t_2)-\xi(t_1) \in{\mathcal{C}} (\bigcup_{t\in[t_1,t_2]} {\mathcal{R}}_t(\xi(t)))\), which comes from the differential inclusion \(\dot{\xi}(t)\in{\mathcal{R}}_t(\xi(t))\). Here, \(X\) is a finite-dimensional Hilbert space, \({\mathcal{R}}_t:{\mathcal{P}}_t\to{X}\) is a reflection-cone-valued map whose domain \({\mathcal{P}}_t\subseteq{X}\) is a polyhedral-set, and \({\mathcal{C}}\) stands for the cone-convex hull. The solutions \(\xi\) are not necessarily continuous, but have both one-sided limits \(\xi(t+)\) and \(\xi(t-)\). The exact description of the problem, a polyhedral sweeping process, is rather intricate. This is an extension of the Skorokhod problem, where there are considered particular items \({\mathcal{P}}_t\) and \({\mathcal{R}}_t\), and continuous solutions \(\xi\).
scientific article; zbMATH DE number 1901532