A new comparison principle for discrete Volterra equations with an application to convex sweeping processes with infinite delays

TL;DR

This paper introduces a new comparison principle for discrete Volterra equations, providing uniform bounds and compactness results crucial for analyzing convex sweeping processes with infinite delays, which are challenging due to the infinite distribution of delays.

Contribution

It develops a resolvent-free comparison principle for discrete Volterra equations, enabling uniform bounds and compactness in the presence of infinite delays, and applies these to convex sweeping processes.

Findings

Established a uniform $L^ ablafty(0,T)$-bound for non-negative kernels.

Proved a general energy decay estimate leading to compactness.

Numerical simulations show the delay can cause the projected point to be far from the boundary.

Abstract

Comparison principles for Volterra equations play a role analogous to maximum principles in PDEs: they provide positivity and stability information on the solution and allow one to control the output of bounded inputs. In the continuous setting, such results often rely on Laplace-transform or spectral methods (see Gripenberg, Londen, and Staffans, Volterra Integral and Functional Equations, 1990). However, these tools are not uniform in the discretization step $h$ hence fail in discrete or semi-discrete approximations. The present note introduces a resolvent-free argument yielding uniform $L^\infty(0,T)$-bounds for non-negative kernels. Compactness is a key ingredient in order to show existence of sweeping processes. While in the classical framework it is well established, adding an infinite distribution of delays complicates greatly the obtaining of such a result. In a first step we show a general energy decay estimate, which is then used to establish compactness. The argument is carried out in the discrete setting and that necessitates the introduction of the new comparison principle. In the classical sweeping process the previous position of the particle lies on the boundary of the constraint set, staying $O(h)$ close to the next projection point ($h$ is the discretization step). Our delay model projects the particle's averaged (by a unit measure kernel) past positions to the constraint set. Numerical simulations show that the projected point can lie at $O(1)$ distance from the convex set's boundary.