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

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.

Figures (5)

• Figure 1: (left) Adhesive memory cell dynamics. (right) Dynamics of a constraint set moving in circles. The projection distance (red arrow) is not infinitesimal, which is an inherent property of the system and not a time-stepping artifact.
• Figure 2: Simulation of the delayed sweeping process with a moving circular constraint following a Lissajous curve.
• Figure 3: Simulation of the delayed sweeping process with a moving stadium-shaped constraint following a Lissajous curve.
• Figure 5: $L^2$ convergence of the discrete solution. Compatible past ($X_p \equiv 0 \in C^0$, blue) shows second-order convergence; incompatible past ($X_p \equiv (2,0) \notin C^0$, orange) degenerates to first order. Reference slopes $h$ and $h^2$ are shown.
• Figure 6: Projection-step product $\langle \delta P^{n+\nud}, \delta X^{n+\nud} \rangle$ for varying step size $h$, comparing circular and stadium-shaped constraints. See Section \ref{['sec:numerical_examples']} for full specifications.

Theorems & Definitions (42)

• theorem 1: Comparison principle for continuous Volterra equations
• theorem 2: Uniform discrete comparison principle
• proposition 1: Energy dissipation
• theorem 3: Limit of the discrete sweeping process
• remark 1
• theorem 4: Convergence for moving circular sets
• remark 2
• theorem 5
• theorem 6
• proof : Proof of Theorem \ref{['thm.w']}