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

1. Problem Statement and Motivation

The paper addresses the mathematical analysis and numerical approximation of delayed sweeping processes, a class of differential inclusions modeling systems with memory and non-overlapping constraints.

• Context: The study is motivated by the modeling of multi-cellular systems where cells exhibit "adhesive memory" (sticking to past positions via transient bonds) and must satisfy non-overlap constraints (moving within a feasible convex set $C(t)$).
• The Model: Unlike classical sweeping processes (Moreau's model) which involve a time-derivative $\dot{X}(t)$, this model replaces the derivative with a Volterra integral operator representing an infinite distribution of delays:
  $$X(t) - \int_0^\infty \varrho(a) X(t-a) \, da \in -N_{C(t)}(X(t))$$
  Here, $X(t)$ is the state, $\varrho$ is a non-negative memory kernel, and $N_{C(t)}$ is the normal cone to the moving convex set $C(t)$.
• Key Challenge:
  1. Non-infinitesimal Projections: In classical sweeping processes, the projection distance is infinitesimal ($O(h)$). In this delayed model, the projection distance can be $O(1)$ because the state depends on the entire past history. This breaks standard compactness arguments used in classical theory.
  2. Discretization Uniformity: Existing stability and convergence tools for Volterra equations (e.g., Laplace transforms, spectral decomposition of resolvents) often fail to provide uniform bounds with respect to the discretization step $h$. This makes proving the existence of solutions for the continuous limit via discrete approximations difficult, especially for kernels that are not exponentially decaying (e.g., algebraic decay).

2. Methodology

The authors develop a novel analytical framework that avoids spectral methods and resolvent functions, focusing instead on comparison principles and energy estimates.

A. A New Comparison Principle (Continuous and Discrete)

The core theoretical contribution is a comparison principle for Volterra equations with non-negative, non-exponential kernels.

• The Obstacle: Standard super-solutions constructed for Volterra equations often fail because the "tail" of the kernel (the integral from $t$ to $\infty$) creates a negative term that cannot be bounded uniformly for general kernels (e.g., polynomial decay).
• The Solution: The authors introduce an initial layer corrector ($w(t)$).
  • They construct a super-solution $S(t) = U(t) + \|f\|_\infty w(t)$, where $U(t)$ is a standard profile and $w(t)$ solves a specific auxiliary problem designed to cancel the negative tail contributions.
  • This allows them to prove that if the source term is bounded, the solution is bounded in $L^\infty$, independent of the spectral properties of the kernel.
• Discrete Adaptation: They rigorously translate this to the discrete setting (piecewise constant approximations). They define a discrete initial layer corrector $W^n$ and prove it remains bounded and non-negative uniformly with respect to the step size $h$.

B. Energy Estimates and Compactness

To establish the existence of the sweeping process, the authors use the minimizing movements (De Giorgi) approach.

• Reformulation: The discrete sweeping step is shown to be equivalent to minimizing an energy functional:
  $$E_n(W) = \frac{h}{2} \sum_{j \ge 1} (W - X_{n-j})^2 R_j$$
  where $R_j$ are discrete kernel weights.
• Dissipation: They derive a discrete energy dissipation inequality. The key difficulty is that the "distance" between the unconstrained average position and the projected position is not small ($O(1)$ rather than $O(h)$).
• Uniform Bounds: Using the new comparison principle, they establish uniform $H^1$ bounds on the discrete solutions. This is crucial because, without the time-derivative term, standard compactness (Ascoli-Arzelà) cannot be applied directly without these bounds.

C. Convergence Analysis

• Moving Circular Sets: They prove convergence specifically for the case where the constraint set is a moving ball $C(t) = B(c(t), R)$.
• Geometric Handling: They analyze the "projection-step product" $\langle \delta P, \delta X \rangle$. In moving sets, this term is not automatically non-negative. The authors decompose it to show it remains bounded, relying on the geometry of the moving center $c(t)$.
• Limit: By combining the uniform $H^1$ bounds (from the comparison principle) and the convergence of discrete convolutions (Appendix B), they prove that the discrete sequence converges to a solution of the continuous delayed sweeping process.

3. Key Contributions

1. Resolvent-Free Comparison Principle: The introduction of an initial layer corrector to derive uniform $L^\infty$ bounds for Volterra equations with non-exponential, non-negative kernels. This bypasses the need for Laplace transforms or spectral decomposition, which are not uniform in the discretization step.
2. Existence for Delayed Sweeping Processes: Proving the existence of solutions for sweeping processes driven by infinite-delay memory operators (Volterra type) rather than differential operators. This fills a gap in the literature where compactness was previously lost due to the lack of a time-derivative.
3. Uniform Discrete Stability: Establishing that the discrete approximation remains stable and bounded uniformly in $h$, even when the projection distance is $O(1)$.
4. Numerical Validation: Providing a robust numerical algorithm (Newton-based projection with tangential refinement) and demonstrating second-order convergence for compatible initial data.

4. Main Results

• Theorem 1 & 2 (Comparison Principles): Proves that solutions to both continuous and discrete Volterra equations with non-negative, monotone kernels are bounded in $L^\infty$, with bounds independent of the discretization step $h$.
• Proposition 3 (Energy Dissipation): Establishes that the discrete sweeping process satisfies a discrete energy dissipation law, controlling the growth of the solution.
• Theorem 6 (Convergence for Circular Sets): Under specific kernel conditions (including algebraic decay like $\varrho(a) \sim (1+a)^{-\alpha}$), the discrete sweeping sequence converges to a continuous solution $X(t)$ satisfying the integral inclusion:
  $$X(t) - \int_0^\infty \varrho(s) X(t-s) \, ds \in -N_{C(t)}(X(t))$$
• Numerical Results: Simulations with Lissajous curves and stadium-shaped constraints confirm that the projection distance is indeed $O(1)$ (a system property, not a numerical artifact) and that the scheme converges with order 2 for compatible initial conditions.

5. Significance and Impact

• Theoretical Advancement: This work extends the theory of sweeping processes beyond the classical Moreau framework (differential inclusions) to integro-differential and purely integral settings. It provides a rigorous foundation for systems where memory effects dominate instantaneous dynamics.
• Biological Modeling: The results directly support the modeling of cell migration and adhesion in structured environments, where cells interact with their past positions through transient bonds. The ability to handle infinite delays and non-infinitesimal projections is essential for accurate biological simulation.
• Numerical Robustness: The developed comparison principle offers a new tool for analyzing numerical schemes for Volterra equations, particularly when the kernel lacks analyticity or exponential decay, a common scenario in physical and biological models.
• Future Directions: The authors note that extending these results to general convex sets (beyond balls) and prox-regular (non-convex) sets remains an open challenge, as the geometric decomposition of the projection-step product becomes more complex.

In summary, the paper successfully bridges the gap between the theory of Volterra equations and sweeping processes by introducing a robust, resolvent-free comparison principle that ensures stability and convergence in the presence of infinite delays and non-infinitesimal projections.