Inexact Catching-Up Algorithm for Moreau's Sweeping Processes

TL;DR

This work introduces an inexact catching-up algorithm for sweeping processes by defining an $\varepsilon-\eta$ approximate projection that aligns with arbitrary numerical projection methods. It proves convergence under three moving-set regimes: $\rho$-uniformly prox-regular, ball-compact uniformly subsmooth, and ball-compact fixed sets, enabling existence and stability results for a broad class of sweeping dynamics. The authors also apply the framework to complementarity dynamical systems, implementing a primal-dual projection scheme to realize the approximate projections, and demonstrate the method on electrical circuits with ideal diodes. Overall, the approach generalizes the classical catching-up algorithm, offers rigorous convergence guarantees for diverse constraint geometries, and provides practical tools for numerically simulating sweeping processes in complex applications.

Abstract

In this paper, we develop an inexact version of the catching-up algorithm for sweeping processes. We define a new notion of approximate projection, which is compatible with any numerical method for approximating exact projections, as this new notion is not restricted to remain strictly within the set. We provide several properties of the new approximate projections, which enable us to prove the convergence of the inexact catching-up algorithm in three general frameworks: prox-regular moving sets, subsmooth moving sets, and merely closed sets. Additionally, we apply our numerical results to address complementarity dynamical systems, particularly electrical circuits with ideal diodes. In this context, we implement the inexact catching-up algorithm using a primal-dual optimization method, which typically does not necessarily guarantee a feasible point. Our results are illustrated through an electrical circuit with ideal diodes. Our results recover classical existence results in the literature and provide new insights into the numerical simulation of sweeping processes.

Figures (3)

• Figure 1: Electrical circuit with ideal diodes.
• Figure 2: On the left solution $x_1$ and on the right solutions $x_2$ (black) and $x_3$ (blue) for $R_1 = 1, R_2 = 2, R_3 = 1$, $L_2 = 1, L_3 = 2, C_4 = 1$, $u(t) = 16\sin(6\pi t) - 0.5$ and $x_0 = (0,0,0)$.
• Figure 3: On the left solution $x_1$ and on the right solutions $x_2$ (black) and $x_3$ (blue) for $R_1 = 1, R_2 = 2, R_3 = 1$, $L_2 = 1, L_3 = 2, C_4 = 1$, $G=(0,1)^t$, $u(t) = \operatorname{sign}(\sin(4 \pi t))$ and $x_0 = (0,0,0)$.

Theorems & Definitions (28)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Lemma 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Proposition 2.7
• proof
• Lemma 2.8