Non-convex sweeping processes in the space of regulated functions

Convex valued geodesics and applications to sweeping processes with bounded retraction

In this paper we provide a formulation for sweeping processes with arbitrary locally bounded retraction, not necessarily left or right continuous. Moreover we provide a proof of the existence and uniqueness of solutions for this formulation which relies on the reduction to the $1$-Lipschitz continuous case by using a suitable family of geodesics for the asymmetric Hausdorff-like distance called "excess".

Global asymptotic stability of nonconvex sweeping processes

Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraint. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prov…

BV-norm continuity of sweeping processes driven by a set with constant shape

We prove the BV-norm well posedness of sweeping processes driven by a moving convex set with constant shape, namely the BV-norm continuity of the so-called play operator of elasto-plasticity.

Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.

A continuation principle for periodic BV-continuous state-dependent sweeping processes

We consider a Caratheodory differential equation with a state-dependent convex constraint that changes BV-continuously in time (a perturbed BV-continuous state-dependent sweeping processes). By setting up an appropriate catching-up algorithm we prove solvability of the initial value problem. Then, for sweeping processes with $T$-periodic right-hand-sides, we prove the existence of at least one $T$-periodic solution. Finally, we further consider a $T$-periodic sweeping process which is close to an autonomous sweeping process with a constant constraint and prove the existence of a $T$-periodic s…

积分公式与Brézis-Ekeland-Nayroles型原理在近正则扫掠过程中的应用

Integral Formulation and the Brézis-Ekeland-Nayroles-Type Principle for Prox-Regular Sweeping Processes

本文提出了一种新的积分公式和Brézis-Ekeland-Nayroles型变分原理，用于分析近正则扫掠过程的稳定性与解的统一性。