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We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí's theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron χ of a fixed shape. We provide examples of elastoplastic spring models leading to structurally stable situations where χ is a one- or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory. [ABSTRACT FROM AUTHOR]

We use the ideas of Adly, Attouch, and Cabot [in Nonsmooth Mechanics and Analysis, Adv. Mech. Math. 12, Springer, New York, 2006, pp. 289-304] on finite-time stabilization of dry friction oscillators to establish a theorem on finite-time stabilization of differential inclusions with a moving polyhedral constraint (known as polyhedral sweeping processes) of the form C + c(t). We then employ the ideas of Moreau [in New Variational Techniques in Mathematical Physics (Centro Internaz. Mat. Estivo (CIME), II Ciclo, Bressanone, 1973), Edizioni Cremonese, Rome, 1974, pp. 171-322] to apply our theorem to a system of elastoplastic springs with a displacement-controlled loading. We show that verifying the condition of the theorem ultimately leads to the following two problems: (i) identifying the active vertex "A"" or the active face "A"" of the polyhedron that the vector c\prime (t) points at; (ii) computing the distance from c\prime (t) to the normal cone to the polyhedron at "A."" We provide a computational guide for solving problems (i)--(ii) in the case of an arbitrary elastoplastic system and apply it to a particular example. Due to the simplicity of the particular example, we can solve (i)--(ii) by the methods of linear algebra and basic combinatorics. [ABSTRACT FROM AUTHOR]

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 process). 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 investigate 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 solution specifically located near the boundary switched equilibrium of the autonomous sweeping process. [ABSTRACT FROM AUTHOR]

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 the conditions for global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraints. 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 prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result. [ABSTRACT FROM AUTHOR]

We develop a theory which allows making qualitative conclusions about the dynamics of both monotone and non-monotone Moreau sweeping processes. Specifically, we first prove that any sweeping processes with almost periodic monotone right-hand-sides admits a globally exponentially stable almost periodic solution. And then we describe the extent to which such a globally stable solution persists under non-monotone perturbations. [ABSTRACT FROM AUTHOR]

We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove the bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable of concluding the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system. [ABSTRACT FROM AUTHOR]

A significant progress in the study of the dynamics of elastoplastic lattice spring models has been recently due to the development of a sweeping process framework where the dynamics of the stresses of springs is linked to a solution of the associated sweeping process. By using the sweeping process framework, a recent result by Gudoshnikov et al. (2022) ensures finite-time convergence of a lattice spring model under assumption that the vector g ′ (t) of the applied displacement controlled-loading lies strictly inside a suitable normal cone of the sweeping process. In this paper we drop the latter assumption by proving that g ′ (t) always belongs to a suitable normal cone when g ′ (t) = c o n s t (i.e. the loading is uni-directional) and when certain genericity condition holds. By using this result we establish that every elastoplastic lattice spring model with a generic associated polyhedron reaches a unique distribution of plastic deformations in finite time. [ABSTRACT FROM AUTHOR]

Networks of elastoplastic springs (elastoplastic systems) have been linked to differential equations with polyhedral constraints in the pioneering paper by Moreau (1974). Periodic loading of an elastoplastic system, therefore, corresponds to a periodic motion of the polyhedral constraint. According to Krejci (1996), every solution of a sweeping process with a periodically moving constraint asymptotically converges to a periodic orbit. Understanding whether such an asymptotic periodic orbit is unique or there can be an entire family of asymptotic periodic orbits (that form a periodic attractor) has been an open problem since then. Since suitable small perturbation of a polyhedral constraint seems to be always capable to destroy a potential family of periodic orbits, it is expected that none of potential periodic attractor is structurally stable. In the present paper we give a simple example to prove that even though the periodic attractor (of non-stationary periodic solutions) can be destroyed by a little perturbation of the moving constraint, the periodic attractor resists perturbations of the physical parameters of the mechanical model (i.e. the parameters of the network of elastoplastic springs). • A concise formulation of the sweeping process of a general network of elastoplastic springs. • A guide to compute the quantities of the sweeping process from a network of elastoplastic springs. • Proof of structural stability of a family of periodic solutions for a sample elastoplastic system. [ABSTRACT FROM AUTHOR]