Pointwise asymptotic convergence of solutions for a phase separation model
Pavel Krejčí Songmu Zheng
Discrete & Continuous Dynamical Systems - A 2006, 16(1): 1-18 doi: 10.3934/dcds.2006.16.1
A new technique, combining the global energy and entropy balance equations with the local stability theory for dynamical systems, is used for proving that every solution to a non-smooth temperature-driven phase separation model with conserved energy converges pointwise in space to an equilibrium as time tends to infinity. Three main features are observed: the limit temperature is uniform in space, there exists a partition of the physical body into at most three constant limit phases, and the phase separation process has a hysteresis-like character.
keywords: Phase-field system asymptotic phase separation entropy. energy
Lipschitz continuous data dependence of sweeping processes in BV spaces
Pavel Krejčí Thomas Roche
Discrete & Continuous Dynamical Systems - B 2011, 15(3): 637-650 doi: 10.3934/dcdsb.2011.15.637
For a rate independent sweeping process with a time dependent smooth convex constraint, we prove that the Kurzweil solution for possibly discontinuous inputs depends locally Lipschitz continuously on the data in terms of the $BV$-norm.
keywords: Rate independence discontinuous sweeping process Kurzweil integral.
An asymptotic convergence result for a system of partial differential equations with hysteresis
Michela Eleuteri Pavel Krejčí
Communications on Pure & Applied Analysis 2007, 6(4): 1131-1143 doi: 10.3934/cpaa.2007.6.1131
A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.
keywords: hysteresis Partial differential equations asymptotic convergence Preisach operator.
Weak differentiability of scalar hysteresis operators
Martin Brokate Pavel Krejčí
Discrete & Continuous Dynamical Systems - A 2015, 35(6): 2405-2421 doi: 10.3934/dcds.2015.35.2405
Rate independent evolutions can be formulated as operators, called hysteresis operators, between suitable function spaces. In this paper, we present some results concerning the existence and the form of directional derivatives and of Hadamard derivatives of such operators in the scalar case, that is, when the driving (input) function is a scalar function.
keywords: Preisach rate independence evolution variational inequalities play Prandtl-Ishlinskii differentiability Hysteresis gliding maximum. accumulated maximum
The Preisach hysteresis model: Error bounds for numerical identification and inversion
Pavel Krejčí
Discrete & Continuous Dynamical Systems - S 2013, 6(1): 101-119 doi: 10.3934/dcdss.2013.6.101
A structure analysis of the Preisach model in a variational setting is carried out by means of an auxiliary hyperbolic equation with memory variable playing the role of time, and amplitude of cycles as spatial variable. Using this representation, we propose an algorithm and derive error estimates for the identification of the Preisach density function and for an approximate inversion of the Preisach operator.
keywords: Preisach model error bounds. Hysteresis
Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators
Pavel Krejčí Jürgen Sprekels
Discrete & Continuous Dynamical Systems - S 2008, 1(2): 283-292 doi: 10.3934/dcdss.2008.1.283
We consider a model for one-dimensional transversal oscillations of an elastic-ideally plastic beam. It is based on the von Mises model of plasticity and leads after a dimensional reduction to a fourth-order partial differential equation with a hysteresis operator of Prandtl-Ishlinskii type whose weight function is given explicitly. In this paper, we study the case of clamped beams involving a kinematic hardening in the stress-strain relation. As main result, we prove the existence and uniqueness of a weak solution. The method of proof, based on spatially semidiscrete approximations, strongly relies on energy dissipation properties of one-dimensional hysteresis operators.
keywords: von Mises model. Elastoplasticity beam equation Prandtl-Ishlinskii model hysteresis operators
Peter E. Kloeden Alexander M. Krasnosel'skii Pavel Krejčí Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - B 2013, 18(2): i-iii doi: 10.3934/dcdsb.2013.18.2i
Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62. For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

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Kurzweil integral representation of interacting Prandtl-Ishlinskii operators
Pavel Krejčí Harbir Lamba Sergey Melnik Dmitrii Rachinskii
Discrete & Continuous Dynamical Systems - B 2015, 20(9): 2949-2965 doi: 10.3934/dcdsb.2015.20.2949
We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators. This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling between the operators. We show that under quite general assumptions the coupled system is equivalent to one effective Prandtl-Ishlinskii operator or, more precisely, to a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative of the state function. This effective operator is described constructively in terms of the parameters of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation. This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii operators. The composition rule, which underpins the analysis of the coupled model, then establishes that a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.
keywords: hysteresis operator network model. composition formula Kurzweil integral discontinuous Prandtl-Ishlinskii operator Regulated function substitution formula
Long time behaviour of a singular phase transition model
Pavel Krejčí Jürgen Sprekels
Discrete & Continuous Dynamical Systems - A 2006, 15(4): 1119-1135 doi: 10.3934/dcds.2006.15.1119
A phase-field system, non-local in space and non-smooth in time, with heat flux proportional to the gradient of the inverse temperature, is shown to admit a unique strong thermodynamically consistent solution on the whole time axis. The temperature remains globally bounded both from above and from below, and its space gradient as well as the time derivative of the order parameter asymptotically vanish in $L^2$-norm as time tends to infinity.
keywords: long-time behaviour. Phase transition nonlocal model integrodifferential heat equation
Optimal control of ODE systems involving a rate independent variational inequality
Martin Brokate Pavel Krejčí
Discrete & Continuous Dynamical Systems - B 2013, 18(2): 331-348 doi: 10.3934/dcdsb.2013.18.331
This paper is concerned with an optimal control problem for a system of ordinary differential equations with rate independent hysteresis modelled as a rate independent evolution variational inequality with a closed convex constraint $Z\subset \mathbb{R}^m$. We prove existence of optimal solutions as well as necessary optimality conditions of first order. In particular, under certain regularity assumptions we completely characterize the jump behaviour of the adjoint.
keywords: hysteresis optimal control necessary optimality conditions. Evolution variational inequalities

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