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A remark on the local Lipschitz continuity of vector hysteresis operators

Pavel Krejčí (2001)

Applications of Mathematics

It is known that the vector stop operator with a convex closed characteristic Z of class C 1 is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping n is Lipschitz continuous on the boundary Z of Z . We prove that in the regular case, this condition is also necessary.

Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity

Olaf Klein (2004)

Applications of Mathematics

The asymptotic behaviour for t of the solutions to a one-dimensional model for thermo-visco-plastic behaviour is investigated in this paper. The model consists of a coupled system of nonlinear partial differential equations, representing the equation of motion, the balance of the internal energy, and a phase evolution equation, determining the evolution of a phase variable. The phase evolution equation can be used to deal with relaxation processes. Rate-independent hysteresis effects in the strain-stress...

Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

Petra Kordulová (2012)

Applications of Mathematics

This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of...

Continuous dependence on parameters and boundedness of solutions to a hysteresis system

Alexander M. Kamachkin, Dmitriy K. Potapov, Victoria V. Yevstafyeva (2022)

Applications of Mathematics

We analyze an ordinary differential system with a hysteresis-relay nonlinearity in two cases when the system is autonomous or nonautonomous. Sufficient conditions for both the continuous dependence on the system parameters and the boundedness of the solutions to the system are obtained. We give a supporting example for the autonomous system.

Dynamics of systems with Preisach memory near equilibria

Stephen McCarthy, Dmitrii Rachinskii (2014)

Mathematica Bohemica

We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear...

Resonance in Preisach systems

Pavel Krejčí (2000)

Applications of Mathematics

This paper deals with the asymptotic behavior as t of solutions u to the forced Preisach oscillator equation w ¨ ( t ) + u ( t ) = ψ ( t ) , w = u + 𝒫 [ u ] , where 𝒫 is a Preisach hysteresis operator, ψ L ( 0 , ) is a given function and t 0 is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function ψ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...

Stress-controlled hysteresis and long-time dynamics of implicit differential equations arising in hypoplasticity

Victor A. Kovtunenko, Ján Eliaš, Pavel Krejčí, Giselle A. Monteiro, Judita Runcziková (2023)

Archivum Mathematicum

A long-time dynamic for granular materials arising in the hypoplastic theory of Kolymbas type is investigated. It is assumed that the granular hardness allows exponential degradation, which leads to the densification of material states. The governing system for a rate-independent strain under stress control is described by implicit differential equations. Its analytical solution for arbitrary inhomogeneous coefficients is constructed in closed form. Under cyclic loading by periodic pressure, finite...

The Kurzweil integral in financial market modeling

Pavel Krejčí, Harbir Lamba, Giselle Antunes Monteiro, Dmitrii Rachinskii (2016)

Mathematica Bohemica

Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities...

Tracking with prescribed transient behaviour

Achim Ilchmann, E. P. Ryan, C. J. Sangwin (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Universal tracking control is investigated in the context of a class 𝒮 of M -input, M -output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary M -valued reference signal r of class W 1 , (absolutely...

Tracking with prescribed transient behaviour

Achim Ilchmann, E. P. Ryan, C. J. Sangwin (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary M -valued reference signal r of class W1,∞ (absolutely...

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