# American Institute of Mathematical Sciences

November  2015, 20(9): 2949-2965. doi: 10.3934/dcdsb.2015.20.2949

## Kurzweil integral representation of interacting Prandtl-Ishlinskii operators

 1 Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567 Praha 1 2 Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030 3 MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland 4 Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75025, United States

Received  June 2014 Revised  July 2015 Published  September 2015

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.
Citation: Pavel Krejčí, Harbir Lamba, Sergey Melnik, Dmitrii Rachinskii. Kurzweil integral representation of interacting Prandtl-Ishlinskii operators. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2949-2965. doi: 10.3934/dcdsb.2015.20.2949
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