Properties of solutions to a class of differential models incorporating Preisach hysteresis operator

We consider a class of equations which have been recently proposed as a mathematical tool for modelling dynamics of hydrological, economic and biological systems exhibiting hysteresis and related memory effects. The detail of the modelling approach is illustrated by an example from the hydrological context where a balance equation is coupled with a hysteretic constitutive relationship between the water content 9 in the soil and the matric potential psi(m) of the soil matrix and where the Preisach hysteresis operator is used as a model of this constitutive relationship. In particular, we present assumptions which eliminate spatial variation and lead to balance equations in the form of ODEs; two examples of such hydrological models are considered followed by a less detailed discussion of applications of similar modelling approach and equations in economics and biology. In the proposed formalism, the closed system is described by an operator-differential equation where the rate of change of the output of the Preisach operator is a function of its input and time. In the main part of the paper, for such operator-differential equations, we study the initial value problem: uniqueness, existence, extendability of solutions, their dependence on initial data, and the structure of the projection of the phase portrait onto the (t, psi(m))-plane. Solutions are characterised by jumps of the derivative induced by either of the two reasons; one is the memory of past extremum values of the solution; the other is a singularity at the lines of zero flow. We analyse the singularity and calculate the value of the jumps thus providing an important input to numerical solution of the equation. Furthermore, we identify possible non-uniqueness points and points of sensitivity to small perturbations of initial data as well as conditions that ensure uniqueness and stability to such perturbations. Regularisation of the equation and natural monotonicity conditions ensuring global stability of a periodic solution for the equation with periodic input are discussed. Rigorous analysis of the well-posedness of the models is presented for the first time. (C) 2011 Elsevier B.V. All rights reserved.