Differential inequalities for hysteresis systems
Let \(W\) denote the so-called deterministic transducer i.e. a dynamical system having inputs and outputs. Denote by \(\Omega(W)\) the state space of the transducer \(W\) and let \({\mathcal F}(F)\) stand for the image of the set \(\Omega(W)\) with respect to the mapping \(F\). The object of the paper is the study of the functional-differential equation (1) \(x'(t)= f(t,x(t),\xi(t))\) with the initial values (2) \(x(t_0)=x_0\), \(w_0(t_0)= w_0\in\Omega(W)\). Here \(x(t)\in \mathbb{R}^n\), \(f:\mathbb{R}\times \mathbb{R}^n\times {\mathcal F}(F)\to\mathbb{R}^n\) is a continuous function and \(\xi(t)\) is an output of the hysteresis nonlinearity \(W\) corresponding to the input \(x(t)\). With help of the concept of semiorder of the state space \(\Omega(W)\) and the concept of monotonicity a few results concerning differential inequalities associated to the problem (1)--(2) are derived. The existence of minimal and maximal solutions of the problem also proved.
scientific article; zbMATH DE number 988084