METHOD OF MULTIVALUED OPERATOR SEMIGROUP TO INVESTIGATE THE LONG-TERM FORECASTS FOR CONTROLLED PIEZOELECTRIC FIELDS

Abstract

We study the evolution inclusion of hyperbolic type with a linear damping, which describes a class of piezoelectric controlled fields with non-monotonic potential. Discontinuous on the phase variable interaction function can be represented as the difference of subdifferentials of convex functionals. This system describes a wide class of controlled Continuum Mechanics processes, in particular, the piezoelectric controlled processes with a multivalued "reaction-displacement" law. The representation of "reaction-displacement" law as the difference of subdifferentials of convex functionals allows more flexible control for piezoelectric system. In such processes, the properties of operator presented in the model play the key role. Therefore, we impose conditions on parameters of the problem such that allow investigated model with acceptable accuracy to describe real physical process and, at the same time, provide an opportunity to use existing mathematical apparatus for it. In this paper, using the methods of the theory of global and trajectory attractors for multivalued operator semigroups the finitedimensioness of weak solutions of the model is substantiated up to a small parameter. Furthermore, the results are applied to a piezoelectric problem.