Dynamics of Solutions for Controlled Piezoelectric Fields with Multivalued “Reaction-Displacement” Law

Abstract

We consider the second-order evolution inclusion that describes a class of Continuum Mechanics controlled processes, in particular, the controlled piezoelectric fields with multivalued “reaction-displacement” law. Discontinuous interaction function is represented as the difference of subdifferentials of convex functionals for more flexible control. This case is actual for automatic feedback control problems. We study the dynamics of weak solutions of the investigated problem in terms of the theory of trajectory and global attractors for multivalued semiflows generated by weak solutions of given problem. A priory estimates for weak solutions are obtained. The existence of global and trajectory attractors is proved. The relationship between the global attractor, trajectory attractor, and the space of complete trajectories are provided. Results of this study allow us to direct the states of investigated system to the desired asymptotic levels and may be used for the development of technical equipment based on piezoelectric effect, in particular for design of piezoelectric positioning controller etc.