Automatic feedback control for one class of contact piezoelectric problems
AbstractIn this paper we investigate the dynamics of solutions of the second order evolution inclusion with discontinuous interaction function which can be represented as the difference of subdifferentials. This case is actual for feedback automatic control problems. In particular, we consider mathematical model of contact piezoelectric process between a piezoelectric body and a foundation and for this problem investigate the long-term behavior of state function. We deduce a priory estimates for weak solutions of studied problem in the phase space. The theorem on the existence of a global attractor for multi-valued semiflow generated by weak solutions of the problem and the structural properties of the limit sets is proved. The main results of the paper were applied to the investigated piezoelectric problem.
Liu Z., Migórski S. Noncoercive Damping in Dynamic Hemivariational Inequality with Application to Problem of Piezoelectricity // Discrete and Continuous Dynamical Systems Series B. — 2008. — doi:10.3934/dcdsb.2008.9.129.
Clarke F.H. Optimization and Nonsmooth Analysis. — NY: Wiley, Interscience, 1983. — 380 p.
Ball J.M. Global attaractors for damped semilinear wave equations // DCDS. — 2004. — Vol. 10. — P. 31–52.
Zgurovsky M.Z., Kasyanov P.O., Zadoianchuk N.V. Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem // Applied Mathematics Letters. — 2012. — Vol. 25. — P. 1569–1574.
Zgurovsky M.Z., Kasyanov P.O., Kapustyan O.V., Valero J., Zadoianchuk N.V. Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Berlin: Springer-Verlag, 2012. — 330 p.
Gorban N.V., Kapustyan V.O., Kasyanov P.O., Paliichuk L.S. On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity // Continuous and Distributed Systems: Theory and Applications / V.A. Sadovnichiy, M.Z. Zgurovsky (Eds.). — Springer-Verlag, 2014. — P. 221–237.
Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. — NY: Springer-Verlag, 1988. — 500 р.
Vishik M., Chepyzhov V. Trajectory and Global Attractors of Three-Dimensional Navier-Stokes Systems // Mathematical Notes. — 2002. — doi:10.1023/A:1014190629738.
Ball J.M. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations // Journal of Nonlinear Sciences. — 1997. — doi:10.1007/s003329900037.
Melnik V.S., Valero J. On attractors of multivalued semi-flows and differential inclusions // Set-Valued Analysis. — 1998. — 6, № 1. — P. 83–111.