Trajectory Behavior of Weak Solutions of the Piezoelectric Problem with Discontinuous Interaction Function on the Phase Variable
DOI:
https://doi.org/10.20535/1810-0546.2014.2.60384Keywords:
Trajectory attractor, Controlled piezoelectric fieldAbstract
The autonomous second order inclusion in a bounded domain, which is modeling the behavior of a class of the controlled piezoelectric fields with nonmonotonous potential, is studied. The investigated system describes not only controlled piezoelectric process with multivalued law “reaction-displacement”, but a wide class of controlled processes of Continuum Mechanics. Conditions on the parameters of the problem do not guarantee the uniqueness of solution of the corresponding Cauchy problem. In particular, any conditions on continuity, monotony of the nonlinear term by a phase variable are not assumed. We study the dynamics of weak solutions of the investigated problems in terms of the theory of trajectory and global attractors for multivalued semiflows generated by weak solutions of given problem. By using the well-known abstract results on the existence of trajectory attractor in the space of trajectories, we show the existence of trajectory attractor in the extended phase space for solutions of the considered evolution problem. Its structural properties are studied. Its relationship with the global attractor and space of complete trajectories is provided. Obtained results are applied to the mathematical model which describes the dynamics of the piezoelectric process.References
S. Park et al., “Crack extension in piezoelectric materials,” SPIE. Smart Materials, V.K. Varadan, Ed., vol. 2189, pp. 357—368, 1994.
X.D. Wang et al., “Coupled behaviour of interacting dielectric cracks in piezoelectric materials,” Int. J. Fracture, vol. 132, pp. 115—133, 2005.
Мірошниченко А.П., Шорохов А.Є. Особливості керування параметрами п’єзокерамічних двигунів // Вісник КНУТД. — 2012. — № 3. — С. 33—37.
J. Burns et al., “Representation of Feedback Operators for Hyperbolic Systems,” Computation and Control IV. Progress in Systems and Control Theory, vol. 20, pp. 57—73, 1995.
H. Khalil, Nonlinear systems.New Jersey: Prentice Hall, 2002, 750 p.
C. Rowley et al., “Dynamic and Closed-Loop Control,” Fundamentals and Applications of Modern Flow Control, vol. 231, 40 p., 2009.
Z. Naniewicz, P. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications. Nonconvex Optimization and Its Applications. Pure and Applied Mathematics. A Series of Monographs and Textbooks. New York: Marcel Dekker, Inc., 1995, 267 p.
V. Dem’yanov et al., “Quasidifferentiability and nonsmooth modeling in Mechanics, Engineering and Economics,” Nonconvex Optimization and Its Applications, vol. 10. Dordrecht: Kluwer Academic Publishers, 1996, 348 p.
P.D. Panagiotopoulos et al., “The nonmonotone skin effects in plane elasticity problems obeying to linear elastic and subdifferential laws,” Zeitschrift fur Angewandte Mathematik und Mechanik, vol. 70, іs. 1, pp. 13—21, 1990.
Панагиотопулос П. Неравенства в механике и их приложения. Выпуклые и невыпуклые функции энергии: пер. с англ. — М.: Мир, 1989. — 494 с.
M.Z. Zgurovsky et al., “Automatic feedback control for one class of contact piezoelectric problems,” System research and information technologies, no. 1, pp. 56—68, 2014.
Liu Z. et al., “Noncoercive Damping in Dynamic Hemivariational Inequality with Application to Problem of Piezoelectricity,” Discrete and Continuous Dynamical Systems, vol. 9, no. 1, pp. 129—143, 2008.
Zgurovsky M.Z. et al., “Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem,” Applied Mathematics Letters, vol. 25, pp. 1569—1574, 2012.
M.Z. Zgurovsky et al., Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Long-Time Behavior of Evolution Inclusions Solutions in Earth Data Analysis. Series: Advances in Mechanics and Mathematics. Berlin: Springer-Verlag, 2012, 330 p.
N.V. Gorban et al., “On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity,” in Continuous and Distributed Systems: Theory and Applications. Solid Mechanics and Its Applications, M.Z. Zgurovsky, V.A. Sadovnichiy, Eds., vol. 211, pp. 221—237, 2014.
F.H. Clarke, Optimization and Nonsmooth Analysis. New York: Wiley, 1983, 308 p.
M. Vishik et al., “Trajectory and Global Attractors of Three-Dimensional Navier-Stokes Systems,” Math. Notes, vol. 71, no. 2, pp. 177—193, 2002.
J.M. Ball, “Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,” J. of Nonlinear Sci., vol. 7, no. 5, pp. 475—502, 1997.
V.S. Melnik et al., “On attractors of multivalued semiflows and differential inclusions,” Set-Valued Analysis, vol. 6, no. 1, pp. 83—111, 1998.
Downloads
Published
Issue
Section
License
Copyright (c) 2017 NTUU KPI Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work
