Geodesic
Method of multivalued operator semigroup to investigate the long-term forecasts for controlled piezoelectric fields
Čebyševskij sbornik, Tome 15 (2014) no. 2, pp. 21-32.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. Bibliography: 15 titles.
Keywords: multivalued operator semigroup, controlled piezoelectric field, hyperbolic inclusion, non-monotonic potential. 
@article{CHEB_2014_15_2_a1,
     author = {P. O. Kasyanov and L. S. Paliichuk and A. N. Tkachuk},
     title = {Method of multivalued operator semigroup to investigate the long-term forecasts for controlled piezoelectric fields},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {21--32},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2014_15_2_a1/}
}
TY  - JOUR
AU  - P. O. Kasyanov
AU  - L. S. Paliichuk
AU  - A. N. Tkachuk
TI  - Method of multivalued operator semigroup to investigate the long-term forecasts for controlled piezoelectric fields
JO  - Čebyševskij sbornik
PY  - 2014
SP  - 21
EP  - 32
VL  - 15
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2014_15_2_a1/
LA  - ru
ID  - CHEB_2014_15_2_a1
ER  - 
%0 Journal Article
%A P. O. Kasyanov
%A L. S. Paliichuk
%A A. N. Tkachuk
%T Method of multivalued operator semigroup to investigate the long-term forecasts for controlled piezoelectric fields
%J Čebyševskij sbornik
%D 2014
%P 21-32
%V 15
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2014_15_2_a1/
%G ru
%F CHEB_2014_15_2_a1
P. O. Kasyanov; L. S. Paliichuk; A. N. Tkachuk. Method of multivalued operator semigroup to investigate the long-term forecasts for controlled piezoelectric fields. Čebyševskij sbornik, Tome 15 (2014) no. 2, pp. 21-32. http://geodesic.mathdoc.fr/item/CHEB_2014_15_2_a1/

[1] Liu Z., Migórski S., “Noncoercive Damping in Dynamic Hemivariational Inequality with Application to Problem of Piezoelectricity”, Discrete and Continuous Dynamical Systems Series B, 9:1 (2008), 129–143 | MR | Zbl

[2] 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, eds. V. A. Sadovnichiy, M. Z. Zgurovsky, Springer-Verlag, 2013, 221–237

[3] Zgurovsky M. Z., Kasyanov P. O., Paliichuk L. S., “Automatic feedback control for one class of contact piezoelectric problems”, System Analysis and Information Technologies, 2014, no. 1, 56–68

[4] Kasyanov P. O., Paliichuk L. S., “Trajectory behavior of weak solutions of the piezoelectric problem with discontinuous interaction function on the phase variable”, Research Bulletin of NTUU “KPI”, 2 (2014)

[5] Clarke F. H., Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983, 308 pp. | MR | Zbl

[6] 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, 25 (2012), 1569–1574 | DOI | MR | Zbl

[7] Zgurovsky M. Z., Kasyanov P. O., Kapustyan O. V., Valero J., Zadoianchuk N. V., Evolution Inclusions and Variation Inequalities for Earth Data Processing, v. III, Springer-Verlag, Berlin, 2012, 330 pp. | Zbl

[8] Kalita P., Łukaszewicz G., “Global attractors for multivalued semiflows with weak continuity properties”, Nonlinear Analysis, 101 (2014), 124–143 | DOI | MR | Zbl

[9] Ball J. M., “Global attaractors for damped semilinear wave equations”, DCDS, 10 (2004), 31–52 | DOI | MR | Zbl

[10] Vishik M., Chepyzhov V., “Trajectory and Global Attractors of Three-Dimensional Navier–Stokes Systems”, Mathematical Notes, 71:1–2 (2002), 177–193 | DOI | MR | Zbl

[11] Ball J. M., “Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations”, Journal of Nonlinear Sciences, 7:5 (1997), 475–502 | DOI | MR | Zbl

[12] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988, 500 pp. | MR | Zbl

[13] Melnik V. S., Valero J., “On attractors of multivalued semi-flows and differential inclusions”, Set-Valued Analysis, 6:1 (1998), 83–111 | DOI | MR

[14] Kasyanov P. O., Zadoyanchuk N. V., “Dinamika reshenii klassa avtonomnykh evolyutsionnykh vklyuchenii vtorogo poryadka”, Kibernetika i sistemnyi analiz, 2012, no. 3, 111–126

[15] Kasyanov P. O., “Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity”, Mathematical Notes, 92:1–2 (2012), 205–218 | DOI | MR | Zbl

[16] Kasyanov P. O., Zadoyanchuk N. V., “Svoistva reshenii evolyutsionnykh vklyuchenii vtorogo poryadka s otobrazheniyami psevdomonotonnogo tipa”, Zhurnal vychislitelnoi i prikladnoi matematiki, 2010, no. 3(102), 63–78