Потраекторное поведение класса управляемых пьезоэлектрических полей с немонотонным потенциалом

Исследовано автономное включение второго порядка в ограниченной области, моделирующее поведение класса управляемых пьезоэлектрических полей с немонотонным потенциалом. Исследуемая система описывает не только управляемый пьезоэлектрический процесс с многозначным законом “реакции-перемещения”, но и широкий класс управляемых процессов механики сплошных сред. Условия на параметры задачи не гарантируют единственности решения соответствующей задачи Коши, в частности, не предполагается никаких условий относительно непрерывности, монотонности нелинейного слагаемого по фазовой переменной. Изучена динамика слабых решений исследуемой задачи в смысле теории глобальных и траекторных аттракторов для многозначных полупотоков, порожденных слабыми решениями данной задачи. Применяя известные абстрактные результаты относительно существования траекторного аттрактора в пространстве траекторий, доказано, что для решений рассмотренной эволюционной задачи существует траекторный аттрактор в расширенном фазовом пространстве, исследованы его структурные свойства, установлена его связь с глобальным аттрактором и пространством полных траекторий поставленной задачи. Полученные результаты применены к математической модели, описывающей динамику пьезоэлектрического процесса.

Год издания: 
2014
Номер: 
2
УДК: 
517.9
С. 21–26.
Литература: 

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2. X.D. Wang et al., “Coupled behaviour of interacting dielectric cracks in piezoelectric materials,” Int. J. Fracture, vol. 132, pp. 115—133, 2005.
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4. 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.
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6. C. Rowley et al., “Dynamic and Closed-Loop Control,” Fundamentals and Applications of Modern Flow Control, vol. 231, 40 p., 2009.
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8. 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.
9. 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.
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11. 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.
12. 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.
13. 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.
14. 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.
15. 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.
16. F.H. Clarke, Optimization and Nonsmooth Analysis. New York: Wiley, 1983, 308 p.
17. M. Vishik et al., “Trajectory and Global Attractors of Three-Dimensional Navier-Stokes Systems,” Math. Notes, vol. 71, no. 2, pp. 177—193, 2002.
18. 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.
19. V.S. Melnik et al., “On attractors of multivalued semiflows and differential inclusions,” Set-Valued Analysis, vol. 6, no. 1, pp. 83—111, 1998.

Транслитерированый список литературы: 

1. S. Park et al., “Crack extension in piezoelectric materials,” SPIE. Smart Materials, V.K. Varadan, Ed., vol. 2189, pp. 357–368, 1994.
2. X.D. Wang et al., “Coupled behaviour of interacting dielectric cracks in piezoelectric materials,” Int. J. Fracture, vol. 132, pp. 115–133, 2005.
3. Miroshnychenko A.P., Shorokhov A.I͡e. Osoblyvosti keruvanni͡a parametramy pi͡ezokeramichnykh dvyhuniv // Visnyk KNUTD. – 2012. – # 3. – S. 33–37.
4. 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.
5. H. Khalil, Nonlinear systems. New Jersey: Prentice Hall, 2002, 750 p.
6. C. Rowley et al., “Dynamic and Closed-Loop Control,” Fundamentals and Applications of Modern Flow Control, vol. 231, 40 p., 2009.
7. 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.
8. V. Dem’yanov et al., “Quasidifferentiability and nonsmo¬oth modeling in Mechanics, Engineering and Econo¬mics,” Nonconvex Optimization and Its Applications, vol. 10. Dordrecht: Kluwer Academic Publishers, 1996, 348 p.
9. P.D. Panagiotopoulos et al., “The nonmonotone skin ef¬fects 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.
10. Panagiotopulos P. Neravenstva v mekhanike i ikh prilozhenii͡a. Vypuklye i nevypuklye funkt͡sii ėnergii: per. s angl. – M.: Mir, 1989. – 494 s.
11. 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.
12. Liu Z. et al., “Noncoercive Damping in Dynamic Hemi¬variational Inequality with Application to Problem of Piezoelectricity,” Discrete and Continuous Dynamical Systems, vol. 9, no. 1, pp. 129–143, 2008.
13. 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.
14. 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 Mathema¬tics. Berlin: Springer-Verlag, 2012, 330 p.
15. N.V. Gorban et al., “On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinea¬rity,” 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.
16. F.H. Clarke, Optimization and Nonsmooth Analysis. New York: Wiley, 1983, 308 p.
17. M. Vishik et al., “Trajectory and Global Attractors of Three-Dimensional Navier-Stokes Systems,” Math. No¬tes, vol. 71, no. 2, pp. 177–193, 2002.
18. 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.
19. V.S. Melnik et al., “On attractors of multivalued semi-flows and differential inclusions,” Set-Valued Analysis, vol. 6, no. 1, pp. 83–111, 1998.

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