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Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 8
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Research Article

Existence and feedback control for a class of nonlinear evolutionary equations

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Pages 1459-1481 | Received 30 Oct 2022, Accepted 26 Jul 2023, Published online: 31 Aug 2023

References

  • Barbu V. Nonlinear semigroup and differential equations in banach spaces. Leyden: Noordhoff; 1976.
  • Kačur J. Method of rothe in evolution equations. Leipzig: B.G. Teubner; 1985. (Teubner-Texte zur Mathematik 80).
  • Tanabe H. Equations of evolution. London: Pitman; 1979.
  • Contantin P, Foias C. Navier-Stokes equations. Chicago, IL: University of Chicago Press; 1988. (Chicago Lectures in Mathematics).
  • Francǔ J. Weakly continuous operators, applications to differential equations. Appl Math. 1994;39(1):45–56. doi: 10.21136/AM
  • Kalita P. Regularity and rothe method error estimates for parabolic hemivariational inequality. J Math Anal Appl. 2012;389:618–631. doi: 10.1016/j.jmaa.2011.12.007
  • Kalita P. Convergence of rothe scheme for hemivariational inequalities of parabolic type. Int J Numer Anal Model. 2013;10(2):445–465.
  • Migórski S, Ochal A. Navier-Stokes problems modeled by evolution hemivariational inequalities. Discrete Contin Dyn Syst Supplement. 2007;2007:731–740.
  • Temam R. Navier-Stokes equations: theory and numerical analysis. 2nd ed., Amsterdam: North-Holland; 1979.
  • Anh CT, Trang PT. Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains. P Roy Soc Edinb Sect A. 2013;143:223–251. doi: 10.1017/S0308210511001491
  • Celebi AO, Kalantarov VK, Polat M. Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain. Appl Anal. 2009;88:381–392. doi: 10.1080/00036810902766682
  • García-Luengo J, Marín-Rubio P, Real J. Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations. Nonlinearity. 2012;25:905–930. doi: 10.1088/0951-7715/25/4/905
  • Zeng B. Feedback control for non-stationary 3D Navier-Stokes-Voigt equations. Math Mech Solids. 2020;25(11):2210–2221. doi: 10.1177/1081286520926557
  • DiStefano JJ, Stubberud AR. Theory and problem of feedback and control systems. New York: McGraw-Hill; 1967.
  • Franklin GF, Powell JD, Emami-Naeini A. Feedback control of dynamic systems. New York: Prentice Hall; 2015.
  • Li XJ, Yong JM. Optimal control theory for infinite dimensional systems. Boster: Birkhäuser; 1995.
  • Mees AL. Dynamics of feedback systems. New York: Wiley; 1981.
  • Zeng B. Feedback control for nonlinear evolutionary equations with applications. Nonlinear Anal: Real World Appl. 2022;66:103535.
  • Zeng B. Feedback control systems governed by evolution equations. Optimization. 2019;68:1223–1243. doi: 10.1080/02331934.2019.1578358
  • Zeng B, Liu ZH. Existence results for impulsive feedback control systems. Nonlinear Anal: Hybrid Syst. 2019;33:1–16. doi: 10.1016/j.na.2018.11.016
  • Liu ZH, Zeng SD, Motreanu D. Evolutionary problems driven by variational inequalities. J Differ Equ. 2016;260:6787–6799. doi: 10.1016/j.jde.2016.01.012
  • Migórski S, Ochal A. Boundary hemivariational inequality of parabolic type. Nonlinear Anal Theory Methods Appl. 2004;57:579–596. doi: 10.1016/j.na.2004.03.004
  • Yin B, Zeng B. A note on a very recent paper ‘Feedback control for nonlinear evolutionary equations with applications’. Nonlinear Anal: Real World Appl. 2023;72:103857.
  • Zeng B, Migórski S. Evolutionary subgradient inclusions with nonlinear weakly continuous operators and applications. Comput Math Appl. 2018;75:89–104. doi: 10.1016/j.camwa.2017.08.040
  • Zeidler E. Nonlinear functional analysis and applications. New York: II A/B: Springer; 1990.
  • Roubi c˘ek T. Nonlinear partial differential equations with applications. Basel, Boston, Berlin: Birkhäuser; 2005.
  • Migórski S, Ochal A, Sofonea M. Nonlinear inclusions and hemivariational inequalities. models and analysis of contact problems. New York: Springer; 2013. (Advances in Mechanics and Mathematics 26).
  • Denkowski Z, Migórski S, Papageorgiou NS. An introduction to nonlinear analysis: theory. Boston, Dordrecht, London, New York: Kluwer Academic/Plenum Publishers; 2003.
  • Shen S, Liu F, Chen J, et al. Numerical techniques for the variable order time fractional diffusion equation. Appl Math Comput. 2012;218:10861–10870.
  • Sofonea M, Migórski S. Variational-hemivariational inequalities with applications. Boca Raton: Chapman and Hall/CRC; 2017.
  • Carstensen C, Gwinner J. A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems. Ann Mat Pura Appl. 1999;177:363–394. doi: 10.1007/BF02505918
  • Clarke FH. Optimization and nonsmooth analysis. New York: Wiley; 1983.
  • Migórski S, Ochal A, Sofonea M. A class of variational-hemivariational inequalities in reflexive Banach spaces. J Elast. 2017;127:151–178. doi: 10.1007/s10659-016-9600-7
  • Han JF, Migórski S. A quasistatic viscoelastic frictional contact problem with multivalued normal compliance, unilateral constraint and material damage. J Math Anal Appl. 2016;443:57–80. doi: 10.1016/j.jmaa.2016.05.012
  • Han WM, Sofonea M. Quasistatic contact problems in viscoelasticity and viscoplasticity. Providence, RI: International Press, American Mathematical Society; 2002. (Studies in Advanced Mathematics; vol. 30).
  • Han W, Migórski S, Sofonea M. A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J Math Anal. 2014;46:3891–3912. doi: 10.1137/140963248
  • Motreanu D, Sofonea M. Quasivariational inequalities and applications in frictional contact problems with normal compliance. Adv Math Sci Appl. 2000;10:103–118.
  • Sofonea M, Matei A. Mathematical models in contact mechanics. Cambridge University Press; 2012. (London Mathematical Society, Lecture Note Series 398).

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