• Previous Article
    Loss of derivatives for hyperbolic boundary problems with constant coefficients
  • DCDS-B Home
  • This Issue
  • Next Article
    Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing
May  2018, 23(3): 1325-1345. doi: 10.3934/dcdsb.2018153

Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers

1. 

Department of mathematics, Koç University, Rumelifeneri Yolu, Sariyer 34450, Istanbul, Turkey

2. 

Institute of Matematics and Mechanics, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan

3. 

Department of Mathematics, Texas AM University, 3368 TAMU, College Station, TX 77843-3368, USA

4. 

Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

* Corresponding author: V. K. Kalantarov

Received  May 2017 Revised  July 2017 Published  February 2018

Fund Project: V.K.Kalantarov would like to thank the Weizmann Institute of Science for the generous hospitality during which this work was initiated. E.S.Titi would like to thank the ICERM, Brown University, for the warm and kind hospitality where this work was completed. The work of E.S.Titi was supported in part by the ONR grant N00014-15-1-2333

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier modes, determining volume elements, determining nodes, etc..The algorithm utilizes these finite parameters in the form of feedback control to stabilize the relevant solutions. For the sake of clarity, and in order to fix ideas, we focus in this work on the case of low Fourier modes feedback controller, however, our results and tools are equally valid for using other feedback controllers employing other spatial coarse mesh interpolants.

Citation: Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153
References:
[1]

A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters -a reaction-diffusion paradigm, Evolution Equations and Control Theory, 3 (2014), 579-594. doi: 10.3934/eect.2014.3.579. Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Partial Differential Equations, North -Holland, Amsterdam, London, NewYork, Tokyo, 1992. Google Scholar

[3]

M. J. Balas, Feedback control of dissipative hyperbolic distributed parameter systems with finite-dimensional controllers, J. Math. Anal. Appl., 98 (1984), 1-24. doi: 10.1016/0022-247X(84)90275-0. Google Scholar

[4]

J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim., 5 (1979), 169-179. doi: 10.1007/BF01442552. Google Scholar

[5]

V. Barbu, Stabilization of Navier-Stokes equation, Control and Stabilization of Partial Differential Equations, 1-50, Sémin. Congr., 29, Soc. Math. France, Paris, 2015. Google Scholar

[6]

V. Barbu, The internal stabilization of the Stokes-Oseen equation by feedback point controllers, Systems Control Lett., 62 (2013), 447-450. doi: 10.1016/j.sysconle.2013.02.009. Google Scholar

[7]

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445. Google Scholar

[8]

V. Barbu and G. Wang, Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407. Google Scholar

[9]

G. Bastin and J. -M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control. Birkhäuser/Springer, 2016. Google Scholar

[10]

C. CaoI. Kevrekidis and E. S. Titi, Numerical criterion for the stabilization of steady states of the Navier-Stokes equations, Indiana University Mathematics Journal, 50 (2001), 37-96. Google Scholar

[11]

C. CaoE Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848. Google Scholar

[12]

A. Yu. Chebotarev, Finite-dimensional controllability of systems of Navier-Stokes type, Differ. Equ., 46 (2010), 1498-1506. Google Scholar

[13]

I. D. Chueshov, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems, Russian Math. Surveys, 53 (1998), 731-776. Google Scholar

[14]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. Google Scholar

[15]

I. D. Chueshov and V. K. Kalantarov, Determining functionals for nonlinear damped wave equations, Mat. Fiz. Anal. Geom., 8 (2001), 215-227. Google Scholar

[16]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. Google Scholar

[17]

B. CockburnD. A. Jones and E. S. Titi, Degrés de liberté déterminants pour équations non linéaires dissipatives, C.R. Acad. Sci.-Paris, Sér. I, 321 (1995), 563-568. Google Scholar

[18]

B. CockburnD. A. Jones and E. S. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Math. Comput., 66 (1997), 1073-1087. Google Scholar

[19]

J. -M. Coron and E. Trélat, Feedback stabilization along a path of steady-states for 1-D semilinear heat and wave equations, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, 2005.Google Scholar

[20]

O. ÇelebiV. Kalantarov and M. Polat, Attractors for the Generalized Benjamin-Bona-Mahony Equation, J. of Differential Equations, 157 (1999), 439-451. Google Scholar

[21]

C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. Google Scholar

[22]

C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension deux, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34. Google Scholar

[23]

C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comput., 43 (1984), 117-133. Google Scholar

[24]

C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153. Google Scholar

[25]

A. V. Fursikov and A. A. Kornev, Feedback stabilization for the Navier-Stokes equations: Theory and calculations. Mathematical aspects of fluid mechanics, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 402 (2012), 130-172. Google Scholar

[26]

B. L. Guo, Finite-dimensional behavior for weakly damped generalized KdV-Burgers equations, Northeastern Mathematical Journal, 10 (1994), 309-319. Google Scholar

[27]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. Google Scholar

[28]

J. K. Hale and G. Raugel, Regularity, determining modes and Galerkin methods, J. Math. Pures Appl. (9), 82 (2003), 1075-1136. Google Scholar

[29]

A. Haraux, Syst/ems deinamiques dissipatifs et applications, Masson, Paris, 1991. Google Scholar

[30]

Ch. Jia, Boundary feedback stabilization of the Korteweg-de Vries-Burgers equation posed on a finite interval, J. Math. Anal. Appl., 444 (2016), 624-647. Google Scholar

[31]

D. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana University Mathematics Journal, 42 (1993), 875-887. Google Scholar

[32]

V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 50-54. Google Scholar

[33]

V. K. Kalantarov and E. S. Titi, Finite-parameters feedback control for stabilizing damped nonlinear wave equations. Nonlinear analysis and optimization, Contemp. Math., Amer. Math. Soc. , Providence, RI, 659 (2016), 115-133. Google Scholar

[34]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714. Google Scholar

[35]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. Google Scholar

[36]

O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 27 (1972), 91-115. Google Scholar

[37]

O. A. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. Google Scholar

[38]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Opt., 25 (1992), 189-224. doi: 10.1007/BF01182480. Google Scholar

[39]

J. -L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non linéaires, Dunod et Gauthier-Villars, Paris, 1969. Google Scholar

[40]

S. Lü and Q. Lu, Fourier spectral approximation to long-time behavior of dissipative generalized KdV-Burgers equations, SIAM J. Numer. Anal., 44 (2006), 561-585. doi: 10.1137/S0036142903426671. Google Scholar

[41]

E. Lunasin and E. S. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study, Evol. Equ. Control Theory, 6 (2017), 535-557, arXiv: 1506.03709, [math. AP]. doi: 10.3934/eect.2017027. Google Scholar

[42]

P. Marcati, Decay and stability for nonlinear hyperbolic equations, J. Differential Equations, 55 (1984), 30-58. doi: 10.1016/0022-0396(84)90087-1. Google Scholar

[43]

I. Munteanu, Boundary stabilization of the phase field system by finite-dimensional feedback controllers, J. Math. Anal. Appl., 412 (2014), 964-975. doi: 10.1016/j.jmaa.2013.11.018. Google Scholar

[44]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98-136. Google Scholar

[45]

D. Prazak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dynam. Differential Equations, 14 (2002), 763-776. doi: 10.1023/A:1020756426088. Google Scholar

[46]

R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of Computational Mathematics, Selected papers of a conference held at IMPA, (Eds. F. Cucker and M. Shub), Springer, (1997), 382-391. Google Scholar

[47]

A. SmyshlyaevE. Cerpa and M. Krstic, Boundary stabilization of a 1-D wave equation with in-domain antidamping, SIAM J. Control Optim., 48 (2010), 4014-4031. doi: 10.1137/080742646. Google Scholar

[48]

R. Temam, Infnite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 2nd augmented edition, 1997. Google Scholar

[49]

B. Wang and W. Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A, Math. Gen., 30 (1997), 4877-4885. doi: 10.1088/0305-4470/30/13/035. Google Scholar

[50]

S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Disc. Cont. Dyn. Sys., 11 (2004), 351-392. doi: 10.3934/dcds.2004.11.351. Google Scholar

[51]

B. -Y. Zheng, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, Control of Nonlinear Distributed Parameter Systems, (College Station, TX, 1999), 337-357, Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001. Google Scholar

[52]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-477. doi: 10.1137/0328025. Google Scholar

show all references

References:
[1]

A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters -a reaction-diffusion paradigm, Evolution Equations and Control Theory, 3 (2014), 579-594. doi: 10.3934/eect.2014.3.579. Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Partial Differential Equations, North -Holland, Amsterdam, London, NewYork, Tokyo, 1992. Google Scholar

[3]

M. J. Balas, Feedback control of dissipative hyperbolic distributed parameter systems with finite-dimensional controllers, J. Math. Anal. Appl., 98 (1984), 1-24. doi: 10.1016/0022-247X(84)90275-0. Google Scholar

[4]

J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim., 5 (1979), 169-179. doi: 10.1007/BF01442552. Google Scholar

[5]

V. Barbu, Stabilization of Navier-Stokes equation, Control and Stabilization of Partial Differential Equations, 1-50, Sémin. Congr., 29, Soc. Math. France, Paris, 2015. Google Scholar

[6]

V. Barbu, The internal stabilization of the Stokes-Oseen equation by feedback point controllers, Systems Control Lett., 62 (2013), 447-450. doi: 10.1016/j.sysconle.2013.02.009. Google Scholar

[7]

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445. Google Scholar

[8]

V. Barbu and G. Wang, Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407. Google Scholar

[9]

G. Bastin and J. -M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and their Applications, 88. Subseries in Control. Birkhäuser/Springer, 2016. Google Scholar

[10]

C. CaoI. Kevrekidis and E. S. Titi, Numerical criterion for the stabilization of steady states of the Navier-Stokes equations, Indiana University Mathematics Journal, 50 (2001), 37-96. Google Scholar

[11]

C. CaoE Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848. Google Scholar

[12]

A. Yu. Chebotarev, Finite-dimensional controllability of systems of Navier-Stokes type, Differ. Equ., 46 (2010), 1498-1506. Google Scholar

[13]

I. D. Chueshov, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems, Russian Math. Surveys, 53 (1998), 731-776. Google Scholar

[14]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. Google Scholar

[15]

I. D. Chueshov and V. K. Kalantarov, Determining functionals for nonlinear damped wave equations, Mat. Fiz. Anal. Geom., 8 (2001), 215-227. Google Scholar

[16]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. Google Scholar

[17]

B. CockburnD. A. Jones and E. S. Titi, Degrés de liberté déterminants pour équations non linéaires dissipatives, C.R. Acad. Sci.-Paris, Sér. I, 321 (1995), 563-568. Google Scholar

[18]

B. CockburnD. A. Jones and E. S. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Math. Comput., 66 (1997), 1073-1087. Google Scholar

[19]

J. -M. Coron and E. Trélat, Feedback stabilization along a path of steady-states for 1-D semilinear heat and wave equations, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, 2005.Google Scholar

[20]

O. ÇelebiV. Kalantarov and M. Polat, Attractors for the Generalized Benjamin-Bona-Mahony Equation, J. of Differential Equations, 157 (1999), 439-451. Google Scholar

[21]

C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. Google Scholar

[22]

C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension deux, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34. Google Scholar

[23]

C. Foias and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comput., 43 (1984), 117-133. Google Scholar

[24]

C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153. Google Scholar

[25]

A. V. Fursikov and A. A. Kornev, Feedback stabilization for the Navier-Stokes equations: Theory and calculations. Mathematical aspects of fluid mechanics, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 402 (2012), 130-172. Google Scholar

[26]

B. L. Guo, Finite-dimensional behavior for weakly damped generalized KdV-Burgers equations, Northeastern Mathematical Journal, 10 (1994), 309-319. Google Scholar

[27]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. Google Scholar

[28]

J. K. Hale and G. Raugel, Regularity, determining modes and Galerkin methods, J. Math. Pures Appl. (9), 82 (2003), 1075-1136. Google Scholar

[29]

A. Haraux, Syst/ems deinamiques dissipatifs et applications, Masson, Paris, 1991. Google Scholar

[30]

Ch. Jia, Boundary feedback stabilization of the Korteweg-de Vries-Burgers equation posed on a finite interval, J. Math. Anal. Appl., 444 (2016), 624-647. Google Scholar

[31]

D. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana University Mathematics Journal, 42 (1993), 875-887. Google Scholar

[32]

V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 50-54. Google Scholar

[33]

V. K. Kalantarov and E. S. Titi, Finite-parameters feedback control for stabilizing damped nonlinear wave equations. Nonlinear analysis and optimization, Contemp. Math., Amer. Math. Soc. , Providence, RI, 659 (2016), 115-133. Google Scholar

[34]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B, 30 (2009), 697-714. Google Scholar

[35]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. Google Scholar

[36]

O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 27 (1972), 91-115. Google Scholar

[37]

O. A. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. Google Scholar

[38]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Opt., 25 (1992), 189-224. doi: 10.1007/BF01182480. Google Scholar

[39]

J. -L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non linéaires, Dunod et Gauthier-Villars, Paris, 1969. Google Scholar

[40]

S. Lü and Q. Lu, Fourier spectral approximation to long-time behavior of dissipative generalized KdV-Burgers equations, SIAM J. Numer. Anal., 44 (2006), 561-585. doi: 10.1137/S0036142903426671. Google Scholar

[41]

E. Lunasin and E. S. Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study, Evol. Equ. Control Theory, 6 (2017), 535-557, arXiv: 1506.03709, [math. AP]. doi: 10.3934/eect.2017027. Google Scholar

[42]

P. Marcati, Decay and stability for nonlinear hyperbolic equations, J. Differential Equations, 55 (1984), 30-58. doi: 10.1016/0022-0396(84)90087-1. Google Scholar

[43]

I. Munteanu, Boundary stabilization of the phase field system by finite-dimensional feedback controllers, J. Math. Anal. Appl., 412 (2014), 964-975. doi: 10.1016/j.jmaa.2013.11.018. Google Scholar

[44]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 98-136. Google Scholar

[45]

D. Prazak, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dynam. Differential Equations, 14 (2002), 763-776. doi: 10.1023/A:1020756426088. Google Scholar

[46]

R. Rosa and R. Temam, Finite-dimensional feedback control of a scalar reaction-diffusion equation via inertial manifold theory, in Foundations of Computational Mathematics, Selected papers of a conference held at IMPA, (Eds. F. Cucker and M. Shub), Springer, (1997), 382-391. Google Scholar

[47]

A. SmyshlyaevE. Cerpa and M. Krstic, Boundary stabilization of a 1-D wave equation with in-domain antidamping, SIAM J. Control Optim., 48 (2010), 4014-4031. doi: 10.1137/080742646. Google Scholar

[48]

R. Temam, Infnite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 2nd augmented edition, 1997. Google Scholar

[49]

B. Wang and W. Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A, Math. Gen., 30 (1997), 4877-4885. doi: 10.1088/0305-4470/30/13/035. Google Scholar

[50]

S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Disc. Cont. Dyn. Sys., 11 (2004), 351-392. doi: 10.3934/dcds.2004.11.351. Google Scholar

[51]

B. -Y. Zheng, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, Control of Nonlinear Distributed Parameter Systems, (College Station, TX, 1999), 337-357, Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001. Google Scholar

[52]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-477. doi: 10.1137/0328025. Google Scholar

[1]

Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159

[2]

V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611

[3]

Andrei Fursikov, Alexey V. Gorshkov. Certain questions of feedback stabilization for Navier-Stokes equations. Evolution Equations & Control Theory, 2012, 1 (1) : 109-140. doi: 10.3934/eect.2012.1.109

[4]

Kaïs Ammari, Thomas Duyckaerts, Armen Shirikyan. Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation. Mathematical Control & Related Fields, 2016, 6 (1) : 1-25. doi: 10.3934/mcrf.2016.6.1

[5]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065

[6]

Fabrizio Colombo, Davide Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition. Communications on Pure & Applied Analysis, 2009, 8 (2) : 601-620. doi: 10.3934/cpaa.2009.8.601

[7]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094

[8]

Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227

[9]

Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934/cpaa.2019040

[10]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

[11]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289

[12]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89

[13]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method. Evolution Equations & Control Theory, 2014, 3 (1) : 147-166. doi: 10.3934/eect.2014.3.147

[14]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[15]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[16]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179

[17]

Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015

[18]

Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719

[19]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

[20]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (61)
  • HTML views (258)
  • Cited by (0)

Other articles
by authors

[Back to Top]