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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. [2] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Partial Differential Equations, North -Holland, Amsterdam, London, NewYork, Tokyo, 1992. [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. [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. [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. [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. [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. [8] V. Barbu and G. Wang, Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407. [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. [10] C. Cao, I. 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. [11] C. Cao, E 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. [12] A. Yu. Chebotarev, Finite-dimensional controllability of systems of Navier-Stokes type, Differ. Equ., 46 (2010), 1498-1506. [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. [14] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. [15] I. D. Chueshov and V. K. Kalantarov, Determining functionals for nonlinear damped wave equations, Mat. Fiz. Anal. Geom., 8 (2001), 215-227. [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. [17] B. Cockburn, D. 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. [18] B. Cockburn, D. A. Jones and E. S. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Math. Comput., 66 (1997), 1073-1087. [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. [20] O. Çelebi, V. Kalantarov and M. Polat, Attractors for the Generalized Benjamin-Bona-Mahony Equation, J. of Differential Equations, 157 (1999), 439-451. [21] C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. [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. [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. [24] C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153. [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. [26] B. L. Guo, Finite-dimensional behavior for weakly damped generalized KdV-Burgers equations, Northeastern Mathematical Journal, 10 (1994), 309-319. [27] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. [28] J. K. Hale and G. Raugel, Regularity, determining modes and Galerkin methods, J. Math. Pures Appl. (9), 82 (2003), 1075-1136. [29] A. Haraux, Syst/ems deinamiques dissipatifs et applications, Masson, Paris, 1991. [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. [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. [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. [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. [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. [35] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. [36] O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 27 (1972), 91-115. [37] O. A. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. [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. [39] J. -L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non linéaires, Dunod et Gauthier-Villars, Paris, 1969. [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. [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. [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. [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. [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. [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. [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. [47] A. Smyshlyaev, E. 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. [48] R. Temam, Infnite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 2nd augmented edition, 1997. [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. [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. [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. [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.

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. [2] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Partial Differential Equations, North -Holland, Amsterdam, London, NewYork, Tokyo, 1992. [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. [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. [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. [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. [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. [8] V. Barbu and G. Wang, Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407. [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. [10] C. Cao, I. 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. [11] C. Cao, E 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. [12] A. Yu. Chebotarev, Finite-dimensional controllability of systems of Navier-Stokes type, Differ. Equ., 46 (2010), 1498-1506. [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. [14] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. [15] I. D. Chueshov and V. K. Kalantarov, Determining functionals for nonlinear damped wave equations, Mat. Fiz. Anal. Geom., 8 (2001), 215-227. [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. [17] B. Cockburn, D. 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. [18] B. Cockburn, D. A. Jones and E. S. Titi, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Math. Comput., 66 (1997), 1073-1087. [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. [20] O. Çelebi, V. Kalantarov and M. Polat, Attractors for the Generalized Benjamin-Bona-Mahony Equation, J. of Differential Equations, 157 (1999), 439-451. [21] C. Foias, O. P. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. [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. [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. [24] C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135-153. [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. [26] B. L. Guo, Finite-dimensional behavior for weakly damped generalized KdV-Burgers equations, Northeastern Mathematical Journal, 10 (1994), 309-319. [27] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Survey and Monographs, 25 AMS, Providence, R. I., 1988. [28] J. K. Hale and G. Raugel, Regularity, determining modes and Galerkin methods, J. Math. Pures Appl. (9), 82 (2003), 1075-1136. [29] A. Haraux, Syst/ems deinamiques dissipatifs et applications, Masson, Paris, 1991. [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. [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. [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. [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. [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. [35] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. [36] O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 27 (1972), 91-115. [37] O. A. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. [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. [39] J. -L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non linéaires, Dunod et Gauthier-Villars, Paris, 1969. [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. [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. [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. [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. [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. [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. [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. [47] A. Smyshlyaev, E. 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. [48] R. Temam, Infnite Dimensional Dynamical Systems in Mechanics and Physics, New York: Springer, 2nd augmented edition, 1997. [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. [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. [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. [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.
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