# American Institute of Mathematical Sciences

April  2012, 32(4): 1169-1208. doi: 10.3934/dcds.2012.32.1169

## Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control

 1 Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex

Received  October 2010 Revised  May 2011 Published  October 2011

Let $-A: \mathcal{D}(A)\to H$ be the generator of an analytic semigroup and $B : U \to [\mathcal{D}(A^*)]'$ a relatively bounded control operator such that $(A-\sigma,B)$ is stabilizable for some $\sigma>0$. In this paper, we consider the stabilization of the nonlinear system $y'+Ay+G(y,u)=Bu$ by means of a feedback or a dynamical control $u$. The control is obtained from the solution to a Riccati equation which is related to a low-gain optimal quadratic minimization problem. We provide a general abstract framework to define exponentially stable solutions which is based on the contruction of Lyapunov functions. We apply such a theory to stabilize, around an unstable stationary solution, the 2D or 3D Navier-Stokes equations with a Neumann control and the 2D or 3D Boussinesq equations with a Dirichlet control.
Citation: Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
##### References:
 [1] C. Amrouche and V. Girault, Problèmes généralisés de Stokes,, Portugal. Math., 49 (1992), 463. Google Scholar [2] C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension,, Czechoslovak Math. J., 44 (1994), 109. Google Scholar [3] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties,, In, 440 (2007), 15. Google Scholar [4] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Equ., 9 (2009), 341. doi: 10.1007/s00028-009-0015-9. Google Scholar [5] G. Avalos and R. Triggiani, Coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behaviour of the resolvent operator on the imaginary axis,, Appl. Anal., 88 (2009), 1357. doi: 10.1080/00036810903278513. Google Scholar [6] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. Google Scholar [7] M. Badra, Feedback stabilization of the 3-D Navier-Stokes equations based on an extended system,, In, 202 (2006), 13. Google Scholar [8] M. Badra, Local stabilization of the Navier-Stokes equations with a feedback controller localized in an open subset of the domain,, Numer. Funct. Anal. Optim., 28 (2007), 559. doi: 10.1080/01630560701348434. Google Scholar [9] M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system,, ESAIM Control Optim. Calc. Var., 15 (2009), 934. doi: 10.1051/cocv:2008059. Google Scholar [10] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control Optim., 48 (2009), 1797. doi: 10.1137/070682630. Google Scholar [11] M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system,, SIAM J. Control Optim., 49 (2011), 420. doi: 10.1137/090778146. Google Scholar [12] V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 9 (2003), 197. doi: 10.1051/cocv:2003009. Google Scholar [13] V. Barbu, "Stabilization of Navier-Stokes Flow,", Communications and Control Engineering, (2010). Google Scholar [14] V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, In, 440 (2007), 55. Google Scholar [15] V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model,, Indiana Univ. Math. J., 57 (2008), 1173. doi: 10.1512/iumj.2008.57.3284. Google Scholar [16] V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers,, Nonlinear Anal., 64 (2006), 2704. doi: 10.1016/j.na.2005.09.012. Google Scholar [17] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006). Google Scholar [18] V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, $d=2,3$, via feedback stabilization of its linearization,, In, 155 (2007), 13. Google Scholar [19] V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization for Navier-Stokes equations by means of finite-dimensional distributed controls,, 2010. Available from: \url{http://hal.archives-ouvertes.fr/hal-00455250/fr/}., (2010). Google Scholar [20] V. Barbu and R Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers,, Indiana Univ. Math. J., 53 (2004), 1443. doi: 10.1512/iumj.2004.53.2445. Google Scholar [21] F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52,, Springer-Verlag, (2006). Google Scholar [22] F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions,, Calc. Var. Partial Differential Equations, 37 (2010), 217. Google Scholar [23] F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505. doi: 10.3934/dcdss.2011.4.505. Google Scholar [24] P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988). Google Scholar [25] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003). Google Scholar [26] H. Fujita and H. Morimoto, On fractional powers of the Stokes operator,, Proc. Japan Acad., 46 (1970), 1141. doi: 10.3792/pja/1195526510. Google Scholar [27] A. V. Fursikov, Real process corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary,, In, 206 (2002), 95. Google Scholar [28] A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10 (2004), 289. doi: 10.3934/dcds.2004.10.289. Google Scholar [29] P. Grisvard, Caractérisation de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40. Google Scholar [30] G. Grubb and V. A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods,, Math. Scand., 69 (1991), 217. Google Scholar [31] S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29. Google Scholar [32] E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups,", rev. ed., (1957). Google Scholar [33] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system,, J. Differential Equations, 247 (2009), 1452. doi: 10.1016/j.jde.2009.06.005. Google Scholar [34] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system,, Adv. Differential Equations, 15 (2010), 231. Google Scholar [35] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary,, Nonlinearity, 24 (2011), 159. doi: 10.1088/0951-7715/24/1/008. Google Scholar [36] I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations--Riccati operator approach,, Appl. Anal., 42 (1991), 243. doi: 10.1080/00036819108840045. Google Scholar [37] I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati's feedback synthesis and regularity of optimal solution,, Appl. Math. Optim., 16 (1987), 147. doi: 10.1007/BF01442189. Google Scholar [38] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74,, Cambridge University Press, (2000). Google Scholar [39] I. Lasiecka and A. Tuffaha, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction,, In, 158 (2009), 171. Google Scholar [40] I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction,, Systems Control Lett., 58 (2009), 499. doi: 10.1016/j.sysconle.2009.02.010. Google Scholar [41] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969). Google Scholar [42] J.-L. Lions and E. Magenes, "Problemes aux Limites Non Homognes et Applications," Vol. I,, Dunod, (1968). Google Scholar [43] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726. Google Scholar [44] J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures Appl. (9), 87 (2007), 627. doi: 10.1016/j.matpur.2007.04.002. Google Scholar [45] J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921. Google Scholar [46] J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398. Google Scholar [47] J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers,, Discrete and Continuous Dynamical Systems A, 27 (2010), 1159. Google Scholar [48] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", Second edition, (1995). Google Scholar [49] R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation,, Nonlinear Anal., 71 (2009), 4967. doi: 10.1016/j.na.2009.03.073. Google Scholar [50] R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen eigenproblems,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 645. doi: 10.3934/dcdss.2009.2.645. Google Scholar

show all references

##### References:
 [1] C. Amrouche and V. Girault, Problèmes généralisés de Stokes,, Portugal. Math., 49 (1992), 463. Google Scholar [2] C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension,, Czechoslovak Math. J., 44 (1994), 109. Google Scholar [3] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties,, In, 440 (2007), 15. Google Scholar [4] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Equ., 9 (2009), 341. doi: 10.1007/s00028-009-0015-9. Google Scholar [5] G. Avalos and R. Triggiani, Coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behaviour of the resolvent operator on the imaginary axis,, Appl. Anal., 88 (2009), 1357. doi: 10.1080/00036810903278513. Google Scholar [6] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. Google Scholar [7] M. Badra, Feedback stabilization of the 3-D Navier-Stokes equations based on an extended system,, In, 202 (2006), 13. Google Scholar [8] M. Badra, Local stabilization of the Navier-Stokes equations with a feedback controller localized in an open subset of the domain,, Numer. Funct. Anal. Optim., 28 (2007), 559. doi: 10.1080/01630560701348434. Google Scholar [9] M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system,, ESAIM Control Optim. Calc. Var., 15 (2009), 934. doi: 10.1051/cocv:2008059. Google Scholar [10] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations,, SIAM J. Control Optim., 48 (2009), 1797. doi: 10.1137/070682630. Google Scholar [11] M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system,, SIAM J. Control Optim., 49 (2011), 420. doi: 10.1137/090778146. Google Scholar [12] V. Barbu, Feedback stabilization of Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 9 (2003), 197. doi: 10.1051/cocv:2003009. Google Scholar [13] V. Barbu, "Stabilization of Navier-Stokes Flow,", Communications and Control Engineering, (2010). Google Scholar [14] V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, In, 440 (2007), 55. Google Scholar [15] V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model,, Indiana Univ. Math. J., 57 (2008), 1173. doi: 10.1512/iumj.2008.57.3284. Google Scholar [16] V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers,, Nonlinear Anal., 64 (2006), 2704. doi: 10.1016/j.na.2005.09.012. Google Scholar [17] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006). Google Scholar [18] V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, $d=2,3$, via feedback stabilization of its linearization,, In, 155 (2007), 13. Google Scholar [19] V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization for Navier-Stokes equations by means of finite-dimensional distributed controls,, 2010. Available from: \url{http://hal.archives-ouvertes.fr/hal-00455250/fr/}., (2010). Google Scholar [20] V. Barbu and R Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers,, Indiana Univ. Math. J., 53 (2004), 1443. doi: 10.1512/iumj.2004.53.2445. Google Scholar [21] F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52,, Springer-Verlag, (2006). Google Scholar [22] F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions,, Calc. Var. Partial Differential Equations, 37 (2010), 217. Google Scholar [23] F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505. doi: 10.3934/dcdss.2011.4.505. Google Scholar [24] P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988). Google Scholar [25] R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003). Google Scholar [26] H. Fujita and H. Morimoto, On fractional powers of the Stokes operator,, Proc. Japan Acad., 46 (1970), 1141. doi: 10.3792/pja/1195526510. Google Scholar [27] A. V. Fursikov, Real process corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary,, In, 206 (2002), 95. Google Scholar [28] A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10 (2004), 289. doi: 10.3934/dcds.2004.10.289. Google Scholar [29] P. Grisvard, Caractérisation de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40. Google Scholar [30] G. Grubb and V. A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods,, Math. Scand., 69 (1991), 217. Google Scholar [31] S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29. Google Scholar [32] E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups,", rev. ed., (1957). Google Scholar [33] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system,, J. Differential Equations, 247 (2009), 1452. doi: 10.1016/j.jde.2009.06.005. Google Scholar [34] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system,, Adv. Differential Equations, 15 (2010), 231. Google Scholar [35] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary,, Nonlinearity, 24 (2011), 159. doi: 10.1088/0951-7715/24/1/008. Google Scholar [36] I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations--Riccati operator approach,, Appl. Anal., 42 (1991), 243. doi: 10.1080/00036819108840045. Google Scholar [37] I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati's feedback synthesis and regularity of optimal solution,, Appl. Math. Optim., 16 (1987), 147. doi: 10.1007/BF01442189. Google Scholar [38] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74,, Cambridge University Press, (2000). Google Scholar [39] I. Lasiecka and A. Tuffaha, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction,, In, 158 (2009), 171. Google Scholar [40] I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction,, Systems Control Lett., 58 (2009), 499. doi: 10.1016/j.sysconle.2009.02.010. Google Scholar [41] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969). Google Scholar [42] J.-L. Lions and E. Magenes, "Problemes aux Limites Non Homognes et Applications," Vol. I,, Dunod, (1968). Google Scholar [43] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations,, SIAM J. Control Optim., 45 (2006), 790. doi: 10.1137/050628726. Google Scholar [44] J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations,, J. Math. Pures Appl. (9), 87 (2007), 627. doi: 10.1016/j.matpur.2007.04.002. Google Scholar [45] J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921. Google Scholar [46] J.-P. Raymond, Feedback stabilization of a fluid-structure model,, SIAM J. Control Optim., 48 (2010), 5398. Google Scholar [47] J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers,, Discrete and Continuous Dynamical Systems A, 27 (2010), 1159. Google Scholar [48] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", Second edition, (1995). Google Scholar [49] R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation,, Nonlinear Anal., 71 (2009), 4967. doi: 10.1016/j.na.2009.03.073. Google Scholar [50] R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen eigenproblems,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 645. doi: 10.3934/dcdss.2009.2.645. Google Scholar
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