September  2020, 9(3): 635-672. doi: 10.3934/eect.2020027

Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

Johann Radon Institute for Computational and Applied Mathematics, ÖAW, Altenbergerstraße 69, A-4040 Linz, Austria

Received  April 2019 Revised  June 2019 Published  September 2020 Early access  December 2019

It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depends on the operator norm of the oblique projection, on the polynomial degree of the nonlinearity, on the norm of the initial condition, and on the total volume covered by the actuators. The range of the feedback controller coincides with the range of the oblique projection, which is the linear span of the actuators. The oblique projection is performed along the orthogonal complement of a subspace spanned by a suitable finite number of eigenfunctions of the diffusion operator. For rectangular domains, it is possible to explicitly construct/place the actuators so that the stability of the closed-loop system is guaranteed. Simulations are presented, which show the semiglobal stabilizing performance of the nonlinear feedback.

Citation: Sérgio S. Rodrigues. Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems. Evolution Equations and Control Theory, 2020, 9 (3) : 635-672. doi: 10.3934/eect.2020027
References:
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[2]

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[3]

K. AmmariT. Duyckaerts and A. Shirikyan, Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation, Math. Control Relat. Fields, 6 (2016), 1-25.  doi: 10.3934/mcrf.2016.6.1.

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S. Aniţa and M. Langlais, Stabilization strategies for some reaction-diffusion systems, Nonlinear Anal. Real World Appl., 10 (2009), 345-357.  doi: 10.1016/j.nonrwa.2007.09.003.

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B. Azmi and K. Kunisch, Receding horizon control for the stabilization of the wave equation, Discrete Contin. Dyn. Syst., 38 (2018), 449-484.  doi: 10.3934/dcds.2018021.

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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), 420-463.  doi: 10.1137/090778146.

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V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers, SIAM J. Control Optim., 50 (2012), 2288-2307.  doi: 10.1137/110837164.

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V. Barbu, Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Automat. Control, 58 (2013), 2416-2420.  doi: 10.1109/TAC.2013.2254013.

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V. BarbuI. 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), 2704-2746.  doi: 10.1016/j.na.2005.09.012.

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V. BarbuS. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478.  doi: 10.1137/100785739.

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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.

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T. BreitenK. Kunisch and S. S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh-Nagumo type, SIAM J. Control Optim., 55 (2017), 2684-2713.  doi: 10.1137/15M1038165.

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L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59.  doi: 10.1016/j.jde.2005.08.002.

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S. Chowdhury and S. Ervedoza, Open loop stabilization of incompressible Navier-Stokes equations in a 2d channel using power series expansion, J. Math. Pures Appl., 130 (2019), 301-346.  doi: 10.1016/j.matpur.2019.01.006.

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J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach, Arch. Rational Mech. Anal., 225 (2017), 993-1023.  doi: 10.1007/s00205-017-1119-y.

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E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.

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J. Fourier, Théorie Analytique de la Chaleur, Éditions Jacques Gabay, Paris, 1988.

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A. V. Fursikov and O. Y. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Russian Math. Surveys, 54 (1999), 565-618.  doi: 10.1070/rm1999v054n03ABEH000153.

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A. HalanayC. M. Murea and C. A. Safta, Numerical experiment for stabilization of the heat equation by Dirichlet boundary control, Numer. Funct. Anal. Optim., 34 (2013), 1317-1327.  doi: 10.1080/01630563.2013.808210.

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O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[27]

D. Kalise and K. Kunisch, Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs, SIAM J. Sci. Comput., 40 (2018), A629-A652. doi: 10.1137/17M1116635.

[28]

D. KaliseK. Kunisch and K. Sturm, Optimal actuator design based on shape calculus, Math. Models Methods Appl. Sci., 28 (2018), 2667-2717.  doi: 10.1142/S0218202518500586.

[29]

A. Kröner and S. S. Rodrigues, Internal exponential stabilization to a nonstationary solution for 1D burgers equations with piecewise constant controls, Proceedings of the 2015 European Control Conference (ECC), Linz, Austria, (2015), 2676-2681, doi: 10.1109/ECC.2015.7330942.

[30]

A. Kröner and S. S. Rodrigues, Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations, SIAM J. Control Optim., 53 (2015), 1020-1055.  doi: 10.1137/140958979.

[31]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., 25 (2019). doi: 10.1051/cocv/2018054.

[32]

K. Kunisch and S. S. Rodrigues, Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ODE systems, Discrete Contin. Dyn. Syst., (2018), 2018-2040, URL https://www.ricam.oeaw.ac.at/publications/ricam-reports/,

[33]

C. LaurentF. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in ${L}^2(\mathbb{T})$, Arch. Ration. Mech. Anal., 218 (2015), 1531-1575.  doi: 10.1007/s00205-015-0887-5.

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H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form ${P}u_t = -{A}u+{\mathcal F}(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

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J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. Ⅰ of Die Grundlehren Math. Wiss. Einzeldarstellungen, Springer-Verlag, 1972. doi: 10.1007/978-3-642-65161-8.

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F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51 (1998), 139-196. 

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K. Morris and S. Yang, A study of optimal actuator placement for control of diffusion, 2016 American Control Conference (AAC), Boston, MA, USA, (2016), 2378-5861. doi: 10.1109/ACC.2016.7525303.

[39]

A. MünchP. Pedregal and F. Periago, Optimal internal stabilization of the linear system of elasticity, Arch. Rational Mech. Anal., 193 (2009), 171-193.  doi: 10.1007/s00205-008-0187-4.

[40]

I. Munteanu, Boundary stabilisation to non-stationary solutions for deterministic and stochastic parabolic-type equations, Internat. J. Control, 92 (2019), 1720-1728.  doi: 10.1080/00207179.2017.1407878.

[41]

T. NagataniH. Emmerich and K. Nakanishi, Burgers equation for kinetic clustering in traffic flow, Physica A, 255 (1998), 158-162.  doi: 10.1016/S0378-4371(98)00082-X.

[42]

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[43]

T. Nambu, Feedback stabilization for distributed parameter systems of parabolic type, Ⅱ, Arch. Rational Mech. Anal., 79 (1882), 241-259.  doi: 10.1007/BF00251905.

[44]

E. M. D. NgomA. Sène and D. Y. Le Roux, Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller, Evol. Equ. Control Theory, 4 (2015), 89-106.  doi: 10.3934/eect.2015.4.89.

[45]

D. Phan and S. S. Rodrigues, Gevrey regularity for Navier-Stokes equations under Lions boundary conditions, J. Funct. Anal., 272 (2017), 2865-2898.  doi: 10.1016/j.jfa.2017.01.014.

[46]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Systems, 30 (2018), 50 pp. doi: 10.1007/s00498-018-0218-0.

[47]

Y. PrivatE. Trélat and E. Zuazua, Actuator design for parabolic distributed parameter systems with the moment method, SIAM J. Control Optim., 55 (2017), 1128-1152.  doi: 10.1137/16M1058418.

[48]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.

[49]

J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two-dimensional Navier-Stokes equations with finite-dimensional controllers, Discrete Contin. Dyn. Syst., 27 (2010), 1159-1187.  doi: 10.3934/dcds.2010.27.1159.

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S. S. Rodrigues, Feedback boundary stabilization to trajectories for 3D Navier-Stokes equations, Appl. Math. Optim., (2018), 1-38. doi: 10.1007/s00245-017-9474-5.

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S. S. Rodrigues and K. Sturm, On the explicit feedback stabilisation of one-dimensional linear nonautonomous parabolic equations via oblique projections, IMA J. Math. Control Inform., (2018). doi: 10.1093/imamci/dny045.

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D. L. Russell and B. Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

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show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, van Nostrand, 1965, URL https://bookstore.ams.org/chel-369-h/.

[2]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, 1993, URL https://www.cambridge.org.

[3]

K. AmmariT. Duyckaerts and A. Shirikyan, Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation, Math. Control Relat. Fields, 6 (2016), 1-25.  doi: 10.3934/mcrf.2016.6.1.

[4]

S. Aniţa and M. Langlais, Stabilization strategies for some reaction-diffusion systems, Nonlinear Anal. Real World Appl., 10 (2009), 345-357.  doi: 10.1016/j.nonrwa.2007.09.003.

[5]

B. Azmi and K. Kunisch, Receding horizon control for the stabilization of the wave equation, Discrete Contin. Dyn. Syst., 38 (2018), 449-484.  doi: 10.3934/dcds.2018021.

[6]

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), 420-463.  doi: 10.1137/090778146.

[7]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.

[8]

V. Barbu, Stabilization of Navier-Stokes equations by oblique boundary feedback controllers, SIAM J. Control Optim., 50 (2012), 2288-2307.  doi: 10.1137/110837164.

[9]

V. Barbu, Boundary stabilization of equilibrium solutions to parabolic equations, IEEE Trans. Automat. Control, 58 (2013), 2416-2420.  doi: 10.1109/TAC.2013.2254013.

[10]

V. BarbuI. 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), 2704-2746.  doi: 10.1016/j.na.2005.09.012.

[11]

V. BarbuS. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478.  doi: 10.1137/100785739.

[12]

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.

[13]

F. Brauer, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl., 14 (1966), 198-206.  doi: 10.1016/0022-247X(66)90021-7.

[14]

T. BreitenK. Kunisch and S. S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh-Nagumo type, SIAM J. Control Optim., 55 (2017), 2684-2713.  doi: 10.1137/15M1038165.

[15]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59.  doi: 10.1016/j.jde.2005.08.002.

[16]

S. Chowdhury and S. Ervedoza, Open loop stabilization of incompressible Navier-Stokes equations in a 2d channel using power series expansion, J. Math. Pures Appl., 130 (2019), 301-346.  doi: 10.1016/j.matpur.2019.01.006.

[17]

J.-M. Coron and H.-M. Nguyen, Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach, Arch. Rational Mech. Anal., 225 (2017), 993-1023.  doi: 10.1007/s00205-017-1119-y.

[18]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Universitext, Springer, London, EDP Sciences, Les Ulis, 2012. doi: 10.1007/978-1-4471-2807-6.

[19]

T. DuyckaertsX. Zhang and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1-41.  doi: 10.1016/j.anihpc.2006.07.005.

[20]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.

[21]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.

[22]

J. Fourier, Théorie Analytique de la Chaleur, Éditions Jacques Gabay, Paris, 1988.

[23]

A. V. Fursikov and O. Y. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Russian Math. Surveys, 54 (1999), 565-618.  doi: 10.1070/rm1999v054n03ABEH000153.

[24]

A. HalanayC. M. Murea and C. A. Safta, Numerical experiment for stabilization of the heat equation by Dirichlet boundary control, Numer. Funct. Anal. Optim., 34 (2013), 1317-1327.  doi: 10.1080/01630563.2013.808210.

[25]

P. R. Halmos, Naive Set Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-1-4757-1645-0.

[26]

O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[27]

D. Kalise and K. Kunisch, Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs, SIAM J. Sci. Comput., 40 (2018), A629-A652. doi: 10.1137/17M1116635.

[28]

D. KaliseK. Kunisch and K. Sturm, Optimal actuator design based on shape calculus, Math. Models Methods Appl. Sci., 28 (2018), 2667-2717.  doi: 10.1142/S0218202518500586.

[29]

A. Kröner and S. S. Rodrigues, Internal exponential stabilization to a nonstationary solution for 1D burgers equations with piecewise constant controls, Proceedings of the 2015 European Control Conference (ECC), Linz, Austria, (2015), 2676-2681, doi: 10.1109/ECC.2015.7330942.

[30]

A. Kröner and S. S. Rodrigues, Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations, SIAM J. Control Optim., 53 (2015), 1020-1055.  doi: 10.1137/140958979.

[31]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., 25 (2019). doi: 10.1051/cocv/2018054.

[32]

K. Kunisch and S. S. Rodrigues, Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ODE systems, Discrete Contin. Dyn. Syst., (2018), 2018-2040, URL https://www.ricam.oeaw.ac.at/publications/ricam-reports/,

[33]

C. LaurentF. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in ${L}^2(\mathbb{T})$, Arch. Ration. Mech. Anal., 218 (2015), 1531-1575.  doi: 10.1007/s00205-015-0887-5.

[34]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form ${P}u_t = -{A}u+{\mathcal F}(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[35]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. Ⅰ of Die Grundlehren Math. Wiss. Einzeldarstellungen, Springer-Verlag, 1972. doi: 10.1007/978-3-642-65161-8.

[36]

F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51 (1998), 139-196. 

[37]

K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control, 56 (2011), 113-124.  doi: 10.1109/TAC.2010.2052151.

[38]

K. Morris and S. Yang, A study of optimal actuator placement for control of diffusion, 2016 American Control Conference (AAC), Boston, MA, USA, (2016), 2378-5861. doi: 10.1109/ACC.2016.7525303.

[39]

A. MünchP. Pedregal and F. Periago, Optimal internal stabilization of the linear system of elasticity, Arch. Rational Mech. Anal., 193 (2009), 171-193.  doi: 10.1007/s00205-008-0187-4.

[40]

I. Munteanu, Boundary stabilisation to non-stationary solutions for deterministic and stochastic parabolic-type equations, Internat. J. Control, 92 (2019), 1720-1728.  doi: 10.1080/00207179.2017.1407878.

[41]

T. NagataniH. Emmerich and K. Nakanishi, Burgers equation for kinetic clustering in traffic flow, Physica A, 255 (1998), 158-162.  doi: 10.1016/S0378-4371(98)00082-X.

[42]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 1962, 2061-2070. doi: 10.1109/JRPROC.1962.288235.

[43]

T. Nambu, Feedback stabilization for distributed parameter systems of parabolic type, Ⅱ, Arch. Rational Mech. Anal., 79 (1882), 241-259.  doi: 10.1007/BF00251905.

[44]

E. M. D. NgomA. Sène and D. Y. Le Roux, Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller, Evol. Equ. Control Theory, 4 (2015), 89-106.  doi: 10.3934/eect.2015.4.89.

[45]

D. Phan and S. S. Rodrigues, Gevrey regularity for Navier-Stokes equations under Lions boundary conditions, J. Funct. Anal., 272 (2017), 2865-2898.  doi: 10.1016/j.jfa.2017.01.014.

[46]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Systems, 30 (2018), 50 pp. doi: 10.1007/s00498-018-0218-0.

[47]

Y. PrivatE. Trélat and E. Zuazua, Actuator design for parabolic distributed parameter systems with the moment method, SIAM J. Control Optim., 55 (2017), 1128-1152.  doi: 10.1137/16M1058418.

[48]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl., 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.

[49]

J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two-dimensional Navier-Stokes equations with finite-dimensional controllers, Discrete Contin. Dyn. Syst., 27 (2010), 1159-1187.  doi: 10.3934/dcds.2010.27.1159.

[50]

S. S. Rodrigues, Feedback boundary stabilization to trajectories for 3D Navier-Stokes equations, Appl. Math. Optim., (2018), 1-38. doi: 10.1007/s00245-017-9474-5.

[51]

S. S. Rodrigues and K. Sturm, On the explicit feedback stabilisation of one-dimensional linear nonautonomous parabolic equations via oblique projections, IMA J. Math. Control Inform., (2018). doi: 10.1093/imamci/dny045.

[52]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[53]

D. L. Russell and B. Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

[54]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

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Figure 1.  Uncontrolled solutions. Linear and nonlinear systems
Figure 2.  Linear systems and linear feedback
Figure 3.  Nonlinear systems and linear feedback
Figure 4.  Nonlinear systems and nonlinear feedback
Figure 5.  Nonlinear systems and nonlinear feedback. Bigger initial condition
Figure 6.  Nonlinear systems and nonlinear feedback. Increasing the number of actuators
Figure 7.  Nonlinear systems and linear feedback. Increasing the number of actuators
Figure 8.  Nonlinear systems and nonlinear feedback. $ y(0) = c_{\rm ic}\sin(8\pi x)\in E_{ {\mathbb M}_\sigma}^\perp $
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