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Null Boundary Controllability of the Linear Kuramoto–Sivashinsky Equation with Variable Coefficients

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Abstract

The purpose of this paper is to study the null boundary controllability of the following linear Kuramoto–Sivashinsky equation: \(\rho (x)y_{t}(t,x)+(\sigma (x)y_{xx}(t,x))_{xx}+(q(x)y_x(t,x))_x=0, t>0,~x\in (0,\ell ),\) with mixed boundary conditions, where \(\rho (x),~\sigma (x)>0\) and q(x) is free of sign restrictions. The control is applied to the Dirichlet boundary condition at the left endpoint \(x=0\). Under disconjugacy and disfocal conditions of the second-order equation \((\sigma (x)\phi ')'+q(x)\phi =0\) on the interval \([0,\ell ]\), we show that the system is null controllable at time \(T>0\). This result is an extension of the one obtained by Cazacu et al. (SIAM J Cont Optim 56:2921–2958, 2018, Sect. 4) in the case where the constant coefficients \(\rho \equiv \sigma \equiv 1\) and \(q>0\). Our approach is essentially based on a detailed spectral analysis together with the moments method.

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References

  1. Armaou, A., Christofides, P.D.: Feedback control of the Kuramoto–Sivashinsky equation. Physica D 137, 49–61 (2000)

    Article  MathSciNet  Google Scholar 

  2. Ben Amara, J.: Systems-disconjugacy of a fourth-order differential equation with a middle term. Commun. Appl. Anal. 20, 25–36 (2016)

    MATH  Google Scholar 

  3. Ben Amara, J., Bouzidi, H.: On the simplicity of eigenvalues of two nonhomogeneous Euler–Bernoulli beams connected by a point mass. J. Diff. Equ. 267, 2083–2103 (2019)

    Article  MathSciNet  Google Scholar 

  4. Ben Amara, J., Vladimirov, A.A.: On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions. J. Math. Sci. 150, 2317–2325 (2008)

    Article  MathSciNet  Google Scholar 

  5. Baudouin, L., Cerpa, E., Crépeau, E., Mercado, A.: Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation. Appl. Anal. 92, 2084–2102 (2013)

    Article  MathSciNet  Google Scholar 

  6. Barret, J.H.: Two point boundary Problems for self-adjoint linear differential equations of the fourth order with middle term. Duke Math. J. 29, 543–554 (1962)

    MathSciNet  Google Scholar 

  7. Cazacu, C.M., Ignat, L.I., Pazoto, A.F.: Null controllability of the Kuramoto–Sivashinsky equation on star-shaped trees. SIAM J. Cont. Optim. 56, 2921–2958 (2018)

    Article  MathSciNet  Google Scholar 

  8. Christofides, P.D., Armaou, A.: Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control. Syst. Cont. Lett. 39, 283–294 (2000)

    Article  MathSciNet  Google Scholar 

  9. Cerpa, E.: Null controllability and stabilization of the linear Kuramoto–Sivashinsky equation. Commun. Pure Appl. Anal. 9, 91–102 (2010)

    Article  MathSciNet  Google Scholar 

  10. Cerpa, E., Mercado, A.: Local exact controllability to the trajectories of the 1-D Kuramoto–Sivashinsky equation. J. Differ. Equ. 250, 2024–2044 (2011)

    Article  MathSciNet  Google Scholar 

  11. Cerpa, E., Guzman, P., Mercado, A.: On the control of the linear Kuramoto–Sivashinsky equation. ESAIM: COCV 23, 165–194 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Hu, C., Temam, R.: Robust control of the Kuramoto–Sivashinsky equation. Dyn. Contin. Discret Impuls. Syst. Ser. B 8, 315–338 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Fedoryuk, M.V.: Asymptotic Analysis. Springer-Verlag, Berlin (1983)

    MATH  Google Scholar 

  14. Kuramoto, Y., Tsuzuki, T.: On the formation of dissipative structures in reaction-diffusion systems. Progr. Theor. Phys. 54, 687–699 (1975)

    Article  Google Scholar 

  15. Leighton, W., Nehari, Z.: On the oscillation of solutions of self-adjoint linear differential equations of fourth-order. Trans. Am. Math. Soc. 98, 325–377 (1958)

    Article  MathSciNet  Google Scholar 

  16. Levitan, B.M., Sargsyan, I.S.: Introduction to Spectral Theory. Amer. Math. Soc, USA (1975)

    Google Scholar 

  17. Liu, W.J., Krstić, M.: Stability enhancement by boundary control in the Kuramoto–Sivashinsky equation. Nonlinear Anal. Ser. A. 43, 485–507 (2001)

    Article  MathSciNet  Google Scholar 

  18. Meléndez, P.G.: Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto–Sivashinsky type equation. J. Math. Anal. Appl. 408, 275–290 (2013)

    Article  MathSciNet  Google Scholar 

  19. Naimark, M.A.: Linear Differential Operators. Ungar, New York (1967)

    MATH  Google Scholar 

  20. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)

    Book  Google Scholar 

  21. Russell, D., Fattorini, H.O.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rat. Mech. Anal. 43, 272–292 (1971)

    Article  MathSciNet  Google Scholar 

  22. Russell, D., Fattorini, H.O.: Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quart. Appl. Math. 32, 45–69 (1974)

    Article  MathSciNet  Google Scholar 

  23. Schwartz, L.: Étude des sommes d’exponentielles. Hermann, Paris (1959)

    MATH  Google Scholar 

  24. Sivashinsky, G.I.: Nonlinear analysis of hydrodynamic instability in laminar flames I: derivation of basic equations. Acta Astro. 4, 1177–1206 (1977)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Engineering Laboratory, Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia

    Jamel Ben Amara

  2. LR Analysis and Control of PDEs, LR22ES03, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Hedi Bouzidi

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  1. Jamel Ben Amara
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  2. Hedi Bouzidi
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Correspondence to Jamel Ben Amara.

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Amara, J.B., Bouzidi, H. Null Boundary Controllability of the Linear Kuramoto–Sivashinsky Equation with Variable Coefficients. Appl Math Optim 86, 33 (2022). https://doi.org/10.1007/s00245-022-09901-z

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