Encyclopedia of Systems and Control

2015 Edition
| Editors: John Baillieul, Tariq Samad

Boundary Control of Korteweg-de Vries and Kuramoto–Sivashinsky PDEs

  • Eduardo Cerpa
Reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5058-9_13


The Korteweg-de Vries (KdV) and the Kuramoto-Sivashinsky (KS) partial differential equations are used to model nonlinear propagation of one-dimensional phenomena. The KdV equation is used in fluid mechanics to describe waves propagation in shallow water surfaces, while the KS equation models front propagation in reaction-diffusion systems. In this article, the boundary control of these equations is considered when they are posed on a bounded interval. Different choices of controls are studied for each equation.


Controllability Dispersive equations Higher-order partial differential equations Parabolic equations Stabilizability 
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  1. Armaou A, Christofides PD (2000) Feedback control of the Kuramoto-Sivashinsky equation. Physica D 137:49–61MathSciNetGoogle Scholar
  2. Cerpa E (2010) Null controllability and stabilization of a linear Kuramoto-Sivashinsky equation. Commun Pure Appl Anal 9:91–102MathSciNetGoogle Scholar
  3. Cerpa E (2014) Control of a Korteweg-de Vries equation: a tutorial. Math Control Rel Fields 4:45–99MathSciNetGoogle Scholar
  4. Cerpa E, Coron J-M (2013) Rapid stabilization for a Korteweg-de Vries equation from the left dirichlet boundary condition. IEEE Trans Autom Control 58:1688–1695MathSciNetGoogle Scholar
  5. Cerpa E, Mercado A (2011) Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation. J Differ Equ 250:2024–2044MathSciNetGoogle Scholar
  6. Cerpa E, Rivas I, Zhang B-Y (2013) Boundary controllability of the Korteweg-de Vries equation on a bounded domain. SIAM J Control Optim 51:2976–3010MathSciNetGoogle Scholar
  7. Christofides PD, Armaou A (2000) Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control. Syst Control Lett 39:283–294MathSciNetGoogle Scholar
  8. Coron JM (2007) Control and nonlinearity. American Mathematical Society, ProvidenceGoogle Scholar
  9. Coron J-M, Crépeau E (2004) Exact boundary controllability of a nonlinear KdV equation with critical lengths. J Eur Math Soc 6:367–398Google Scholar
  10. Glass O, Guerrero S (2008) Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit. Asymptot Anal 60:61–100MathSciNetGoogle Scholar
  11. Glass O, Guerrero S (2010) Controllability of the KdV equation from the right Dirichlet boundary condition. Syst Control Lett 59:390–395MathSciNetGoogle Scholar
  12. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag 39:422–443Google Scholar
  13. Krstic M (2009) Delay compensation for nonlinear, adaptive, and PDE systems. Birkhauser, BostonGoogle Scholar
  14. Kuramoto Y, Tsuzuki T (1975) On the formation of dissipative structures in reaction-diffusion systems. Theor Phys 54:687–699Google Scholar
  15. Lin Guo Y-J (2002) Null boundary controllability for a fourth order parabolic equation. Taiwan J Math 6:421–431Google Scholar
  16. Liu W-J, Krstic M (2001) Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation. Nonlinear Anal Ser A Theory Methods 43:485–507MathSciNetGoogle Scholar
  17. Perla Menzala G, Vasconcellos CF, Zuazua E (2002) Stabilization of the Korteweg-de Vries equation with localized damping. Q Appl Math LX:111–129Google Scholar
  18. Rosier L (1997) Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim Calc Var 2:33–55MathSciNetGoogle Scholar
  19. Rosier L, Zhang B-Y (2009) Control and stabilization of the Korteweg-de Vries equation: recent progresses. J Syst Sci Complex 22:647–682MathSciNetGoogle Scholar
  20. Sivashinsky GI (1977) Nonlinear analysis of hydrodynamic instability in laminar flames – I derivation of basic equations. Acta Astronaut 4:1177–1206MathSciNetGoogle Scholar
  21. Smyshlyaev A, Krstic M (2010) Adaptive control of parabolic PDEs. Princeton University Press, PrincetonGoogle Scholar
  22. Zhang BY (1999) Exact boundary controllability of the Korteweg-de Vries equation. SIAM J Control Optim 37:543–565MathSciNetGoogle Scholar

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© Springer-Verlag London 2015

Authors and Affiliations

  • Eduardo Cerpa
    • 1
  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile