Open Access
Issue
ESAIM: COCV
Volume 26, 2020
Article Number 55
Number of page(s) 32
DOI https://doi.org/10.1051/cocv/2019033
Published online 10 September 2020
  1. F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parbolic systems. J. Evol. Equ. 9 (2009) 267–291. [CrossRef] [MathSciNet] [Google Scholar]
  2. F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: a survey. Math. Control Relat. Fields 1 (2011) 267–306. [Google Scholar]
  3. V. Barbu, Local controllability of the phase field system. Nonlinear Anal. 50 (2002) 363–372. [CrossRef] [MathSciNet] [Google Scholar]
  4. C. Bardos and L. Tartar, Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines. Arch. Ration. Mech. Anal. 50 (1973) 10–25. [Google Scholar]
  5. K. Beauchard and F. Marbach, Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations. Preprint arXiv:1712.09790 (2017). [Google Scholar]
  6. C. Caputo, T. Goudon and A.F. Vasseur, Solutions of the 4-species quadratic reaction-diffusion system are bounded and C, in any space dimension. Preprint arXiv:1709.05694 (2017). [Google Scholar]
  7. J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1992) 295–312. [CrossRef] [MathSciNet] [Google Scholar]
  8. J.-M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007), p. 136. [Google Scholar]
  9. J.-M. Coron, S. Guerrero, P. Martin and L. Rosier, Homogeneity applied to the controllability of a system of parabolic equations, in Proceedings 2015 European Control Conference (ECC 2015), Linz, Austria (2015) 2470–2475. [CrossRef] [Google Scholar]
  10. J.-M. Coron, S. Guerrero and L. Rosier, Null controllability of a parabolic system with a cubic coupling term. SIAM J. Control Optim. 48 (2010) 5629–5653. [Google Scholar]
  11. R. Denk, M. Hieber and J. Prüss, Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257 (2007) 193–224. [CrossRef] [MathSciNet] [Google Scholar]
  12. E. Fernández-Cara, A review of basic theoretical results concerning the Navier-Stokes and other similar equations. Bol. Soc. Esp. Mat. Apl. SeMA 32 (2005) 45–73. [Google Scholar]
  13. E. Fernández-Cara and S, Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1399–1446. [Google Scholar]
  14. E. Fernández-Cara, J. Limaco and S.B. de Menezes, Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control. ESAIM: COCV 22 (2016) 1017–1039. [CrossRef] [EDP Sciences] [Google Scholar]
  15. J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems. Arch. Ration. Mech. Anal. 218 (2015) 553–587. [Google Scholar]
  16. A.V. Fursikov and O. Yu. Imanuvilov. Controllability of evolution equations, Vol. 34 of Lecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). [Google Scholar]
  17. P. Gao, Null controllability with constraints on the state for the reaction-diffusion system. Comput. Math. Appl. 70 (2015) 776–788. [Google Scholar]
  18. M. González-Burgos and L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force. Port. Math. 67 (2010) 91–113. [CrossRef] [MathSciNet] [Google Scholar]
  19. O. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. 87 (2007) 408–437. [Google Scholar]
  20. D. Jerison and G. Lebeau. Nodal sets of sums of eigenfunctions, in Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1999) 223–239. [Google Scholar]
  21. J. Kanel, The Cauchy problem for a system of semilinear parabolic equations with balance conditions. Differentsial nye Uravneniya 20 (1984) 1753–1760. [Google Scholar]
  22. J. Kanel, Solvability in the large of a system of reaction-diffusion equations with the balance condition. Differentsial nye Uravneniya, 26 (1990) 448–458. [Google Scholar]
  23. K. Le Balc’h, Null-controllability of two species reaction-diffusion system with nonlinear coupling: a new duality method. Preprint arXiv:1802.09187 (2018). [Google Scholar]
  24. K. Le Balc’h, Controllability of a 4 × 4 quadratic reaction–diffusion system. J. Differ. Equ. 266 (2019) 3100–3188. [Google Scholar]
  25. J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM: COCV 18 (2012) 712–747. [CrossRef] [EDP Sciences] [Google Scholar]
  26. G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. In Séminaire sur les Équations aux Dérivées Partielles, 1994–1995, pages Exp. No. VII, 13. École Polytech., Palaiseau (1995). [Google Scholar]
  27. P. Lissy and E. Zuazua, Internal observability for coupled systems of linear partial differential equations. Preprint hal-01480301 (2018). [Google Scholar]
  28. Y. Liu, T. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model. ESAIM: COCV 19 (2013) 20–42. [CrossRef] [EDP Sciences] [Google Scholar]
  29. S. Micu and T. Takahashi, Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity. J. Differ. Equ. 264 (2018) 3664–3703. [Google Scholar]
  30. B. Perthame, Parabolic equations in biology. Growth, reaction, movement and diffusion. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham (2015). [CrossRef] [Google Scholar]
  31. M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78 (2010) 417–45. [CrossRef] [Google Scholar]
  32. T.I. Seidman, How violent are fast controls? Math. Control Signals Syst. 1 (1988) 89–95. [CrossRef] [Google Scholar]
  33. P. Souplet, Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth. J. Evolut. Equ. 18 (2018) 1713–1720. [CrossRef] [Google Scholar]
  34. Z. Wu, J.Yin and C. Wang, Elliptic & parabolic equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). [CrossRef] [Google Scholar]

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