Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type
- [1] CMLS Ecole Polytechnique 91128 Palaiseau Cedex France
Séminaire Laurent Schwartz — EDP et applications (2011-2012)
- Volume: 2011-2012, page 1-24
- ISSN: 2266-0607
Access Full Article
topAbstract
topHow to cite
topReferences
top- F. Alabau-Boussouira, P. Cannarsa, and G. Fragnelli. Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ., 6(2):161–204, 2006. Zbl1103.35052MR2227693
- S. Alinhac and C. Zuily. Uniqueness and nonuniqueness of the Cauchy problem for hyperbolic operators with double characteristics. Comm. Partial Differential Equations, 6(7):799–828, 1981. Zbl0482.35052MR623646
- Y. Almog. The stability of the normal state of superconductors in the presence of electric currents. SIAM J. Math. Anal., 40(2):824–850, 2008. Zbl1165.82029MR2438788
- K. Beauchard. Null controllability of Kolmogorov-type equations. (preprint), 2012. Zbl1291.93035
- K. Beauchard, P. Cannarsa, and R. Guglielmi. Null controllability of Grushin-type operators in dimension two. J. Eur. Math. Soc (to appear), 2011. Zbl1293.35148
- K. Beauchard and E. Zuazua. Some controllability results for the 2D Kolmogorov equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 26:1793–1815, 2009. Zbl1172.93005MR2566710
- A. Benabdallah, Y. Dermenjian, and J. Le Rousseau. Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem. J. Math. Anal. Appl., 336(2):865–887, 2007. Zbl1189.35349MR2352986
- A. Benabdallah, Y. Dermenjian, and J. Le Rousseau. On the controllability of linear parabolic equations with an arbitrary control location for stratified media. C. R. Math. Acad. Sci. Paris, 344(6):357–362, 2007. Zbl1115.35055MR2310670
- P. Cannarsa and L. de Teresa. Controllability of 1-D coupled degenerate parabolic equations. Electron. J. Differ. Equ., Paper No. 73:21 p., 2009. Zbl1178.35216MR2519898
- P. Cannarsa, G. Fragnelli, and D. Rocchetti. Null controllability of degenerate parabolic operators with drift. Netw. Heterog. Media, 2(4):695–715 (electronic), 2007. Zbl1140.93011MR2357764
- P. Cannarsa, G. Fragnelli, and D. Rocchetti. Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form. J. Evol. Equ., 8:583–616, 2008. Zbl1176.35108MR2460930
- P. Cannarsa, P. Martinez, and J. Vancostenoble. Persistent regional null controllability for a class of degenerate parabolic equations. Commun. Pure Appl. Anal., 3(4):607–635, 2004. Zbl1063.35092MR2106292
- P. Cannarsa, P. Martinez, and J. Vancostenoble. Null controllability of degenerate heat equations. Adv. Differential Equations, 10(2):153–190, 2005. Zbl1145.35408MR2106129
- P. Cannarsa, P. Martinez, and J. Vancostenoble. Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim., 47(1):1–19, 2008. Zbl1168.35025MR2373460
- P. Cannarsa, P. Martinez, and J. Vancostenoble. Carleman estimates and null controllability for boundary-degenerate parabolic operators. C. R. Math. Acad. Sci. Paris, 347(3-4):147–152, 2009. Zbl1162.35330MR2538102
- A. Doubova, E. Fernández-Cara, and E. Zuazua. On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim., 42 (3):798–819, 2002. Zbl1038.93041MR1939871
- A. Doubova, A. Osses, and J.-P. Puel. Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients. ESAIM Control Optim. Calc. Var., 8:621–661, 2002. Zbl1092.93006MR1932966
- T. Duyckaerts, X. Zhang, and E. Zuazua. On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré : Analyse Nonlinéaire, 25:1–41, 2008. Zbl1248.93031MR2383077
- Sylvain Ervedoza. Control and stabilization properties for a singular heat equation with an inverse-square potential. Comm. Partial Differential Equations, 33(10-12):1996–2019, 2008. Zbl1170.35331MR2475327
- C. Fabre, J.P. Puel, and E. Zuazua. Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh, 125A:31–61, 1995. Zbl0818.93032MR1318622
- H.O. Fattorini and D. Russel. Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal., 43:272–292, 1971. Zbl0231.93003MR335014
- E. Fernández-Cara and E. Zuazua. Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 17:583–616, 2000. Zbl0970.93023MR1791879
- E. Fernández-Cara and E. Zuazua. The cost of approximate controllability for heat equations: The linear case. Advances in Differential Equations, 5(4-6):465–514, 2000. Zbl1007.93034MR1750109
- E. Fernández-Cara and E. Zuazua. On the null controllability of the one-dimensional heat equation with BV coefficients. Computational and Applied Mathematics, 12:167–190, 2002. Zbl1119.93311MR2009951
- C. Flores and L. de Teresa. Carleman estimates for degenerate parabolic equations with first order terms and applications. C. R. Math. Acad. Sci. Paris, 348(7-8):391–396, 2010. Zbl1188.35032MR2607025
- A.V. Fursikov and O.Y. Imanuvilov. Controllability of evolution equations. Lecture Notes Series, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 34, 1996. Zbl0862.49004MR1406566
- N. Garofalo. Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension. J. Differential Equations, 104(1):117–146, 1993. Zbl0788.35051MR1224123
- M. González-Burgos and L. de Teresa. Some results on controllability for linear and nonlinear heat equations in unbounded domains. Adv. Differential Equations, 12 (11):1201–1240, 2007. Zbl1170.93007MR2372238
- L. Hörmander. Hypoelliptic second order differential equations. Acta Math., 119:147–171, 1967. Zbl0156.10701MR222474
- O.Y. Imanuvilov. Boundary controllability of parabolic equations. Uspekhi. Mat. Nauk, 48(3(291)):211–212, 1993. MR1243631
- O.Y. Imanuvilov. Controllability of parabolic equations. Mat. Sb., 186(6):109–132, 1995. Zbl0845.35040MR1349016
- O.Y. Imanuvilov and M. Yamamoto. Carleman estimate for a parabolic equation in Sobolev spaces of negative order and its applications. Control of Nonlinear Distributed Parameter Systems, G. Chen et al. eds., Marcel-Dekker, pages 113–137, 2000. Zbl0977.93041MR1817179
- G. Lebeau and L. Robbiano. Contrôle exact de l’équation de la chaleur. Comm. P.D.E., 20:335–356, 1995. Zbl0819.35071MR1312710
- G. Lebeau and J. Le Rousseau. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM:COCV (DOI:10.1051/cocv/2011168), 2011. Zbl1262.35206
- J.-L. Lions. Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Avant propos de P. Lelong. Dunod, Paris, 1968. Zbl0179.41801MR244606
- A. Lopez and E. Zuazua. Uniform null controllability for the one dimensional heat equation with rapidly oscillating periodic density. Annales IHP. Analyse non linéaire, 19 (5):543–580, 2002. Zbl1009.35009MR1922469
- P. Martinez and J. Vancostenoble. Carleman estimates for one-dimensional degenerate heat equations. J. Evol. Equ., 6(2):325–362, 2006. Zbl1179.93043MR2227700
- P. Martinez, J. Vancostonoble, and J.-P. Raymond. Regional null controllability of a linearized Crocco type equation. SIAM J. Control Optim., 42, no. 2:709–728, 2003. Zbl1037.93013MR1982289
- L. Miller. On the null-controllability of the heat equation in unbounded domains. Bulletin des Sciences Mathématiques, 129, 2:175–185, 2005. Zbl1079.35018MR2123266
- L. Miller. On exponential observability estimates for the heat semigroup with explicit rates. Rendiconti Lincei: Matematica e Applicazioni, 17, 4:351–366, 2006. Zbl1150.93006MR2287707
- J. Le Rousseau. Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. J. Differential Equations, 233(2):417–447, 2007. Zbl1128.35020MR2292514
- J. Vancostenoble and E. Zuazua. Null controllability for the heat equation with singular inverse-square potentials. J. Funct. Anal., 254(7):1864–1902, 2008. Zbl1145.93009MR2397877
- C. Villani. Hypocoercivity, volume 202. Mem. Amer. Math. Soc., 2009. Zbl1197.35004MR2562709
- Zabczyk. Mathematical control theory: an introduction. Birkhäuser, 2000. Zbl1071.93500MR2348543
- E. Zuazua. Approximate controllability of the semilinear heat equation: boundary control. International Conference in honour of Prof. R. Glowinski, Computational Sciences for the 21st Century, M.O. Bristeau et al. eds., John Wiley and Sons, pages 738–747, 1997. Zbl0916.93016
- E. Zuazua. Finite dimensional null-controllability of the semilinear heat equation. J.Math. Pures et Appl., 76:237–264, 1997. Zbl0872.93014MR1441986
- C. Zuily. Uniqueness and non-uniqueness in the Cauchy problem. Boston Basel Stuttgart Birkhäuser, 1983. Zbl0521.35003