AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Carleman Estimates, Observability Inequalities and Null Controllability for Interior Degenerate Non Smooth Parabolic Equations
About this Title
Genni Fragnelli, Dipartimento di Matematica, Università di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari, Italy and Dimitri Mugnai, Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
Publication: Memoirs of the American Mathematical Society
Publication Year:
2016; Volume 242, Number 1146
ISBNs: 978-1-4704-1954-7 (print); 978-1-4704-2946-1 (online)
DOI: https://doi.org/10.1090/memo/1146
Published electronically: February 29, 2016
Table of Contents
Chapters
- 1. Introduction
- 2. Mathematical tools and preliminary results
- 3. Carleman estimate for non degenerate parabolic problems with non smooth coefficient
- 4. Carleman estimate for degenerate non smooth parabolic problems
- 5. Observability inequalities and application to null controllability
- 6. Linear and Semilinear Extensions
- 7. Final Comments
- A. Rigorous derivation of Lemma 3.5
Abstract
We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on observability results through Carleman estimates for the associated adjoint problem. The novelties of the present paper are two. First, the coefficient of the leading operator only belongs to a Sobolev space. Second, the degeneracy point is allowed to lie even in the interior of the control region, so that no previous result can be adapted to this situation; however, different cases can be handled, and new controllability results are established as a consequence.- El Mustapha Ait Ben Hassi, Farid Ammar Khodja, Abdelkarim Hajjaj, and Lahcen Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory 2 (2013), no. 3, 441–459. MR 3093224, DOI 10.3934/eect.2013.2.441
- El Mustapha Ait Ben Hassi, Farid Ammar Khodja, Abdelkarim Hajjaj, and Lahcen Maniar, Null controllability of degenerate parabolic cascade systems, Port. Math. 68 (2011), no. 3, 345–367. MR 2832802, DOI 10.4171/PM/1895
- F. Alabau-Boussouira, P. Cannarsa, and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ. 6 (2006), no. 2, 161–204. MR 2227693, DOI 10.1007/s00028-006-0222-6
- Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, Basel, 2001. MR 1886588
- K. Beauchard, Null controllability of Kolmogorov-type equations, Math. Control Signals Systems 26 (2014), no. 1, 145–176. MR 3163490, DOI 10.1007/s00498-013-0110-x
- K. Beauchard, P. Cannarsa, and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 1, 67–101. MR 3141729, DOI 10.4171/JEMS/428
- K. Beauchard and E. Zuazua, Some controllability results for the 2D Kolmogorov equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, 1793–1815. MR 2566710, DOI 10.1016/j.anihpc.2008.12.005
- Mourad Bellassoued and Masahiro Yamamoto, Carleman estimates and an inverse heat source problem for the thermoelasticity system, Inverse Problems 27 (2011), no. 1, 015006, 18. MR 2746409, DOI 10.1088/0266-5611/27/1/015006
- Aziz Belmiloudi, Nonlinear optimal control problems of degenerate parabolic equations with logistic time-varying delays of convolution type, Nonlinear Anal. 63 (2005), no. 8, 1126–1152. MR 2211586, DOI 10.1016/j.na.2005.05.033
- Assia Benabdallah, Yves Dermenjian, and Jérôme 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 (2007), no. 2, 865–887. MR 2352986, DOI 10.1016/j.jmaa.2007.03.024
- A. Bensoussan, G. Da Prato, M. C. Delfout, S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems and Control: Foundations and applications, Birkhäuser, 1993.
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
- Jean-Marie Buchot and Jean-Pierre Raymond, A linearized model for boundary layer equations, Optimal control of complex structures (Oberwolfach, 2000) Internat. Ser. Numer. Math., vol. 139, Birkhäuser, Basel, 2002, pp. 31–42. MR 1901628
- Piermarco Cannarsa and Luz de Teresa, Controllability of 1-D coupled degenerate parabolic equations, Electron. J. Differential Equations (2009), No. 73, 21. MR 2519898
- Piermarco Cannarsa and Genni Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations (2006), No. 136, 20. MR 2276561
- Piermarco Cannarsa, Genni Fragnelli, and Dario Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ. 8 (2008), no. 4, 583–616. MR 2460930, DOI 10.1007/s00028-008-0353-34
- Piermarco Cannarsa, Genni Fragnelli, and Dario Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media 2 (2007), no. 4, 695–715. MR 2357764, DOI 10.3934/nhm.2007.2.695
- P. Cannarsa, G. Fragnelli, and J. Vancostenoble, Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl. 320 (2006), no. 2, 804–818. MR 2225996, DOI 10.1016/j.jmaa.2005.07.006
- P. Cannarsa, G. Fragnelli, and J. Vancostenoble, Linear degenerate parabolic equations in bounded domains: controllability and observability, Systems, control, modeling and optimization, IFIP Int. Fed. Inf. Process., vol. 202, Springer, New York, 2006, pp. 163–173. MR 2241704, DOI 10.1007/0-387-33882-9_{1}5
- Thierry Cazenave and Alain Haraux, An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and its Applications, vol. 13, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors. MR 1691574
- Igor Chueshov, Irena Lasiecka, and Daniel Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations 21 (2009), no. 2, 269–314. MR 2506664, DOI 10.1007/s10884-009-9132-y
- Anna 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 (2002), 621–661. A tribute to J. L. Lions. MR 1932966, DOI 10.1051/cocv:2002047
- Hassan Emamirad, Gisèle Ruiz Goldstein, and Jerome A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc. 140 (2012), no. 6, 2043–2052. MR 2888192, DOI 10.1090/S0002-9939-2011-11069-4
- Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
- L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Unique continuation for Schrödinger evolutions, with applications to profiles of concentration and traveling waves, Comm. Math. Phys. 305 (2011), no. 2, 487–512. MR 2805469, DOI 10.1007/s00220-011-1256-3
- Enrique Fernández-Cara, Manuel González-Burgos, Sergio Guerrero, and Jean-Pierre Puel, Null controllability of the heat equation with boundary Fourier conditions: the linear case, ESAIM Control Optim. Calc. Var. 12 (2006), no. 3, 442–465. MR 2224822, DOI 10.1051/cocv:2006010
- Enrique Fernández-Cara and Enrique Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 5, 583–616 (English, with English and French summaries). MR 1791879, DOI 10.1016/S0294-1449(00)00117-7
- Enrique Fernández-Cara and Enrique Zuazua, On the null controllability of the one-dimensional heat equation with BV coefficients, Comput. Appl. Math. 21 (2002), no. 1, 167–190. Special issue in memory of Jacques-Louis Lions. MR 2009951
- Wendell H. Fleming and Michel Viot, Some measure-valued Markov processes in population genetics theory, Indiana Univ. Math. J. 28 (1979), no. 5, 817–843. MR 542340, DOI 10.1512/iumj.1979.28.28058
- Carmelo Flores and Luz de Teresa, Carleman estimates for degenerate parabolic equations with first order terms and applications, C. R. Math. Acad. Sci. Paris 348 (2010), no. 7-8, 391–396 (English, with English and French summaries). MR 2607025, DOI 10.1016/j.crma.2010.01.007
- Genni Fragnelli, Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), no. 3, 687–701. MR 3010677, DOI 10.3934/dcdss.2013.6.687
- G. Fragnelli, G. Ruiz Goldstein, J.A. Goldstein, S. Romanelli, Generators with interior degeneracy on spaces of $L^2$ type, Electron. J. Differential Equations 2012 (2012), 1–30.
- Genni Fragnelli and Dimitri Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal. 2 (2013), no. 4, 339–378. MR 3199737, DOI 10.1515/anona-2013-0015
- A. V. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. MR 1406566
- Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
- Herbert Koch and Daniel Tataru, Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients, Comm. Partial Differential Equations 34 (2009), no. 4-6, 305–366. MR 2530700, DOI 10.1080/03605300902740395
- Oleg Yu. Imanuvilov and Masahiro Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci. 39 (2003), no. 2, 227–274. MR 1987865
- Oleg Yu. Imanuvilov and Masahiro Yamamoto, An inverse problem and an observability inequality for the Lamé system with stress boundary condition, Appl. Anal. 88 (2009), no. 5, 711–733. MR 2547601, DOI 10.1080/00036810802556779
- Jérôme Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations 233 (2007), no. 2, 417–447. MR 2292514, DOI 10.1016/j.jde.2006.10.005
- Jérôme Le Rousseau and Gilles Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var. 18 (2012), no. 3, 712–747. MR 3041662, DOI 10.1051/cocv/2011168
- Jérôme Le Rousseau and Luc Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 953–990. MR 2591978, DOI 10.1007/s00205-009-0242-9
- G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations 20 (1995), no. 1-2, 335–356 (French). MR 1312710, DOI 10.1080/03605309508821097
- Suzanne M. Lenhart and Jiong Min Yong, Optimal control for degenerate parabolic equations with logistic growth, Nonlinear Anal. 25 (1995), no. 7, 681–698. MR 1341521, DOI 10.1016/0362-546X(94)00179-L
- P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ. 6 (2006), no. 2, 325–362. MR 2227700, DOI 10.1007/s00028-006-0214-6
- Sorin Micu and Enrique Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1635–1659. MR 1806726, DOI 10.1090/S0002-9947-00-02665-9
- B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific & Technical, Harlow, 1990. MR 1069756
- J.-P. Raymond and M. Vanninathan, Null controllability in a heat-solid structure model, Appl. Math. Optim. 59 (2009), no. 2, 247–273. MR 2480782, DOI 10.1007/s00245-008-9053-x
- Mikko Salo and Leo Tzou, Carleman estimates and inverse problems for Dirac operators, Math. Ann. 344 (2009), no. 1, 161–184. MR 2481057, DOI 10.1007/s00208-008-0301-9
- Mikko Salo and Leo Tzou, Inverse problems with partial data for a Dirac system: a Carleman estimate approach, Adv. Math. 225 (2010), no. 1, 487–513. MR 2669360, DOI 10.1016/j.aim.2010.03.003
- Norio Shimakura, Partial differential operators of elliptic type, Translations of Mathematical Monographs, vol. 99, American Mathematical Society, Providence, RI, 1992. Translated and revised from the 1978 Japanese original by the author. MR 1168472
- Andreas Stahel, Degenerate semilinear parabolic equations, Differential Integral Equations 5 (1992), no. 3, 683–691. MR 1157496
- Michael E. Taylor, Partial differential equations I. Basic theory, 2nd ed., Applied Mathematical Sciences, vol. 115, Springer, New York, 2011. MR 2744150
- Bin Wu, Carleman estimate for a strongly damped wave equation and applications to an inverse problem, Math. Methods Appl. Sci. 35 (2012), no. 4, 427–437. MR 2898879, DOI 10.1002/mma.1570
- Masahiro Yamamoto and Ying Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Problems 28 (2012), no. 10, 105010, 10. MR 2987913, DOI 10.1088/0266-5611/28/10/105010