Numerical Exact Controllability of the 1D Heat Equation: Duality and Carleman Weights

详细信息    查看全文

• 作者：Enrique Fernández-Cara ; Arnaud Münch
• 关键词：Heat equation ; Null controllability ; Numerical solution ; Duality ; 35K35 ; 65M12 ; 93B40
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2014
• 出版时间：October 2014
• 年：2014
• 卷：163
• 期：1
• 页码：253-285
• 全文大小：770 KB
• 参考文献：1. Fernández-Cara, E., Münch, A.: Strong convergent approximations of null controls for the 1D heat equations. SeMA J. 61, 49-8 (2013) CrossRef
  2. Russell, D.L.: A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52, 189-21 (1973)
  3. Russell, D.L.: Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev. 20, 639-39 (1978) CrossRef
  4. Lions, J.L.: Contr?labilité Exacte, Perturbations et Sabilisation de Systèmes Distribués, Recherches en Mathématiques Appliquées. Tomes 1 et 2. Masson, Paris (1988)
  5. Coron, J.M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Providence (2007)
  6. Alessandrini, G., Escauriaza, L.: Null-controllability of one-dimensional parabolic equations. ESAIM Control Optim. Calc. Var. 14, 284-93 (2008) CrossRef
  7. Lebeau, G., Robbiano, L.: Contr?le exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20, 335-56 (1995) CrossRef
  8. Imanuvilov, O.Y.: Controllability of parabolic equations (Russian). Mat. Sb. 186, 109-32 (1995)
  9. Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems. Encyclopedia of Mathematics and its Applications, vol. 74. Cambridge University Press, Cambridge (2000) CrossRef
  10. Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34, pp. 1-63. Seoul National University, Seoul (1996)
  11. Carthel, C., Glowinski, R., Lions, J.L.: On exact and approximate boundary controllabilities for the heat equation: a numerical approach. J. Optim. Theory Appl. 82, 429-84 (1994) CrossRef
  12. Glowinski, R., Lions, J.L., He, J.: Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Encyclopedia of Mathematics and its Applications, vol. 117. Cambridge University Press, Cambridge (2008) CrossRef
  13. Fernández-Cara, E., Guerrero, S.: Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45, 1399-446 (2006)
  14. Münch, A., Zuazua, E.: Numerical approximation of null controls for the heat equation: ill-posedness and remedies. Inverse Probl. 26, 39p (2010) CrossRef
  15. Kindermann, S.: Convergence rates of the Hilbert uniqueness method via Tikhonov regularization. J. Optim. Theory Appl. 103, 657-73 (1999) CrossRef
  16. Micu, S., Zuazua, E.: On the regularity of null-controls of the linear 1-D heat equation. C. R. Acad. Sci. Paris Ser. I 349, 673-77 (2011) CrossRef
  17. Ben Belgacem, F., Kaber, S.M.: On the Dirichlet boundary controllability of the 1-D heat equation: semi-analytical calculations and ill-posedness degree. Inverse Probl. 27, 23 (2011)
  18. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, Classics in Applied Mathematics, vol. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999) CrossRef
  19. Glowinski, R., Lions, J.L.: Exact and Approximate Controllability for Distributed Parameter Systems. Acta Numerica, pp. 159-33. Cambridge University Press, Cambridge (1996)
  20. Courant, R.: Variational methods for the solution of problems with equilibrium and vibration. Bull. Am. Math. Soc. 49, 1-3 (1943) CrossRef
  21. Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303-20 (1969) CrossRef
  22. Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283-98. Academic Press, New York (1969)
  23. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)
  24. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer-Verlag, New York (1999)
  25. Tr?ltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010)
  26. López, A., Zuazua, E.: Some New Results to the Null Controllability of the 1-D Heat Equation. Séminaire sur les Equations aux dérivées Partielles, Exp. No. VIII. Ecole Polytechnique, Palaiseau (1998)
  27. Zuazua, E.: Control and numerical approximation of the wave and heat equations. In: ICM2006, vol. III, pp. 1389-417. Madrid (2006)
  28. Boyer, F., Hubert, F., Le Rousseau, J.: Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications. SIAM J. Control Optim. 48, 5357-397 (2010) CrossRef
  29. Labbé, S., Trélat, E.: Uniform controllability of semi-discrete approximations of parabolic control systems. Syst. Control Lett. 55, 597-09 (2006) CrossRef
  30. Boyer, F., Hubert, F., Le Rousseau, J.: Uniform null-controllability properties for space/time-discretized parabolic equations. Numer. Math. 118, 601-61 (2011) CrossRef
  31. Ervedoza, S., Valein, J.: On the observability of abstract time-discrete linear parabolic equations. Rev. Mat. Complut. 23, 163-90 (2010) CrossRef
  32. Münch, A., Pedregal, P.: Numerical null controllability of the heat equation through a least squares and variational approach. Eur. J. Appl. Math. (2011). http://hal.archives-ouvertes.fr/hal-00724035
  33. Münch, A., Periago, F.: Optimal distribution of the internal null control for the 1D heat equation. J. Differ. Equ. 250, 95-11 (2011) CrossRef
  34. Benabdallah, A., Dermanjan, Y., Le Rousseau, J.: 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, 865-87 (2007) CrossRef
  35. Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35, 1297-327 (1997) CrossRef
  36. Fursikov, A.V.: Optimal Control of Distributed Systems: Theory and Applications. Translations of Mathematical Monographs, vol. 187. American Mathematical Society, Providence (2000)
  37. Münch, A., Pedregal, P.: A least-squares formulation for the approximation of null controls for the Stokes system. C. R. Acad. Sci. Paris Ser. I 351, 545-50 (2013) CrossRef
  38. Puel, J.P.: Global Carleman inequalities for the wave equations and applications to controllability and inverse problems. Cours Udine Mod1-06-04-542968, Udine (2011)
  39. Zhang, X.: Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control. Optim. 39, 812-34 (2010) CrossRef
  40. C?ndea, N., Fernández-Cara, E., Münch, A.: Numerical null controllability of the wave equation through primal methods and Carleman estimates. ESAIM Control Optim. Calc. Var. 19, 1076-108 (2013) CrossRef
  41. Fernández-Cara, E., Münch, A.: Numerical null controllability of a semi-linear 1D heat equation via a least squares reformulation. C. R. Acad. Sci. Paris Ser. I 349, 867-71 (2011) CrossRef
  42. Fernández-Cara, E., Münch, A.: Numerical null controllability of semi-linear 1D heat equations : fixed point, least squares and Newton methods. Math. Control Relat. Fields (AIMS) 3, 217-46 (2012)
  
• 作者单位：Enrique Fernández-Cara (1)
  Arnaud Münch (2)
  
  1. Departamento EDAN, University of Sevilla, Aptdo.?1160, 41080?, Sevilla, Spain
  2. Laboratoire de Mathématiques, UMR CNRS 6620, Université Blaise Pascal (Clermont-Ferrand?2), Campus des Cézeaux, 63177?, Aubière, France
  
• ISSN：1573-2878

文摘

This article is devoted to the numerical computation of distributed null controls for the 1D?heat equation. The goal is to compute a control that drives (a numerical approximation of) the solution from a prescribed initial state exactly to zero. We extend the earlier contribution of Carthel, Glowinski, and?Lions, which is devoted to the computation of controls of minimal square-integrable norm. We start from some constrained extremal problems (involving unbounded weights in time), introduced by?Fursikov and?Imanuvilov, and we apply appropriate duality techniques. Then, we provide numerical approximations of the associated dual problems, and apply conjugate gradient algorithms. Finally, several experiments are presented, and we highlight the influence of the weights and analyze this approach in terms of robustness and efficiency. Also, the results are compared with others in a previous article of the authors, where primal methods were considered.