Abstract
We numerically examine the cost of the null boundary control for the transport diffusion equation y t − εy xx + My x = 0, x ∈ (0, L), t ∈ (0, T) with respect to the positive parameter ε. It is known that this cost is uniformly bounded with respect to ε if T ≥ T M with \(T_M\in [1, 2\sqrt {3}]L/M\) if M > 0 and if \(T_M\in [2\sqrt {2},2(1+\sqrt {3})]L/\vert M\vert \) if M < 0. We propose a method to approximate the underlying observability constant and then conjecture, through numerical computations, the minimal time of controllability T M leading to a uniformly bounded cost. Several experiments for M ∈{−1, 1} are performed and discussed.
Dedicated to Prof. Enrique Fernández-Cara on the occasion of his 60th birthday.
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References
Amirat, Y., Munch, A.: On the controllability of an advection-diffusion equation with respect to the diffusion parameter: asymptotic analysis and numerical simulations. Acta Math. Appl. Sin. (to appear)
Boyer, F.: On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems. In: CANUM 2012, 41e Congrès National d’Analyse Numérique. ESAIM Proceedings, vol. 41, pp. 15–58. EDP Science, Les Ulis (2013)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)
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, 429–484 (1994)
Chapelle, D., Bathe, K.-J.: The inf-sup test. Comput. Struct. 47, 537–545 (1993)
Chatelin, F.: Eigenvalues of Matrices. Classics in Applied Mathematics, vol. 71. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2012). With exercises by Mario Ahués and the author, Translated with additional material by Walter Ledermann, Revised reprint of the 1993 edition [ MR1232655]
Cîndea, N., Münch, A.: A mixed formulation for the direct approximation of the control of minimal L 2-norm for linear type wave equations. Calcolo 52, 245–288 (2015)
Coron, J.-M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Providence (2007)
Coron, J.-M., Guerrero, S.: Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44, 237–257 (2005)
Duprez, M., Münch, A.: Numerical estimations of the cost of boundary controls for the one dimensional heat equation (in preparation)
Fernández-Cara, E., Münch, A.: Numerical exact controllability of the 1D heat equation: duality and Carleman weights. J. Optim. Theory Appl. 163, 253–285 (2014)
Fernández-Cara, E., González-Burgos, M., de Teresa, L.: Boundary controllability of parabolic coupled equations. J. Funct. Anal. 259, 1720–1758 (2010)
Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996)
Glass, O.: A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Anal. 258, 852–868 (2010)
Labbé, S., Trélat, E.: Uniform controllability of semidiscrete approximations of parabolic control systems. Syst. Control Lett. 55, 597–609 (2006)
Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20, 335–356 (1995)
Lions, J.-L.: Perturbations singulières dans les problèmes aux limites et en contrôle optimal. Lecture Notes in Mathematics, vol. 323. Springer, Berlin (1973)
Lissy, P.: A link between the cost of fast controls for the 1-d heat equation and the uniform controllability of a 1-d transport-diffusion equation. C.R. Math. 350, 591–595 (2012)
Lissy, P.: Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation. J. Differ. Equ. 259, 5331–5352 (2015)
Montaner, S., Munch, A.: Approximation of controls for the linear wave equation: a first order mixed formulation (submitted)
Münch, A., Souza, D.: A mixed formulation for the direct approximation of L 2-weighted controls for the linear heat equation. Adv. Comput. Math. 42, 85–125 (2016)
Münch, A., Souza, D.A.: Inverse problems for linear parabolic equations using mixed formulations – Part 1: theoretical analysis. J. Inverse Ill-Posed Probl. 25, 445–468 (2017)
Münch, A., Zuazua, E.: Numerical approximation of null controls for the heat equation: ill-posedness and remedies. Inverse Prob. 26, 085018, 39 (2010)
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Münch, A. (2018). Numerical Estimations of the Cost of Boundary Controls for the Equation y t − εy xx + My x = 0 with Respect to ε . In: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. (eds) Recent Advances in PDEs: Analysis, Numerics and Control. SEMA SIMAI Springer Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97613-6_9
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