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Numerical null controllability of the heat equation through a least squares and variational approach

Published online by Cambridge University Press:  13 February 2014

ARNAUD MÜNCH  and
PABLO PEDREGAL
ARNAUD MÜNCH
Affiliation:
Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand 2), UMR CNRS 6620, Campus des Cézeaux, 63177 Aubière, France email: arnaud.munch@math.univ-bpclermont.fr
PABLO PEDREGAL
Affiliation:
E.T.S. Ingenieros Industriales, Universidad de Castilla La Mancha, Campus de Ciudad Real, Spain email: pablo.pedregal@uclm.es
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Abstract

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This work is concerned with the numerical computation of null controls for the heat equation. The goal is to compute an approximation of controls that drives the solution from a prescribed initial state at t=0 to zero at t=T. In spite of the diffusion of the heat equation, recent developments indicate that this issue is difficult and still largely open. Most of the existing literature, concerned with controls of minimal L2-norm, make use of dual convex arguments and introduce backward adjoint system. In practice, the null control problem is then reduced to the minimization of a dual conjugate function with respect to the final condition of the adjoint state. As a consequence of the highly regularizing property of the heat kernel, this final condition – which may be seen as the Lagrange multiplier for the null controllability condition – does not belong to L2, but to a much larger space than can hardly be approximated by finite (discrete) dimensional basis. This phenomenon, unavoidable whatever be the numerical approximation used, strongly deteriorates the efficiency of minimization algorithms. In this work, we do not use duality arguments and in particular do not introduce any backward heat equation. For the boundary case, the approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time, in particular the null controllability condition at time T, and then finding among this class one element satisfying the heat equation. This second step is done by minimizing a convex functional among the admissible corrector functions of the heat equation. The inner case is performed in a similar way. We present the (variational) approach, discuss the main features of it and then describe some numerical experiments highlighting the interest of the method. The method holds in any dimension but, for the sake of simplicity, we provide details in the one-space dimensional case.

Keywords

Heat equationNull controllabilityNumerical approximationVariational approach
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 3 , June 2014 , pp. 277 - 306
Copyright
Copyright © Cambridge University Press 2014 

References

[1]Alessandrini, G. & Escauriaza, L. (2008) Null-controllability of one-dimensional parabolic equations. ESAIM Control Optim. Calc. Var. 14 (2), 284–293.CrossRefGoogle Scholar
[2]Ben Belgacem, F. & Kaber, S. M. (2011) On the Dirichlet boundary controllability of the 1-D heat equation: Semi-analytical calculations and ill-posedness degre. Inverse Probl. 27 (5), 055012, 19 pp.CrossRefGoogle Scholar
[3]Boyer, F., Hubert, F. & Le Rousseau, J. (2011) Uniform null-controllability properties for space/time-discretized parabolic equations. Numer. Math. 118 (4), 601–661.CrossRefGoogle Scholar
[4]Carthel, C., Glowinski, R. & Lions, J.-L. (1994) On exact and approximate boundary controllability for the heat equation: A numerical approach. J. Optim. Theory Appl. 82 (3), 429–484.CrossRefGoogle Scholar
[5]Coron, J. M. (2007) Control and Nonlinearity, AMS Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, Providence, RI.Google Scholar
[6]Coron, J. M. & Trélat, E. (2004) Global steady-state controllability of one-dimensional semi-linear heat equations. SIAM J. Control Optim. 43, 549–569.CrossRefGoogle Scholar
[7]Ervedoza, S. & Valein, J. (2010) On the observability of abstract time-discrete linear parabolic equations. Rev. Mat. Comput. 23 (1), 163–190.CrossRefGoogle Scholar
[8]Fattorini, H. O. & Russel, D. L. (1971) Exact controllability theorems for linear parabolic equation in one space dimension. Arch. Ration. Mech. 43, 272–292.CrossRefGoogle Scholar
[9]Fernández-Cara, E. & Münch, A. (2011) Numerical null controllability of a semi-linear 1D heat equation via a least squares reformulation. C.R. Acad. Sci. Paris Sér. I 349, 867–871.CrossRefGoogle Scholar
[10]Fernández-Cara, E. & Münch, A. (2012) Numerical null controllability of semi-linear 1D heat equations: Fixed points, least squares and Newton methods. Math. Control Relat. Fields 2 (3), 217–246.CrossRefGoogle Scholar
[11]Fernández-Cara, E. & Münch, A. (2013) Numerical null controllability of the 1D heat equation: Primal algorithms. Séma J. 61 (1), 49–78.Google Scholar
[12]Fernández-Cara, E. & Münch, A. (to appear) Numerical null controllability of the 1D heat equation: Carleman weights and duality. J. Opt. Th. Appl. doi: 10.1007/s10957-013-0517-z.Google Scholar
[13]Fernández-Cara, E. & Zuazua, E. (2000) Null and approximate controllability for weakly blowing up semilinear heat equations Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (5), 583–616.CrossRefGoogle Scholar
[14]Fursikov, A. V. & Yu, O. Imanuvilov (1996) Controllability of Evolution Equations, Lecture Notes Series, No. 34. Seoul National University, Korea, 163 pp.Google Scholar
[15]Glowinski, R. (1983) Numerical Methods for Nonlinear Variational Problems, Springer series in Computational Physics, Springer, New York, NY.Google Scholar
[16]Glowinski, R., Lions, J. L. & He, J. (2008) Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
[17]Hào, D. N. (1998) Methods for Inverse Heat Conduction Problems, Methods and Procedures in Mathematical Physics, 43, Peter Lang, Frankfurt am Main, Germany.CrossRefGoogle Scholar
[18]Kindermann, S. (1999) Convergence rates of the Hilbert Uniqueness Method via Tikhonov regularization. J. Optim. Theory Appl. 103 (3), 657–673.CrossRefGoogle Scholar
[19]Labbé, S. & Trélat, E. (2006) Uniform controllability of semi-discrete approximations of parabolic control systems. Syst. Control Lett. 55, 597–609.CrossRefGoogle Scholar
[20]Laroche, B., Martin, P. & Rouchon, P. (2000) Motion planning for the heat equation. Int. J. Robust Nonlinear Control 10, 629–643.3.0.CO;2-N>CrossRefGoogle Scholar
[21]Lasiecka, I. & Triggiani, R. (1991) Exact controllability of semilinear abstract systems with applications to waves and plates boundary control. Appl. Math. Optim. 23, 109–154.CrossRefGoogle Scholar
[22]Lasiecka, I. & Triggiani, R. (2000) Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems. Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, UK.Google Scholar
[23]Lebeau, G. & Robbiano, L. (1995) Contrôle exact de l'équation de la chaleur. Comm. Partial Differ. Equ. 20 (1–2), 335–356.CrossRefGoogle Scholar
[24]Lions, J. L. (1971) Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, NY.CrossRefGoogle Scholar
[25]Lions, J. L. (1988) Exact controllability, stabilizability and perturbations for distributed systems. SIAM Rev. 30, 1–68.CrossRefGoogle Scholar
[26]Micu, S. & Zuazua, E. (2011) On the regularity of null-controls of the linear 1-D heat equation. C. R. Acad. Sci. Paris, Ser. I 349, 673–677.CrossRefGoogle Scholar
[27]Micu, S. & Zuazua, E. (2011) Regularity issues for the null-controllability of the linear 1-D heat equation. Syst. Cont. Lett. 60 (6), 406–413.CrossRefGoogle Scholar
[28]Münch, A. & Pedregal, P. (2013) A least-squares formulation for the approximation of null controls for the Stokes system. C. R. Acad. Sci. Paris, Ser. I 351, 545–550.CrossRefGoogle Scholar
[29]Münch, A. & Zuazua, E. (2010) Numerical approximation of null controls for the heat equation: Ill-posedness and remedies. Inverse Probl. 26 (8), 085018, 39pp.CrossRefGoogle Scholar
[30]Pedregal, P. (2010) A variational perspective on controllability. Inverse Probl. 26 (1), 015004, 17pp.CrossRefGoogle Scholar
[31]Pedregal, P. (to appear) Erratum: “A variational perspective on controllability.” [Inverse Probl. 26 (1), 015004].CrossRefGoogle Scholar
[32]Russell, D. L. (1978) Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev. 20, 639–739.CrossRefGoogle Scholar