The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1)\{½}. It follows that, under this assumption, our starting question has a positive answer.
Mots-clés : wave equation, null-controllability, vanishing viscosity, moment problem, biorthogonals
@article{COCV_2014__20_1_116_0,
author = {Bugariu, Ioan Florin and Micu, Sorin},
title = {A singular controllability problem with vanishing viscosity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {116--140},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013057},
mrnumber = {3182693},
zbl = {1282.93047},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2013057/}
}
TY - JOUR AU - Bugariu, Ioan Florin AU - Micu, Sorin TI - A singular controllability problem with vanishing viscosity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 116 EP - 140 VL - 20 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013057/ DO - 10.1051/cocv/2013057 LA - en ID - COCV_2014__20_1_116_0 ER -
%0 Journal Article %A Bugariu, Ioan Florin %A Micu, Sorin %T A singular controllability problem with vanishing viscosity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 116-140 %V 20 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013057/ %R 10.1051/cocv/2013057 %G en %F COCV_2014__20_1_116_0
Bugariu, Ioan Florin; Micu, Sorin. A singular controllability problem with vanishing viscosity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 116-140. doi : 10.1051/cocv/2013057. http://www.numdam.org/articles/10.1051/cocv/2013057/
[1] and , Families of exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press (1995). | MR | Zbl
[2] , , and , A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013) 39-71. | MR | Zbl
[3] and , An extension problem related to the fractional Laplacian. Commun. Partial Differ. Eqs. 32 (2007) 1245-1260. | MR | Zbl
[4] and , An Introduction to Semilinear Evolution Equation. Oxford University Press Inc., New York (1998). | MR | Zbl
[5] and , Proof of Extensions of Two Conjectures on Structural Damping for Elastic Systems. Pacific J. Math. 136 (1989) 15-55. | MR | Zbl
[6] and , Characterization of Domains of Fractional Powers of Certain Operators Arising in Elastic Systems and Applications. J. Differ. Eqs. 88 (1990) 279-293. | MR | Zbl
[7] , Control and nonlinearity, Mathematical Surveys and Monographs. Amer. Math. Soc. Providence, RI 136 (2007). | MR | Zbl
[8] and , Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237-257. | MR | Zbl
[9] and , Boundary problems for fractional Laplacians. Stoch. Dyn. 5 (2005) 385-424. | MR | Zbl
[10] , Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983) 27-70. | MR | Zbl
[11] , Ingham-type inequalities for complex frequencies and applications to control theory. J. Math. Appl. 324 (2006) 941-954. | MR | Zbl
[12] and , Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Q. Appl. Math. 32 (1974/75) 45-69. | MR | Zbl
[13] and , Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971) 272-292. | MR | Zbl
[14] , A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Anal. 258 (2010) 852-868. | MR | Zbl
[15] , Bounds on Functions Biorthogonal to Sets of Complex Exponentials; Control of Dumped Elastic Systems. J. Math. Anal. Appl. 158 (1991) 487-508. | MR | Zbl
[16] and , Dispersive Properties of Numerical Schemes for Nonlinear Schrödinger Equation, Foundations of Computational Mathematics, Santander 2005, London Math.l Soc. Lect. Notes. Edited by L.M. Pardo. Cambridge University Press 331 (2006) 181-207. | MR | Zbl
[17] and , Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47 (2009) 1366-1390. | MR | Zbl
[18] , A non-local regularization of first order Hamilton-Jacobi equations, J. Differ. Eqs. 211 (2005) 218-246. | MR | Zbl
[19] , A note on Fourier transform. J. London Math. Soc. 9 (1934) 29-32. | MR | Zbl
[20] , Some trigonometric inequalities with applications to the theory of series Math. Zeits. 41 (1936) 367-379. | MR | Zbl
[21] , Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls. ESAIM: COCV 4 (1999) 83-98. | Numdam | MR | Zbl
[22] and , Fourier Series in Control Theory. Springer-Verlag, New-York (2005). | MR | Zbl
[23] , Uniform controllability of scalar conservation laws in the vanishing viscosity limit. SIAM J. Control Optim. 50 (2012) 1661-1699. | MR | Zbl
[24] , and , Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equation. J. Math. Pures Appl. 79 (2000) 741-808. | MR | Zbl
[25] , and , Null-controllability of a Hyperbolic Equation as Singular Limit of Parabolic Ones. J. Fourier Anal. Appl. 41 (2010) 991-1007. | MR | Zbl
[26] and , Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity. ESAIM: COCV 18 (2012) 277-293. | Numdam | MR | Zbl
[27] and , A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle, Asymptot. Anal. 66 (2010) 139-160. | MR | Zbl
[28] , Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425-444. | MR | Zbl
[29] and , Fourier Transforms in Complex Domains. AMS Colloq. Publ. Amer. Math. Soc. New-York 19 (1934). | MR | Zbl
[30] and , On the Controllability of a Wave Equation with Structural Damping. Int. J. Tomogr. Stat. 5 (2007) 79-84. | MR
[31] , A unified boundary controllability theory for hyperbolic and parabolic partial differential equation. Stud. Appl. Math. 52 (1973) 189-221. | MR | Zbl
[32] , On uniform nullcontrollability and blow-up estimates, Chapter 15 in Control Theory of Partial Differential Equations, edited by O. Imanuvilov, G. Leugering, R. Triggiani and B.Y. Zhang. Chapman and Hall/CRC, Boca Raton (2005) 215-227. | MR | Zbl
[33] , Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen. Math. Ann. 77 (1916) 482-496. | JFM | MR
[34] and , Observation and Control for Operator Semigroups. Birkhuser Advanced Texts. Springer, Basel (2009). | Zbl
[35] , An Introduction to Nonharmonic Fourier Series. Academic Press, New-York (1980). | MR | Zbl
[36] , Mathematical Control Theory: An Introduction. Birkhuser, Basel (1992). | MR | Zbl
[37] , Propagation, Observation, Control and Numerical Approximation of Waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. | Zbl
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