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A singular controllability problem with vanishing viscosity

Published online by Cambridge University Press:  10 December 2013

Ioan Florin Bugariu  and
Sorin Micu
Ioan Florin Bugariu
Affiliation:
Facultatea de Stiinte Exacte, Universitatea din Craiova, 200585, Romania. florinbugariu86@yahoo.com; sdmicu@yahoo.com
Sorin Micu
Affiliation:
Facultatea de Stiinte Exacte, Universitatea din Craiova, 200585, Romania. florinbugariu86@yahoo.com; sdmicu@yahoo.com
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Abstract

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.

Keywords

Wave equationnull-controllabilityvanishing viscositymoment problembiorthogonals
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 116 - 140
Copyright
© EDP Sciences, SMAI 2013

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References

S.A. Avdonin and S.A. Ivanov, Families of exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press (1995).
Brändle, C., Colorado, E., de Pablo, A. and Sánchez, U., A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013) 39–71. Google Scholar
Caffarelli, L. and Silvestre, L., An extension problem related to the fractional Laplacian. Commun. Partial Differ. Eqs. 32 (2007) 1245–1260. Google Scholar
T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equation. Oxford University Press Inc., New York (1998).
Chen, S. and Triggiani, R., Proof of Extensions of Two Conjectures on Structural Damping for Elastic Systems. Pacific J. Math. 136 (1989) 15–55. Google Scholar
Chen, S. and Triggiani, R., Characterization of Domains of Fractional Powers of Certain Operators Arising in Elastic Systems and Applications. J. Differ. Eqs. 88 (1990) 279–293. Google Scholar
Coron, J.M., Control and nonlinearity, Mathematical Surveys and Monographs. Amer. Math. Soc. Providence, RI 136 (2007). Google Scholar
Coron, J.M. and Guerrero, S., Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237–257. Google Scholar
Guan, Q.-Y. and Ma, Z.-M., Boundary problems for fractional Laplacians. Stoch. Dyn. 5 (2005) 385–424. Google Scholar
DiPerna, R.J., Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983) 27–70. Google Scholar
Edward, J., Ingham-type inequalities for complex frequencies and applications to control theory. J. Math. Appl. 324 (2006) 941–954. Google Scholar
Fattorini, H.O. and Russell, D.L., 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. Google Scholar
Fattorini, H.O. and Russell, D.L., Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971) 272–292. Google Scholar
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 (2010) 852–868. Google Scholar
Hansen, S.W., Bounds on Functions Biorthogonal to Sets of Complex Exponentials; Control of Dumped Elastic Systems. J. Math. Anal. Appl. 158 (1991) 487–508. Google Scholar
L. Ignat and E. Zuazua, 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.
Ignat, L. and Zuazua, E., Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47 (2009) 1366–1390. Google Scholar
Imbert, C., A non-local regularization of first order Hamilton-Jacobi equations, J. Differ. Eqs. 211 (2005) 218–246. Google Scholar
Ingham, A.E., A note on Fourier transform. J. London Math. Soc. 9 (1934) 29–32. CrossRefGoogle Scholar
Ingham, A.E., Some trigonometric inequalities with applications to the theory of series Math. Zeits. 41 (1936) 367–379. Google Scholar
Khapalov, A., Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls. ESAIM: COCV 4 (1999) 83–98. Google Scholar
V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer-Verlag, New-York (2005).
Léautaud, M., Uniform controllability of scalar conservation laws in the vanishing viscosity limit. SIAM J. Control Optim. 50 (2012) 1661–1699. Google Scholar
López, A., Zhang, X. and Zuazua, E., 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. Google Scholar
Micu, S., Ortega, J.H. and Pazoto, A.F., Null-controllability of a Hyperbolic Equation as Singular Limit of Parabolic Ones. J. Fourier Anal. Appl. 41 (2010) 991–1007. Google Scholar
Micu, S. and Rovenţa, I., Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity. ESAIM: COCV 18 (2012) 277–293. Google Scholar
Micu, S. and de Teresa, L., A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle, Asymptot. Anal. 66 (2010) 139–160. Google Scholar
Miller, L., Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425–444. Google Scholar
Paley, R.E.A.C. and Wiener, N., Fourier Transforms in Complex Domains. AMS Colloq. Publ. Amer. Math. Soc. New-York 19 (1934). Google Scholar
Rosier, L. and Rouchon, P., On the Controllability of a Wave Equation with Structural Damping. Int. J. Tomogr. Stat. 5 (2007) 79–84. Google Scholar
Russel, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equation. Stud. Appl. Math. 52 (1973) 189–221. Google Scholar
T.I. Seidman, 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.
Szász, O., Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen. Math. Ann. 77 (1916) 482–496. Google Scholar
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhuser Advanced Texts. Springer, Basel (2009).
R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, New-York (1980).
J. Zabczyk, Mathematical Control Theory: An Introduction. Birkhuser, Basel (1992).
Zuazua, E., Propagation, Observation, Control and Numerical Approximation of Waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. Google Scholar