Abstract
This article considers a hyperbolic equation perturbed by a vanishing viscosity term depending on a small parameter ε>0. We show that the resulting parabolic equation is null-controllable. Moreover, we provide uniform estimates, with respect to ε, for the parabolic controls and we prove their convergence to a control of the limit hyperbolic equation. The method we use is based on Fourier expansion of solutions and the analysis of a biorthogonal sequence to a family of complex exponential functions.
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Avdonin, S.A., Ivanov, S.A.: Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, Cambridge (1995)
Coron, J.-M., Guerrero, S.: Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44, 237–257 (2005)
Fattorini, H.O., 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, 45–69 (1974/75)
Fattorini, H.O., Russell, D.L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43, 272–292 (1971)
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)
Ingham, A.E.: A note on Fourier transform. J. Lond. Math. Soc. 9, 29–32 (1934)
Kahane, J.P.: Pseudo-Périodicité et Séries de Fourier Lacunaires. Ann. Sci. Ecole Norm. Super. 37, 93–95 (1962)
Komornik, V., Loreti, P.: Fourier Series in Control Theory. Springer, New York (2005)
Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)
López, A., Zhang, X., Zuazua, E.: Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations. J. Math. Pures Appl. 79, 741–808 (2000)
Micu, S.: Uniform boundary controllability of the semi-discrete 1-D wave equation with vanishing viscosity. SIAM J. Control Optim. 47, 2857–2885 (2008)
Micu, S., de Teresa, L.: A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle. Asymptot. Anal. 66, 139–160 (2010)
Miller, L.: Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218, 425–444 (2005)
Miller, L.: Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time. J. Differ. Equ. 204, 202–226 (2004)
Miller, L.: The control transmutation method and the cost of fast controls. SIAM J. Control Optim. 45, 762–772 (2006)
Paley, R.E.A.C., Wiener, N.: Fourier Transforms in Complex Domains. AMS Colloq. Publ., vol. 19. Am. Math. Soc., Providence (1934)
Russell, D.L.: A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52, 189–221 (1973)
Redheffer, R.M.: Completeness of sets of complex exponentials. Adv. Math. 24, 1–62 (1977)
Tucsnak, M., Tenenbaum, G.: New blow-up rates of fast controls for the Schrödinger and heat equations. J. Differ. Equ. 243, 70–100 (2007)
Tucsnak, M., Weiss, G.: Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts. Birkhäuser, Basel (2009)
Young, R.: An Introduction to Nonharmonic Fourier Series. Academic Press, San Diego (1980)
Zabczyk, J.: Mathematical Control Theory: An Introduction. Birkhäuser, Basel (1992)
Zuazua, E.: Propagation, Observation, control and numerical approximation of Waves approximated by finite difference methods. SIAM Rev. 47, 197–243 (2005)
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Communicated by L. Vega.
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Micu, S., Ortega, J.H. & Pazoto, A.F. Null-controllability of a Hyperbolic Equation as Singular Limit of Parabolic Ones. J Fourier Anal Appl 17, 991–1007 (2011). https://doi.org/10.1007/s00041-010-9168-8
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DOI: https://doi.org/10.1007/s00041-010-9168-8