Free Access
Issue |
ESAIM: COCV
Volume 20, Number 1, January-March 2014
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Page(s) | 116 - 140 | |
DOI | https://doi.org/10.1051/cocv/2013057 | |
Published online | 10 December 2013 |
- 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). [Google Scholar]
- C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013) 39–71. [Google Scholar]
- L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Eqs. 32 (2007) 1245–1260. [CrossRef] [Google Scholar]
- T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equation. Oxford University Press Inc., New York (1998). [Google Scholar]
- S. Chen and R. Triggiani, Proof of Extensions of Two Conjectures on Structural Damping for Elastic Systems. Pacific J. Math. 136 (1989) 15–55. [CrossRef] [MathSciNet] [Google Scholar]
- S. Chen and R. Triggiani, Characterization of Domains of Fractional Powers of Certain Operators Arising in Elastic Systems and Applications. J. Differ. Eqs. 88 (1990) 279–293. [CrossRef] [Google Scholar]
- J.M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs. Amer. Math. Soc. Providence, RI 136 (2007). [Google Scholar]
- J.M. Coron and S. Guerrero, Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237–257. [MathSciNet] [Google Scholar]
- Q.-Y. Guan and Z.-M. Ma, Boundary problems for fractional Laplacians. Stoch. Dyn. 5 (2005) 385–424. [CrossRef] [MathSciNet] [Google Scholar]
- R.J. DiPerna, Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983) 27–70. [CrossRef] [Google Scholar]
- J. Edward, Ingham-type inequalities for complex frequencies and applications to control theory. J. Math. Appl. 324 (2006) 941–954. [Google Scholar]
- H.O. Fattorini and D.L. Russell, 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]
- H.O. Fattorini and D.L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971) 272–292. [Google Scholar]
- O. Glass, 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. [CrossRef] [MathSciNet] [Google Scholar]
- S.W. Hansen, Bounds on Functions Biorthogonal to Sets of Complex Exponentials; Control of Dumped Elastic Systems. J. Math. Anal. Appl. 158 (1991) 487–508. [CrossRef] [MathSciNet] [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. [Google Scholar]
- L. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47 (2009) 1366–1390. [CrossRef] [MathSciNet] [Google Scholar]
- C. Imbert, A non-local regularization of first order Hamilton-Jacobi equations, J. Differ. Eqs. 211 (2005) 218–246. [CrossRef] [Google Scholar]
- A.E. Ingham, A note on Fourier transform. J. London Math. Soc. 9 (1934) 29–32. [CrossRef] [Google Scholar]
- A.E. Ingham, Some trigonometric inequalities with applications to the theory of series Math. Zeits. 41 (1936) 367–379. [Google Scholar]
- A. Khapalov, Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls. ESAIM: COCV 4 (1999) 83–98. [CrossRef] [EDP Sciences] [Google Scholar]
- V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer-Verlag, New-York (2005). [Google Scholar]
- M. Léautaud, Uniform controllability of scalar conservation laws in the vanishing viscosity limit. SIAM J. Control Optim. 50 (2012) 1661–1699. [Google Scholar]
- A. López, X. Zhang and E. Zuazua, 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. [CrossRef] [MathSciNet] [Google Scholar]
- S. Micu, J.H. Ortega and A.F. Pazoto, Null-controllability of a Hyperbolic Equation as Singular Limit of Parabolic Ones. J. Fourier Anal. Appl. 41 (2010) 991–1007. [Google Scholar]
- S. Micu and I. Rovenţa, Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity. ESAIM: COCV 18 (2012) 277–293. [CrossRef] [EDP Sciences] [Google Scholar]
- S. Micu and L. de Teresa, 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]
- L. Miller, Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425–444. [CrossRef] [MathSciNet] [Google Scholar]
- R.E.A.C. Paley and N. Wiener, Fourier Transforms in Complex Domains. AMS Colloq. Publ. Amer. Math. Soc. New-York 19 (1934). [Google Scholar]
- L. Rosier and P. Rouchon, On the Controllability of a Wave Equation with Structural Damping. Int. J. Tomogr. Stat. 5 (2007) 79–84. [MathSciNet] [Google Scholar]
- D.L. Russel, 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. [Google Scholar]
- O. Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen. Math. Ann. 77 (1916) 482–496. [CrossRef] [MathSciNet] [Google Scholar]
- M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhuser Advanced Texts. Springer, Basel (2009). [Google Scholar]
- R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, New-York (1980). [Google Scholar]
- J. Zabczyk, Mathematical Control Theory: An Introduction. Birkhuser, Basel (1992). [Google Scholar]
- E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. [CrossRef] [MathSciNet] [Google Scholar]
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