F. Ammar-khodja, A. Benabdallah, M. Gonzalez-burgos, L. De-teresa, and L. , The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, Journal de Math??matiques Pures et Appliqu??es, vol.96, issue.6, pp.96-555, 2011.
DOI : 10.1016/j.matpur.2011.06.005
URL : https://hal.archives-ouvertes.fr/hal-00539825

J. Coron, Control and nonlinearity, of Mathematical Surveys and Monographs, 2007.
DOI : 10.1090/surv/136

H. O. Fattorini, R. , and D. L. , Exact controllability theorems for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis, vol.43, issue.4, pp.272-292, 1971.
DOI : 10.1007/BF00250466

E. Güichal, A lower bound of the norm of the control operator for the heat equation, Journal of Mathematical Analysis and Applications, vol.110, issue.2, p.519527, 1985.
DOI : 10.1016/0022-247X(85)90313-0

L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Mathematics, 2003.

L. Ho and D. Russell, Admissible Input Elements for Systems in Hilbert Space and a Carleson Measure Criterion, SIAM Journal on Control and Optimization, vol.21, issue.4, pp.614-640, 1983.
DOI : 10.1137/0321037

X. Guo and M. Xu, Some physical applications of fractional Schr??dinger equation, Journal of Mathematical Physics, vol.47, issue.8, p.82104, 2006.
DOI : 10.1063/1.2235026

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Mathematische Zeitschrift, vol.3, issue.2, pp.367-379, 1936.
DOI : 10.1112/plms/s2-38.1.458

P. Lissy, A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation, Comptes Rendus Mathematique, vol.350, issue.11-12, pp.11-12, 2012.
DOI : 10.1016/j.crma.2012.06.004
URL : https://hal.archives-ouvertes.fr/hal-00702497

P. Lissy, An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection???diffusion equation in the vanishing viscosity limit, Systems & Control Letters, vol.69, pp.98-102, 2014.
DOI : 10.1016/j.sysconle.2014.04.011

P. Lissy, On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension, SIAM Journal on Control and Optimization, vol.52, issue.4, pp.2651-2676
DOI : 10.1137/140951746
URL : https://hal.archives-ouvertes.fr/hal-00859458

P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport???diffusion equation, Journal of Differential Equations, vol.259, issue.10, pp.5331-5352, 2015.
DOI : 10.1016/j.jde.2015.06.031
URL : https://hal.archives-ouvertes.fr/hal-01133526

S. Mandelbrojt, Analytic functions and classes of infinitely differentiable functions, Rice Inst. Pamphlet, vol.29, issue.1, p.142, 1942.

R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A: Mathematical and General, vol.37, issue.31, pp.37-161, 2004.
DOI : 10.1088/0305-4470/37/31/R01

L. Miller, How Violent are Fast Controls for Schr???dinger and Plate Vibrations?, Archive for Rational Mechanics and Analysis, vol.172, issue.3, pp.429-456, 2004.
DOI : 10.1007/s00205-004-0312-y

L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, Journal of Differential Equations, vol.204, issue.1, pp.202-226, 2004.
DOI : 10.1016/j.jde.2004.05.007
URL : https://hal.archives-ouvertes.fr/hal-00002088

L. Miller, On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian, Mathematics of Control, Signals, and Systems, vol.65, issue.1, pp.260-271, 2006.
DOI : 10.1007/s00498-006-0003-3
URL : https://hal.archives-ouvertes.fr/hal-00008809

R. M. Redheffer, Completeness of sets of complex exponentials Advances in Math, pp.1-62, 1977.

L. Robino, Linear partial differential operators in Gevrey spaces, 1993.

W. Rudin, Real and complex analysis, 1966.

T. Seidman, Two results on exact boundary control of parabolic equations, Applied Mathematics & Optimization, vol.6, issue.1, p.145152, 1984.
DOI : 10.1007/BF01442174

G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schr??dinger and heat equations, Journal of Differential Equations, vol.243, issue.1, pp.70-100, 2007.
DOI : 10.1016/j.jde.2007.06.019