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2017, 7(1): 21-40. doi: 10.3934/mcrf.2017002

Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory

Ceremade, Université Paris-Dauphine & CNRS, UMR 7534, PSL, 75016 Paris, France

* Corresponding author:Pierre Lissy

Received  December 2015 Revised  October 2016 Published  December 2016

In this paper, we construct some interesting Gevrey functions of order $α$ for every $α>1$ with compact support by a clever use of the Bray-Mandelbrojt iterative process. We then apply these results to the moment method, which will enable us to derive some upper bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type that partially improve the existing results proved in [P. Lissy, On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension SIAM J. Control Optim., 52(4), 2651-2676]. However this construction fails to improve the results of [G. Tenenbaum and M. Tucsnak, New blow-up rates of fast controls for the Schrödinger and heat equations, Journal of Differential Equations, 243 (2007), 70-100] in the precise case of the usual heat and Schrödinger equation.
Citation: Pierre Lissy. Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory. Mathematical Control & Related Fields, 2017, 7 (1) : 21-40. doi: 10.3934/mcrf.2017002
References:
[1]

F. Ammar-Khodja, A. Benabdallah, M. Gonzalez-Burgos, L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590. doi: 10.1016/j.matpur.2011.06.005.

[2]

J. -M. Coron, Control and Nonlinearity Volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, 2007.

[3]

H. O. Fattorini, D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., 43 (1971), 272-292.

[4]

E. Güichal, A lower bound of the norm of the control operator for the heat equation, J. Math. Anal. Appl., 110 (1985), 519-527. doi: 10.1016/0022-247X(85)90313-0.

[5]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation J. Math. Phys. 47 (2006), 082104, 9pp.

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators, I. Distribution Theory and Fourier Analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003. x+440 pp.

[7]

L. Ho, D. Russell, Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM J. Control Optim., 21 (1983), 614-640. doi: 10.1137/0321037.

[8]

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426.

[9]

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, C. R. Math. Acad. Sci. , Paris, 350 (2012), 591-595.

[10]

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 and Control Letters, 69 (2014), 98-102. doi: 10.1016/j.sysconle.2014.04.011.

[11]

P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676. doi: 10.1137/140951746.

[12]

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, J. Differential Equations, 259 (2015), 5331-5352. doi: 10.1016/j.jde.2015.06.031.

[13]

S. Mandelbrojt, Analytic functions and classes of infinitely differentiable functions, Rice Inst. Pamphlet, 29 (1942), 142 pp.

[14]

R. Metzler, J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161-R208. doi: 10.1088/0305-4470/37/31/R01.

[15]

L. Miller, How Violent are Fast Controls for Schrödinger and Plate Vibrations?, Arch. Ration. Mech. Anal., 172 (2004), 429-456. doi: 10.1007/s00205-004-0312-y.

[16]

L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226. doi: 10.1016/j.jde.2004.05.007.

[17]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Mathematics of Control, Signals and Systems, 18 (2006), 260-271. doi: 10.1007/s00498-006-0003-3.

[18]

R. M. Redheffer, Completeness of sets of complex exponentials, Advances in Math., 24 (1977), 1-62.

[19]

L. Robino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co. Inc. , River Edge, NJ, 1993.

[20]

W. Rudin, Real and Complex Analysis, Third edition. McGraw-Hill Book Co. , New York, 1987.

[21]

T. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim., 11 (1984), 145-152. doi: 10.1007/BF01442174.

[22]

T. Seidman, S. A. Avdonin, S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254. doi: 10.1007/BF02511154.

[23]

G. Tenenbaum, M. Tucsnak, New blow-up rates of fast controls for the Schrödinger and heat equations, Journal of Differential Equations, 243 (2007), 70-100. doi: 10.1016/j.jde.2007.06.019.

show all references

References:
[1]

F. Ammar-Khodja, A. Benabdallah, M. Gonzalez-Burgos, L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590. doi: 10.1016/j.matpur.2011.06.005.

[2]

J. -M. Coron, Control and Nonlinearity Volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, 2007.

[3]

H. O. Fattorini, D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., 43 (1971), 272-292.

[4]

E. Güichal, A lower bound of the norm of the control operator for the heat equation, J. Math. Anal. Appl., 110 (1985), 519-527. doi: 10.1016/0022-247X(85)90313-0.

[5]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation J. Math. Phys. 47 (2006), 082104, 9pp.

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators, I. Distribution Theory and Fourier Analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003. x+440 pp.

[7]

L. Ho, D. Russell, Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM J. Control Optim., 21 (1983), 614-640. doi: 10.1137/0321037.

[8]

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426.

[9]

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, C. R. Math. Acad. Sci. , Paris, 350 (2012), 591-595.

[10]

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 and Control Letters, 69 (2014), 98-102. doi: 10.1016/j.sysconle.2014.04.011.

[11]

P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676. doi: 10.1137/140951746.

[12]

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, J. Differential Equations, 259 (2015), 5331-5352. doi: 10.1016/j.jde.2015.06.031.

[13]

S. Mandelbrojt, Analytic functions and classes of infinitely differentiable functions, Rice Inst. Pamphlet, 29 (1942), 142 pp.

[14]

R. Metzler, J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161-R208. doi: 10.1088/0305-4470/37/31/R01.

[15]

L. Miller, How Violent are Fast Controls for Schrödinger and Plate Vibrations?, Arch. Ration. Mech. Anal., 172 (2004), 429-456. doi: 10.1007/s00205-004-0312-y.

[16]

L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226. doi: 10.1016/j.jde.2004.05.007.

[17]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Mathematics of Control, Signals and Systems, 18 (2006), 260-271. doi: 10.1007/s00498-006-0003-3.

[18]

R. M. Redheffer, Completeness of sets of complex exponentials, Advances in Math., 24 (1977), 1-62.

[19]

L. Robino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co. Inc. , River Edge, NJ, 1993.

[20]

W. Rudin, Real and Complex Analysis, Third edition. McGraw-Hill Book Co. , New York, 1987.

[21]

T. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim., 11 (1984), 145-152. doi: 10.1007/BF01442174.

[22]

T. Seidman, S. A. Avdonin, S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254. doi: 10.1007/BF02511154.

[23]

G. Tenenbaum, M. Tucsnak, New blow-up rates of fast controls for the Schrödinger and heat equations, Journal of Differential Equations, 243 (2007), 70-100. doi: 10.1016/j.jde.2007.06.019.

Figure 1.  Difference between $C_S(\alpha)$ and the upper bound of [11] with respect to $\alpha$
Figure 2.  Difference between $C_S(\alpha)$ and the lower bound of [12] with respect to $\alpha$
Figure 3.  Difference between $C_H(\alpha)$ and the upper bound of [11] with respect to $\alpha$
Figure 4.  Difference between $C_H(\alpha)$ and the lower bound of [12] with respect to $\alpha$
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