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March  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-KhodjaA. BenabdallahM. Gonzalez-Burgos and 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation J. Math. Phys., 47 (2006), 082104, 9pp. doi: 10.1063/1.2235026.  Google Scholar

[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.  Google Scholar

[7]

L. Ho and 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.  Google Scholar

[8]

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

[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. doi: 10.1016/j.crma.2012.06.004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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, J. Phys. A, 37 (2004), R161-R208.  doi: 10.1088/0305-4470/37/31/R01.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

L. Robino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/9789814360036.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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.  doi: 10.1016/j.jde.2007.06.019.  Google Scholar

show all references

References:
[1]

F. Ammar-KhodjaA. BenabdallahM. Gonzalez-Burgos and 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation J. Math. Phys., 47 (2006), 082104, 9pp. doi: 10.1063/1.2235026.  Google Scholar

[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.  Google Scholar

[7]

L. Ho and 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.  Google Scholar

[8]

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

[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. doi: 10.1016/j.crma.2012.06.004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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, J. Phys. A, 37 (2004), R161-R208.  doi: 10.1088/0305-4470/37/31/R01.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

L. Robino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/9789814360036.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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.  doi: 10.1016/j.jde.2007.06.019.  Google Scholar

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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