# American Institute of Mathematical Sciences

• Previous Article
A discrete hierarchy of double bracket equations and a class of negative power series
• MCRF Home
• This Issue
• Next Article
Optimal size of business and dividend strategy in a nonlinear model with refinancing and liquidation value
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-Khodja, A. Benabdallah, M. 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. Seidman, S. 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-Khodja, A. Benabdallah, M. 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. Seidman, S. 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
Difference between $C_S(\alpha)$ and the upper bound of [11] with respect to $\alpha$
Difference between $C_S(\alpha)$ and the lower bound of [12] with respect to $\alpha$
Difference between $C_H(\alpha)$ and the upper bound of [11] with respect to $\alpha$
Difference between $C_H(\alpha)$ and the lower bound of [12] with respect to $\alpha$
 [1] Víctor Hernández-Santamaría, Luz de Teresa. Robust Stackelberg controllability for linear and semilinear heat equations. Evolution Equations & Control Theory, 2018, 7 (2) : 247-273. doi: 10.3934/eect.2018012 [2] Zhen Wang, Xiong Li, Jinzhi Lei. Second moment boundedness of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2963-2991. doi: 10.3934/dcdsb.2014.19.2963 [3] Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039 [4] Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661 [5] Yanqing Wang, Donghui Yang, Jiongmin Yong, Zhiyong Yu. Exact controllability of linear stochastic differential equations and related problems. Mathematical Control & Related Fields, 2017, 7 (2) : 305-345. doi: 10.3934/mcrf.2017011 [6] Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074 [7] Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167 [8] Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001 [9] A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003 [10] Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 [11] Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029 [12] Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583 [13] Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203 [14] J. Húska, Peter Poláčik. Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 81-113. doi: 10.3934/dcds.2008.20.81 [15] Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493 [16] Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13 [17] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [18] Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe, Atsushi Yagi. An $L^p$-approach to singular linear parabolic equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 989-1008. doi: 10.3934/dcds.2008.22.989 [19] Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008 [20] Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004

2018 Impact Factor: 1.292

## Metrics

• PDF downloads (16)
• HTML views (8)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]