• 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-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 $\alpha$
[1]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[2]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[3]

Julian Koellermeier, Giovanni Samaey. Projective integration schemes for hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (2) : 353-387. doi: 10.3934/krm.2021008

[4]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[5]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

[6]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017

[7]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617

[8]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[9]

Hongjie Dong, Xinghong Pan. On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021049

[10]

Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021104

[11]

Miguel R. Nuñez-Chávez. Controllability under positive constraints for quasilinear parabolic PDEs. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021024

[12]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016

[13]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[14]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[15]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

[16]

Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021082

[17]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[18]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222

[19]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[20]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (61)
  • HTML views (51)
  • Cited by (2)

Other articles
by authors

[Back to Top]