May  2020, 13(5): 1441-1472. doi: 10.3934/dcdss.2020082

Precise estimates for biorthogonal families under asymptotic gap conditions

1. 

Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 1, 00133 Roma, Italy

2. 

Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier Toulouse Ⅲ, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France

* Corresponding author

Received  June 2017 Revised  February 2018 Published  June 2019

Fund Project: This research was partly supported by the Institut Mathematique de Toulouse and Istituto Nazionale di Alta Matematica through funds provided by the national group GNAMPA and the GDRE CONEDP

A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell [12, 13], which is based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behavior: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic one. The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and thus to measure the influence of the 'bad' gap condition and the good influence of the 'good' asymptotic one. To achieve our goals, we extend some of the general results by Fattorini-Russell [12, 13] concerning biorthogonal families, using complex analysis techniques that were developed by Seidman [36], Güichal [20], Tenenbaum-Tucsnak [37] and Lissy [27, 28].

Citation: Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. Precise estimates for biorthogonal families under asymptotic gap conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1441-1472. doi: 10.3934/dcdss.2020082
References:
[1]

F. Ammar KhodjaA. BenabdallahM. González-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.

[2]

K. BeauchardP. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101.  doi: 10.4171/JEMS/428.

[3]

K. BeauchardL. Miller and M. Morancey, 2D Grushin-type equations: Minimal time and null controllable data, J. Differ. Equations, 259 (2015), 5813-5845.  doi: 10.1016/j.jde.2015.07.007.

[4]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.

[5]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Mem. Am. Math. Soc., 239 (2016), ix+209 pp. doi: 10.1090/memo/1133.

[6]

P. CannarsaP. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.

[7]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, to appear in ESAIM, Control Optim. Calc. Var..

[8]

J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257. 

[9]

J. Dardé and S. Ervedoza, On the reachable set for the one-dimensional heat equation, SIAM J. Control Optim., 56 (2018), 1692–1715, arXiv: 1609.02692. doi: 10.1137/16M1093215.

[10]

S. Ervedoza and E. Zuazua, Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., 202 (2011), 975-1017.  doi: 10.1007/s00205-011-0445-8.

[11]

W. N. Everitt, A catalogue of Sturm-Liouville differential equations, in Sturm-Liouville Theory, Birkhäuser, (2005), 271–331. doi: 10.1007/3-7643-7359-8_12.

[12]

H. O. Fattorini and D. L. Russel, Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[13]

H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.

[14]

H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, 13 (1975), 1-13.  doi: 10.1137/0313001.

[15]

E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. 

[16]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University, Seoul, 1996.

[17]

O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035.

[18]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.

[19]

M. Gueye and P. Lissy, Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation, ESAIM Control Optim. Calc. Var., 22 (2016), 1184-1203.  doi: 10.1051/cocv/2016036.

[20]

E. N. 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.

[21]

S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U.

[22]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations dune plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. 

[23]

E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Band 1: Gewöhnliche Differentialgleichungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977.

[24]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005.

[25]

N. N. Lebedev, Special Functions and Their Applications, Dover Publications, New York, 1972.

[26]

J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249.  doi: 10.1007/BF02788145.

[27]

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.

[28]

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

[29]

P. MartinL. Rosier and P. Rouchon, Null controllability of one-dimensional parabolic equations using flatness, Automatica J. IFAC, 50 (2014), 3067-3076.  doi: 10.1016/j.automatica.2014.10.049.

[30]

P. MartinL. Rosier and P. Rouchon, On the reachable states for the boundary control of the heat equation, Applied Mathematics Research eXpress, 2016 (2016), 181-216.  doi: 10.1093/amrx/abv013.

[31]

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

[32]

R. M. Redheffer, Elementary remarks on completeness, Duke Math. J., 35 (1968), 103-116.  doi: 10.1215/S0012-7094-68-03511-4.

[33]

L. Schwartz, Étude des Sommes D'exponentielles, deuxième édition, Hermann, Paris, 1959.

[34]

T. I. Seidman, Time-invariance of the reachable set for linear control problems, J. Math. Anal. Appl., 72 (1979), 17-20.  doi: 10.1016/0022-247X(79)90271-3.

[35]

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

[36]

T. I. 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.

[37]

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

[38]

G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, 1944.

[39]

R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.

show all references

References:
[1]

F. Ammar KhodjaA. BenabdallahM. González-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.

[2]

K. BeauchardP. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101.  doi: 10.4171/JEMS/428.

[3]

K. BeauchardL. Miller and M. Morancey, 2D Grushin-type equations: Minimal time and null controllable data, J. Differ. Equations, 259 (2015), 5813-5845.  doi: 10.1016/j.jde.2015.07.007.

[4]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.

[5]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Mem. Am. Math. Soc., 239 (2016), ix+209 pp. doi: 10.1090/memo/1133.

[6]

P. CannarsaP. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.

[7]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, to appear in ESAIM, Control Optim. Calc. Var..

[8]

J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257. 

[9]

J. Dardé and S. Ervedoza, On the reachable set for the one-dimensional heat equation, SIAM J. Control Optim., 56 (2018), 1692–1715, arXiv: 1609.02692. doi: 10.1137/16M1093215.

[10]

S. Ervedoza and E. Zuazua, Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., 202 (2011), 975-1017.  doi: 10.1007/s00205-011-0445-8.

[11]

W. N. Everitt, A catalogue of Sturm-Liouville differential equations, in Sturm-Liouville Theory, Birkhäuser, (2005), 271–331. doi: 10.1007/3-7643-7359-8_12.

[12]

H. O. Fattorini and D. L. Russel, Exact Controllability Theorems for Linear Parabolic Equations in One Space Dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[13]

H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.

[14]

H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, 13 (1975), 1-13.  doi: 10.1137/0313001.

[15]

E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential Equations, 5 (2000), 465-514. 

[16]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University, Seoul, 1996.

[17]

O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035.

[18]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.

[19]

M. Gueye and P. Lissy, Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation, ESAIM Control Optim. Calc. Var., 22 (2016), 1184-1203.  doi: 10.1051/cocv/2016036.

[20]

E. N. 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.

[21]

S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U.

[22]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations dune plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. 

[23]

E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Band 1: Gewöhnliche Differentialgleichungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977.

[24]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005.

[25]

N. N. Lebedev, Special Functions and Their Applications, Dover Publications, New York, 1972.

[26]

J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249.  doi: 10.1007/BF02788145.

[27]

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.

[28]

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

[29]

P. MartinL. Rosier and P. Rouchon, Null controllability of one-dimensional parabolic equations using flatness, Automatica J. IFAC, 50 (2014), 3067-3076.  doi: 10.1016/j.automatica.2014.10.049.

[30]

P. MartinL. Rosier and P. Rouchon, On the reachable states for the boundary control of the heat equation, Applied Mathematics Research eXpress, 2016 (2016), 181-216.  doi: 10.1093/amrx/abv013.

[31]

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

[32]

R. M. Redheffer, Elementary remarks on completeness, Duke Math. J., 35 (1968), 103-116.  doi: 10.1215/S0012-7094-68-03511-4.

[33]

L. Schwartz, Étude des Sommes D'exponentielles, deuxième édition, Hermann, Paris, 1959.

[34]

T. I. Seidman, Time-invariance of the reachable set for linear control problems, J. Math. Anal. Appl., 72 (1979), 17-20.  doi: 10.1016/0022-247X(79)90271-3.

[35]

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

[36]

T. I. 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.

[37]

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

[38]

G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, 1944.

[39]

R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.

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