September & December  2018, 8(3&4): 899-933. doi: 10.3934/mcrf.2018040

Carleman commutator approach in logarithmic convexity for parabolic equations

Institut Denis Poisson, CNRS, UMR 7013, Université d’Orléans, BP 6759, 45067 Orléans Cedex 2, France

Received  August 2017 Revised  December 2017 Published  September 2018

Fund Project: This work is supported by the Région Centre (France) - THESPEGE Project.

In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.

Citation: Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040
References:
[1]

S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach Space, Comm. Pure Appl. Math., 16 (1963), 121-239.  doi: 10.1002/cpa.3160160204.

[2]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.

[3]

C. Bardos and K. D. Phung, Observation estimate for kinetic transport equations by diffusion approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 640-664.  doi: 10.1016/j.crma.2017.04.017.

[4]

C. Bardos and L. Tartar, Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal., 50 (1973), 10-25.  doi: 10.1007/BF00251291.

[5]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal., 7 (2002), 585-599.  doi: 10.1155/S108533750220408X.

[6]

F. Chaves-Silva and G. Lebeau, Spectral inequality and optimal cost of controllability for the Stokes system, ESAIM Control Optim. Calc. Var., 22 (2016), 1137-1162.  doi: 10.1051/cocv/2016034.

[7]

L. EscauriazaF. J. Fernandez and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.  doi: 10.1080/00036810500277082.

[8]

L. EscauriazaC. KenigG. Ponce and L. Vega, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay, Math. Res. Lett., 15 (2008), 957-971.  doi: 10.4310/MRL.2008.v15.n5.a10.

[9]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.

[10]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy uncertainty principle, convexity and parabolic evolutions, Comm. Math. Phys., 346 (2016), 667-678.  doi: 10.1007/s00220-015-2500-z.

[11]

L. EscauriazaS. Montaner and C. Zhang, Observation from measurable sets for parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015), 837-867.  doi: 10.1016/j.matpur.2015.05.005.

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[13]

P. Gao, The Lebeau-Robbiano inequality for the one-dimensional fourth order elliptic operator and its application, ESAIM Control Optim. Calc. Var., 22 (2016), 811-831.  doi: 10.1051/cocv/2015030.

[14]

A. Grigor'yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 353-362.  doi: 10.1017/S0308210500028511.

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.

[16]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (Chicago, IL, 1996), Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, (1999), 223-239.

[17]

J. Le RousseauM. Léautaud and L. Robbiano, Controllability of a parabolic system with a diffuse interface, J. Eur. Math. Soc. (JEMS), 15 (2013), 1485-1574.  doi: 10.4171/JEMS/397.

[18]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.

[19]

J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.

[20]

J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Ration. Mech. Anal., 195 (2010), 953-990.  doi: 10.1007/s00205-009-0242-9.

[21]

J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math., 183 (2011), 245-336.  doi: 10.1007/s00222-010-0278-3.

[22]

J. Le Rousseau and L. Robbiano, Spectral inequality and resolvent estimate for the bi-Laplace operator, preprint, arXiv: 1509.02098.

[23]

M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., 258 (2010), 2739-2778.  doi: 10.1016/j.jfa.2009.10.011.

[24]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[25]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.  doi: 10.1007/s002050050078.

[26]

X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.

[27]

F. Lin, Remarks on a backward parabolic problem, Methods Appl. Anal., 10 (2003), 245-252.  doi: 10.4310/MAA.2003.v10.n2.a5.

[28]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.

[29]

L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465.

[30]

L. Payne, Improperly Posed Problems in Partial Differential Equations, Regional Conference Series in Applied Mathematics, Vol. 22, SIAM, 1975.

[31]

K. D. Phung, Note on the cost of the approximate controllability for the heat equation with potential, J. Math. Anal. Appl., 295 (2004), 527-538.  doi: 10.1016/j.jmaa.2004.03.059.

[32]

K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.  doi: 10.1016/j.jfa.2010.04.015.

[33]

K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc. (JEMS), 15 (2013), 681-703.  doi: 10.4171/JEMS/371.

[34]

K. D. PhungG. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, J. Differential Equations, 263 (2017), 5012-5041.  doi: 10.1016/j.jde.2017.06.008.

[35]

K.D. PhungL. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 477-499.  doi: 10.1016/j.anihpc.2013.04.005.

[36]

C. C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539.  doi: 10.1080/03605309608821195.

[37]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.

[38]

S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.

[39]

T. M. N. Vo, The local backward heat problem, preprint arXiv: 1704.05314.

[40]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.  doi: 10.1137/15M1051907.

[41]

X. Yu and L. Zhang, The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian, ESAIM: COCV. doi: 10.1051/cocv/2017075.

[42]

Y. Zhang, Unique continuation estimates for the Kolmogorov equation in the whole space, C. R. Math. Acad. Sci. Paris, 354 (2016), 389-393.  doi: 10.1016/j.crma.2016.01.009.

show all references

References:
[1]

S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach Space, Comm. Pure Appl. Math., 16 (1963), 121-239.  doi: 10.1002/cpa.3160160204.

[2]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.

[3]

C. Bardos and K. D. Phung, Observation estimate for kinetic transport equations by diffusion approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 640-664.  doi: 10.1016/j.crma.2017.04.017.

[4]

C. Bardos and L. Tartar, Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal., 50 (1973), 10-25.  doi: 10.1007/BF00251291.

[5]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal., 7 (2002), 585-599.  doi: 10.1155/S108533750220408X.

[6]

F. Chaves-Silva and G. Lebeau, Spectral inequality and optimal cost of controllability for the Stokes system, ESAIM Control Optim. Calc. Var., 22 (2016), 1137-1162.  doi: 10.1051/cocv/2016034.

[7]

L. EscauriazaF. J. Fernandez and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.  doi: 10.1080/00036810500277082.

[8]

L. EscauriazaC. KenigG. Ponce and L. Vega, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay, Math. Res. Lett., 15 (2008), 957-971.  doi: 10.4310/MRL.2008.v15.n5.a10.

[9]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.

[10]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy uncertainty principle, convexity and parabolic evolutions, Comm. Math. Phys., 346 (2016), 667-678.  doi: 10.1007/s00220-015-2500-z.

[11]

L. EscauriazaS. Montaner and C. Zhang, Observation from measurable sets for parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015), 837-867.  doi: 10.1016/j.matpur.2015.05.005.

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[13]

P. Gao, The Lebeau-Robbiano inequality for the one-dimensional fourth order elliptic operator and its application, ESAIM Control Optim. Calc. Var., 22 (2016), 811-831.  doi: 10.1051/cocv/2015030.

[14]

A. Grigor'yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 353-362.  doi: 10.1017/S0308210500028511.

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.

[16]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (Chicago, IL, 1996), Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, (1999), 223-239.

[17]

J. Le RousseauM. Léautaud and L. Robbiano, Controllability of a parabolic system with a diffuse interface, J. Eur. Math. Soc. (JEMS), 15 (2013), 1485-1574.  doi: 10.4171/JEMS/397.

[18]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.

[19]

J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.

[20]

J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Ration. Mech. Anal., 195 (2010), 953-990.  doi: 10.1007/s00205-009-0242-9.

[21]

J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math., 183 (2011), 245-336.  doi: 10.1007/s00222-010-0278-3.

[22]

J. Le Rousseau and L. Robbiano, Spectral inequality and resolvent estimate for the bi-Laplace operator, preprint, arXiv: 1509.02098.

[23]

M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., 258 (2010), 2739-2778.  doi: 10.1016/j.jfa.2009.10.011.

[24]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[25]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.  doi: 10.1007/s002050050078.

[26]

X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.

[27]

F. Lin, Remarks on a backward parabolic problem, Methods Appl. Anal., 10 (2003), 245-252.  doi: 10.4310/MAA.2003.v10.n2.a5.

[28]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.

[29]

L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465.

[30]

L. Payne, Improperly Posed Problems in Partial Differential Equations, Regional Conference Series in Applied Mathematics, Vol. 22, SIAM, 1975.

[31]

K. D. Phung, Note on the cost of the approximate controllability for the heat equation with potential, J. Math. Anal. Appl., 295 (2004), 527-538.  doi: 10.1016/j.jmaa.2004.03.059.

[32]

K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.  doi: 10.1016/j.jfa.2010.04.015.

[33]

K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc. (JEMS), 15 (2013), 681-703.  doi: 10.4171/JEMS/371.

[34]

K. D. PhungG. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, J. Differential Equations, 263 (2017), 5012-5041.  doi: 10.1016/j.jde.2017.06.008.

[35]

K.D. PhungL. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 477-499.  doi: 10.1016/j.anihpc.2013.04.005.

[36]

C. C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539.  doi: 10.1080/03605309608821195.

[37]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.

[38]

S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.

[39]

T. M. N. Vo, The local backward heat problem, preprint arXiv: 1704.05314.

[40]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.  doi: 10.1137/15M1051907.

[41]

X. Yu and L. Zhang, The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian, ESAIM: COCV. doi: 10.1051/cocv/2017075.

[42]

Y. Zhang, Unique continuation estimates for the Kolmogorov equation in the whole space, C. R. Math. Acad. Sci. Paris, 354 (2016), 389-393.  doi: 10.1016/j.crma.2016.01.009.

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