doi: 10.3934/mcrf.2022026
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Impulse null approximate controllability for heat equation with dynamic boundary conditions

Cadi Ayyad University, Faculty of Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), B.P. 2390, Marrakesh, Morocco

* Corresponding author: Lahcen Maniar

Received  May 2021 Revised  February 2022 Early access June 2022

The main purpose of this article is to prove a logarithmic convexity estimate for the solution of a linear heat equation subject to dynamic boundary conditions in a bounded convex domain. As an application, we prove the impulsive null approximate controllability for an impulsive heat equation with dynamic boundary conditions.

Citation: Salah-Eddine Chorfi, Ghita El Guermai, Lahcen Maniar, Walid Zouhair. Impulse null approximate controllability for heat equation with dynamic boundary conditions. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022026
References:
[1]

E. M. Ait Ben HassiS.-E. Chorfi and L. Maniar, An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions, J. Inverse Ill-Posed Probl., 30 (2022), 363-378.  doi: 10.1515/jiip-2020-0067.

[2]

E. M. Ait Ben HassiS.-E. Chorfi and L. Maniar, Identification of source terms in heat equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 45 (2022), 2364-2379.  doi: 10.1002/mma.7933.

[3]

E. M. Ait Ben HassiS.-E. ChorfiL. Maniar and O. Oukdach, Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions, Evol. Equ. Control Theory, 10 (2021), 837-859.  doi: 10.3934/eect.2020094.

[4]

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.

[5]

A. Ben Aissa and W. Zouhair, Qualitative properties for the $1-D$ impulsive wave equation: Controllability and observability, Quaestiones Mathematicae, (2021).  doi: 10.2989/16073606.2021.1940346.

[6]

I. BoutaayamouS.-E. ChorfiL. Maniar and O. Oukdach, The cost of approximate controllability of heat equation with general dynamical boundary conditions, Port. Math., 78 (2021), 65-99.  doi: 10.4171/pm/2061.

[7]

R. Buffe and K. D. Phung, Observation estimate for the heat equations with Neumann boundary condition via logarithmic convexity, preprint, 2021, arXiv: 2105.12977

[8]

O. Camacho and H. Leiva, Impulsive semilinear heat equation with delay in control and in state, Asian J. Control, 22 (2020), 1075-1089.  doi: 10.1002/asjc.2017.

[9]

A. CarrascoC. Guevara and H. Leiva, Controllability of the impulsive semilinear beam equation with memory and delay, IMA J. Math. Control Inform., 36 (2019), 213-223.  doi: 10.1093/imamci/dnx042.

[10]

C. DuqueJ. UzcateguiH. Leiva and O. Camacho, Approximate controllability of semilinear strongly damped wave equation with impulses, delays, and nonlocal conditions, J. Math. Comput. Sci, 20 (2019), 108-121. 

[11]

Z. J. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[12]

A. FaviniG. R. GoldsteinJ. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.  doi: 10.1007/s00028-002-8077-y.

[13]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696.

[14]

M. Frigon and D. O'Regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl., 193 (1995), 96-113.  doi: 10.1006/jmaa.1995.1224.

[15]

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

[16]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Diff. Equ., 11 (2006), 457-480. 

[17]

L. Hörmander, Notions of Convexity, Progress in Mathematics, 127, Birkhäuser, Basel, 1994.

[18]

J. Jost, Riemannian Geometry and Geometric Analysis, 5th edition, Springer-Verlag, Berlin, 2008.

[19]

A. Y. Khapalov, Exact controllability of second-order hyperbolic equations with impulse controls, Appl. Anal., 63 (1996), 223-238.  doi: 10.1080/00036819608840505.

[20]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control Theory, 9 (2020), 535-559.  doi: 10.3934/eect.2020023.

[21]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, to appear in Mathematische Nachrichten, 2021.

[22]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., 35 (1932), 260-275. 

[23]

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

[24]

H. LeivaW. Zouhair and D. Cabada, Existence, uniqueness and controllability analysis of Benjamin-Bona-Mahony equation with non instantaneous impulses, delay and non local conditions, J. Math. Control Sci., 7 (2021), 91-108. 

[25]

M. Malik and A. Kumar, Existence and controllability results to second order neutral differential equation with non-instantaneous impulses, J. Control Decis., 7 (2020), 286-308.  doi: 10.1080/23307706.2019.1571449.

[26]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusive type, Evol. Equat. and Cont. Theo., 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[27]

B. M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.

[28]

L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, 1975.

[29]

K. D. Phung, Carleman commutator approach in logarithmic convexity for parabolic equations, Math. Control Rel. Fields, 8 (2018), 899-933.  doi: 10.3934/mcrf.2018040.

[30]

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.

[31]

S. Qin and G. Wang, Controllability of impulse controlled systems of heat equations coupled by constant matrices, J. Differential Equations, 263 (2017), 6456-6493.  doi: 10.1016/j.jde.2017.07.018.

[32]

N. Sauer, Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer, Partial Differ. Equ. Appl., 1 (2020), Paper No. 48, 20 pp. doi: 10.1007/s42985-020-00050-y.

[33]

V. ShahR. K. GeorgeJ. Sharma and P. Muthukumar, Existence and uniqueness of classical and mild solutions of generalized impulsive evolution equation, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 775-780.  doi: 10.1515/ijnsns-2018-0042.

[34]

M. E. Taylor, Partial Differential Equations I. Basic Theory, 2nd edition, Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[35]

J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012.

[36]

T. M. N. Vo, The local backward heat problem, preprint, 2017, arXiv: 1704.05314.

[37]

T. Yang, Impulsive Control Theory, Springer, Science and Business Media, 272, 2001.

show all references

References:
[1]

E. M. Ait Ben HassiS.-E. Chorfi and L. Maniar, An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions, J. Inverse Ill-Posed Probl., 30 (2022), 363-378.  doi: 10.1515/jiip-2020-0067.

[2]

E. M. Ait Ben HassiS.-E. Chorfi and L. Maniar, Identification of source terms in heat equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 45 (2022), 2364-2379.  doi: 10.1002/mma.7933.

[3]

E. M. Ait Ben HassiS.-E. ChorfiL. Maniar and O. Oukdach, Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions, Evol. Equ. Control Theory, 10 (2021), 837-859.  doi: 10.3934/eect.2020094.

[4]

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.

[5]

A. Ben Aissa and W. Zouhair, Qualitative properties for the $1-D$ impulsive wave equation: Controllability and observability, Quaestiones Mathematicae, (2021).  doi: 10.2989/16073606.2021.1940346.

[6]

I. BoutaayamouS.-E. ChorfiL. Maniar and O. Oukdach, The cost of approximate controllability of heat equation with general dynamical boundary conditions, Port. Math., 78 (2021), 65-99.  doi: 10.4171/pm/2061.

[7]

R. Buffe and K. D. Phung, Observation estimate for the heat equations with Neumann boundary condition via logarithmic convexity, preprint, 2021, arXiv: 2105.12977

[8]

O. Camacho and H. Leiva, Impulsive semilinear heat equation with delay in control and in state, Asian J. Control, 22 (2020), 1075-1089.  doi: 10.1002/asjc.2017.

[9]

A. CarrascoC. Guevara and H. Leiva, Controllability of the impulsive semilinear beam equation with memory and delay, IMA J. Math. Control Inform., 36 (2019), 213-223.  doi: 10.1093/imamci/dnx042.

[10]

C. DuqueJ. UzcateguiH. Leiva and O. Camacho, Approximate controllability of semilinear strongly damped wave equation with impulses, delays, and nonlocal conditions, J. Math. Comput. Sci, 20 (2019), 108-121. 

[11]

Z. J. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[12]

A. FaviniG. R. GoldsteinJ. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.  doi: 10.1007/s00028-002-8077-y.

[13]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696.

[14]

M. Frigon and D. O'Regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl., 193 (1995), 96-113.  doi: 10.1006/jmaa.1995.1224.

[15]

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

[16]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Diff. Equ., 11 (2006), 457-480. 

[17]

L. Hörmander, Notions of Convexity, Progress in Mathematics, 127, Birkhäuser, Basel, 1994.

[18]

J. Jost, Riemannian Geometry and Geometric Analysis, 5th edition, Springer-Verlag, Berlin, 2008.

[19]

A. Y. Khapalov, Exact controllability of second-order hyperbolic equations with impulse controls, Appl. Anal., 63 (1996), 223-238.  doi: 10.1080/00036819608840505.

[20]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control Theory, 9 (2020), 535-559.  doi: 10.3934/eect.2020023.

[21]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, to appear in Mathematische Nachrichten, 2021.

[22]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., 35 (1932), 260-275. 

[23]

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

[24]

H. LeivaW. Zouhair and D. Cabada, Existence, uniqueness and controllability analysis of Benjamin-Bona-Mahony equation with non instantaneous impulses, delay and non local conditions, J. Math. Control Sci., 7 (2021), 91-108. 

[25]

M. Malik and A. Kumar, Existence and controllability results to second order neutral differential equation with non-instantaneous impulses, J. Control Decis., 7 (2020), 286-308.  doi: 10.1080/23307706.2019.1571449.

[26]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusive type, Evol. Equat. and Cont. Theo., 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[27]

B. M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.

[28]

L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, 1975.

[29]

K. D. Phung, Carleman commutator approach in logarithmic convexity for parabolic equations, Math. Control Rel. Fields, 8 (2018), 899-933.  doi: 10.3934/mcrf.2018040.

[30]

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.

[31]

S. Qin and G. Wang, Controllability of impulse controlled systems of heat equations coupled by constant matrices, J. Differential Equations, 263 (2017), 6456-6493.  doi: 10.1016/j.jde.2017.07.018.

[32]

N. Sauer, Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer, Partial Differ. Equ. Appl., 1 (2020), Paper No. 48, 20 pp. doi: 10.1007/s42985-020-00050-y.

[33]

V. ShahR. K. GeorgeJ. Sharma and P. Muthukumar, Existence and uniqueness of classical and mild solutions of generalized impulsive evolution equation, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 775-780.  doi: 10.1515/ijnsns-2018-0042.

[34]

M. E. Taylor, Partial Differential Equations I. Basic Theory, 2nd edition, Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[35]

J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012.

[36]

T. M. N. Vo, The local backward heat problem, preprint, 2017, arXiv: 1704.05314.

[37]

T. Yang, Impulsive Control Theory, Springer, Science and Business Media, 272, 2001.

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