Impulse null approximate controllability for heat equation with dynamic boundary conditions

Salah-Eddine Chorfi; Ghita El Guermai; Lahcen Maniar; Walid Zouhair

doi:10.3934/mcrf.2022026

Article Contents

2023, Volume 13, Issue 3: 1023-1046. Doi: 10.3934/mcrf.2022026

This issue Previous Article Identifying a space-dependent source term in distributed order time-fractional diffusion equations Next Article Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates

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
^* Corresponding author: Lahcen Maniar

Received: May 2021

Revised: February 2022

Early access: June 2022

Published: September 2023

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Abstract

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.

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Mathematics Subject Classification: Primary: 35R12, 49N25; Secondary: 93C27.

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[1]	E. M. Ait Ben Hassi, S.-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 Hassi, S.-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 Hassi, S.-E. Chorfi, L. 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. Boutaayamou, S.-E. Chorfi, L. 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. Carrasco, C. 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. Duque, J. Uzcategui, H. 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. Favini, G. R. Goldstein, J. 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, 5^th 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. Leiva, W. 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. Maniar, M. 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. Phung, G. 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. Shah, R. K. George, J. 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, 2^nd 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.