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An inverse problem of finding a time-dependent parameter in a bilinear heat equation

  • Redouane El Mezegueldy ORCID logo EMAIL logo , Youssef Ouakrim ORCID logo and Mohamed Ouzahra ORCID logo
Published/Copyright: March 28, 2025

Abstract

This paper investigates the inverse problem of estimating an unknown bilinear control u, applied locally in a reaction-diffusion equation. Our goal is to achieve a temperature profile close to a desired reference at the final time. We formulate the problem as an optimal control framework and analyze the existence, optimality conditions and stability of the solution with respect to the data. An algorithm and some numerical experiments are proposed to show the effectiveness of our approach in steering the system towards a desired state.

References

[1] L. F. Abbott, E. Farhi and S. Gutmann, The path integral for dendritic trees, Biol. Cybernetics 66 (1991), no. 1, 49–60. 10.1007/BF00196452Search in Google Scholar PubMed

[2] J. M. Ball, J. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim. 20 (1982), no. 4, 575–597. 10.1137/0320042Search in Google Scholar

[3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Search in Google Scholar

[4] P. Cannarsa, A. Duca and C. Urbani, Exact controllability to eigensolutions of the bilinear heat equation on compact networks, Discrete Contin. Dyn. Syst. Ser. S 15 (2022), no. 6, 1377–1401. 10.3934/dcdss.2022011Search in Google Scholar

[5] P. Cannarsa, G. Floridia and A. Y. Khapalov, Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign, J. Math. Pures Appl. (9) 108 (2017), no. 4, 425–458. 10.1016/j.matpur.2017.07.002Search in Google Scholar

[6] P. Cannarsa and A. Khapalov, Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 4, 1293–1311. 10.3934/dcdsb.2010.14.1293Search in Google Scholar

[7] J. R. Cannon and Y. P. Lin, An inverse problem of finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl. 145 (1990), no. 2, 470–484. 10.1016/0022-247X(90)90414-BSearch in Google Scholar

[8] J. R. Cannon, Y. P. Lin and S. Wang, Determination of source parameter in parabolic equations, Meccanica 27 (1992), no. 2, 85–94. 10.1007/BF00420586Search in Google Scholar

[9] J. R. Cannon, Y. P. Lin and S. Xu, Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations, Inverse Problems 10 (1994), no. 2, 227–243. 10.1088/0266-5611/10/2/004Search in Google Scholar

[10] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57 (2004), no. 11, 1413–1457. 10.1002/cpa.20042Search in Google Scholar

[11] M. Dehghan, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model. 25 (2001), no. 9, 743–754. 10.1016/S0307-904X(01)00010-5Search in Google Scholar

[12] M. Dehghan, Finding a control parameter in one-dimensional parabolic equations, Appl. Math. Comput. 135 (2003), no. 2–3, 491–503. 10.1016/S0096-3003(02)00063-2Search in Google Scholar

[13] M. Dehghan and M. Tatari, Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math. Comput. Model. 44 (2006), no. 11–12, 1160–1168. 10.1016/j.mcm.2006.04.003Search in Google Scholar

[14] Z.-C. Deng, K. Qian, X.-B. Rao, L. Yang and G.-W. Luo, An inverse problem of identifying the source coefficient in a degenerate heat equation, Inverse Probl. Sci. Eng. 23 (2015), no. 3, 498–517. 10.1080/17415977.2014.922079Search in Google Scholar

[15] Z.-C. Deng, L. Yang, J.-N. Yu and G.-W. Luo, An inverse problem of identifying the coefficient in a nonlinear parabolic equation, Nonlinear Anal. 71 (2009), no. 12, 6212–6221. 10.1016/j.na.2009.06.014Search in Google Scholar

[16] Z.-C. Deng, L. Yang, J.-N. Yu and G.-W. Luo, Identifying the radiative coefficient of an evolutional type heat conduction equation by optimization method, J. Math. Anal. Appl. 362 (2010), no. 1, 210–223. 10.1016/j.jmaa.2009.08.042Search in Google Scholar

[17] Z.-C. Deng, J.-N. Yu and L. Yang, Optimization method for an evolutional type inverse heat conduction problem, J. Phys. A 41 (2008), no. 3, Article ID 035201. 10.1088/1751-8113/41/3/035201Search in Google Scholar

[18] A. L. Hodgkin, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiolosy 52 (1952), no. 1, 25–71. 10.1016/S0092-8240(05)80004-7Search in Google Scholar

[19] A. L. Hodgkin and W. A. H. Rushton, The electrical constants of a crustacean nerve fibre, Proc. Roy. Soc. Lond. Ser. B 133 (1946), no. 873, 444–479. 10.1098/rspb.1946.0024Search in Google Scholar PubMed

[20] A. Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, Lecture Notes in Math. 1995, Springer, Berlin, 2010. 10.1007/978-3-642-12413-6Search in Google Scholar

[21] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Math. Comput. Biology Ser., Chapman & Hall/CRC, Boca Raton, 2007. Search in Google Scholar

[22] X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems Control Found. Appl., Birkhäuser, Boston, 1995. Search in Google Scholar

[23] Y. Liu, D. Jiang and M. Yamamoto, Inverse source problem for a double hyperbolic equation describing the three-dimensional time cone model, SIAM J. Appl. Math. 75 (2015), no. 6, 2610–2635. 10.1137/15M1018836Search in Google Scholar

[24] R. R. Mohler, Bilinear control processes, IEEE Trans. Syst. Sci. Cybernetics 6 (1970), no. 3, 192–197. 10.1109/TSSC.1970.300341Search in Google Scholar

[25] M. Ouzahra, Approximate and exact controllability of a reaction-diffusion equation governed by bilinear control, Eur. J. Control 32 (2016), 32–38. 10.1016/j.ejcon.2016.05.004Search in Google Scholar

[26] M. Ouzahra, Finite-time control for the bilinear heat equation, Eur. J. Control 57 (2021), 284–293. 10.1016/j.ejcon.2020.06.010Search in Google Scholar

[27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983. 10.1007/978-1-4612-5561-1Search in Google Scholar

[28] W. Rall, Branching dendritic trees and motoneuron membrane resistivity, Exp. Neurology 1 (1959), no. 5, 491–527. 10.1016/0014-4886(59)90046-9Search in Google Scholar PubMed

[29] W. Rundell, The determination of a parabolic equation from initial and final data, Proc. Amer. Math. Soc. 99 (1987), no. 4, 637–642. 10.1090/S0002-9939-1987-0877031-4Search in Google Scholar

[30] S. Yahyaoui, L. Lafhim and M. Ouzahra, Problem of optimal control for bilinear systems with endpoint constraint, Internat. J. Control 96 (2023), no. 8, 1961–1970. 10.1080/00207179.2022.2079005Search in Google Scholar

[31] L. Yang, J.-N. Yu and Z.-C. Deng, An inverse problem of identifying the coefficient of parabolic equation, Appl. Math. Model. 32 (2008), no. 10, 1984–1995. 10.1016/j.apm.2007.06.025Search in Google Scholar

[32] E. H. Zerrik and N. El Boukhari, Optimal bounded controls problem for bilinear systems, Evol. Equ. Control Theory 4 (2015), no. 2, 221–232. 10.3934/eect.2015.4.221Search in Google Scholar

Received: 2022-10-28
Revised: 2024-04-22
Accepted: 2025-03-05
Published Online: 2025-03-28
Published in Print: 2025-08-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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