This study addresses the problem of parameter estimation for a one-dimensional reaction-diffusion equation, involving both unknown domain parameters and unknown boundary parameters. The proposed approach utilizes the least-squares method to design an adaptive law for parameter estimation. The convergence analysis demonstrates that under persistent excitation conditions, the adaptive law converges exponentially to zero, indicating that the estimated parameters converge exponentially to their true values. Numerical simulations confirm the effectiveness. Furthermore, it is shown that within a certain range of the reaction coefficient, the auxiliary system acts as a state observer, providing an accurate estimate of the system state at an exponential rate.
References
[1]
Hogea, C. Davatzikos, C. and Biros, G. (2008) An Image-Driven Parameter Estimation Problem for a Reaction-Diffusion Glioma Growth Model with Mass Effects. Journal of Mathematical Biology, 56, 793-825. https://doi.org/10.1007/s00285-007-0139-x
[2]
Chen, Y.J., Wei, Y. and Pock, T. (2015) On Learning Optimized Reaction Diffusion Processes for Effective Image Restoration. 2015 IEEE Conference on Computer Vision and Pattern Recognition, Boston, 7-12 June 2015, 5261-5269. https://doi.org/10.1109/CVPR.2015.7299163
[3]
Konukoglu, E., Clatz, O., Menze, B.H., Stieltjes, B., Weber, M.A., Mandonnet, E., Delingette, H. and Ayache, N. (2010) Image Guided Personalization of Reaction-Diffusion Type Tumor Growth Models Using Modified Anisotropic Eikonal Equations. IEEE Transactions on Medical Imaging, 29, 77-95. https://doi.org/10.1109/TMI.2009.2026413
[4]
Hormuth II, D.A., Weis, J.A., Barnes, S.L., Miga, M.I., Rericha, E.C., Quaranta, V. and Yankeelov, T.E. (2015) Predicting in Vivo Glioma Growth with the Reaction Diffusion Equation Constrained by Quantitative Magnetic Resonance Imaging Data. Physical Biology, 12, Article 046006. https://doi.org/10.1088/1478-3975/12/4/046006
[5]
Soh, S., Byrska, M., Kandere-Grzybowska, K. and Grzybowski, B.A. (2010) Reaction-Diffusion Systems in Intracellular Molecular Transport and Control. Angewandte Chemie International Edition, 49, 4170-4198. https://doi.org/10.1002/anie.200905513
[6]
Shoji, H., Yokogawa, S., Iwamoto, R. and Yamada, K. (1990) Bifurcation Points of Periodic Triangular Patterns Obtained in Reaction-Diffusion System with Anisotropic Diffusion. Journal of Applied Mathematics and Physics, 10, 2341-2355. https://doi.org/10.4236/jamp.2022.107159
[7]
Ma, S. (2021) Hopf Bifurcation of a Gene-Protein Network Module with Reaction Diffusion and Delay Effects. International Journal of Modern Nonlinear Theory and Application, 10, 91-105. https://doi.org/10.4236/ijmnta.2021.103007
[8]
Duan, X. (2017) The Prolongation Structures for the System of the Reaction-Diffusion Type. Journal of Applied Mathematics and Physics, 5, 92-100. https://doi.org/10.4236/jamp.2017.51010
[9]
Gholami, A., Mang, A. and Biros, G. (2016) An Inverse Problem Formulation for Parameter Estimation of a Reaction Diffusion Model of Low Grade Gliomas. Journal of Mathematical Biology, 72, 409-433. https://doi.org/10.1007/s00285-015-0888-x
[10]
Sgura, I., Lawless, A.S. and BBozzini, B. (2018) Parameter Estimation for a Morphochemical Reaction-Diffusion Model of Electrochemical Pattern Formation. Inverse Problems in Science & Engineering, 27, 1-30. https://doi.org/10.1080/17415977.2018.1490278
[11]
Zhao, Z.X., Banda, M.K. and Guo, B.Z. (2017) Simultaneous Identification of Diffusion Coefficient, Spacewise Dependent Source and Initial Value for One-Dimensional Heat Equation. Mathematical Methods in the Applied Sciences, 40, 3552-3565. https://doi.org/10.1002/mma.4245
[12]
Ji, C.T. and Zhang, Z.Q. (2022) Adaptive Boundary Observer Design for Coupled Parabolic PDEs with Different Diffusions and Parameter Uncertainty. IEEE Transactions on Circuits and Systems I: Regular Papers, 69, 3037-3047. https://doi.org/10.1109/TCSI.2022.3159645
[13]
Ahmed-Ali, T., Giri, F., Krstic, M., Burlion, L. and Lamnabhi-Lagarrigue, F. (2017) Adaptive Observer Design with Heat PDE Sensor. Automatica, 82, 93-100. https://doi.org/10.1016/j.automatica.2017.04.030
[14]
Ahmed-Ali, T., Giri, F., Krstic, M., Burlion, L. and Lamnabhi-Lagarrigue, F. (2015) Adaptive Observer for Parabolic PDEs with Uncertain Parameter in the Boundary Condition. 2015 European Control Conference, Linz, 15-17 July 2015, 1343-1348. https://doi.org/10.1109/ECC.2015.7330725
[15]
Ahmed-Ali, T., Giri, F., Krstic, M., Lamnabhi-Lagarrigue, F. and Burlion, L. (2016) Adaptive Observer for a Class of Parabolic PDEs. IEEE Transactions on Automatic Control, 61, 3083-3090. https://doi.org/10.1109/TAC.2015.2500237
[16]
Garvieand, M.R. and Trenchea, C. (2014) Identification of Space-Time Distributed Parameters in the Gierer—Meinhardt Reaction-Diffusion System. SIAM Journal on Applied Mathematics, 74, 147-166. https://doi.org/10.1137/120885784
[17]
Soubeyrand, S. and Roques, L. (2014) Parameter Estimation for Reaction-Diffusion Models of Biological Invasions. Population Ecology, 56 427-434. https://doi.org/10.1007/s10144-013-0415-0
[18]
Zhao, Z.X., Guo, B.Z. and Han, Z.J. (2022) Parameter Identification and Source Term Inversion of One-Dimensional Reaction-Diffusion Equations. Control Theory & Applications, 39, Article 1601.
[19]
Deutscher, J. and Kerschbaum, S. (2018) Backstepping Control of Coupled Linear Parabolic PIDEs with Spatially Varying Coefficients. IEEE Transactions on Automatic Control, 63, 4218-4233.
[20]
Deutscher, J. (2015) A Backstepping Approach to the Output Regulation of Boundary Controlled Parabolic PDEs. Automatica, 57, 56-64. https://doi.org/10.1016/j.automatica.2015.04.008
[21]
Smyshlyaev, A. and Krstic, M. (2005) Backstepping Observers for a Class of Parabolic PDEs. Systems & Control Letters, 54, 613-625. https://doi.org/10.1016/j.sysconle.2004.11.001
[22]
Bokovi, D.M., Balogh, A. and Krstic, M. (2003) Backstepping in Infinite Dimension for a Class of Parabolic Distributed Parameter Systems. Mathematics of Control, Signals and Systems, 16, 44-75. https://doi.org/10.1007/s00498-003-0128-6
[23]
Bernard, P. and Krstic, M. (2014) Adaptive Output-Feedback Stabilization of Non-Local Hyperbolic PDEs. Automatica, 50, 2692-2699. https://doi.org/10.1016/j.automatica.2014.09.001
[24]
Krstic, M., Moura, S.J. and Chaturvedi, N.A. (2014) Adaptive Partial Differential Equation Observer for Battery State-Of-Charge/State-of-Health Estimation via an Electrochemical Model. Journal of Dynamic Systems, Measurement, and Control, 136, Article 011015.
[25]
Ahmed-Ali, T., Giri, F., Krstic, M., Burlion, L. and Lamnabhi-Lagarrigue, F. (2016) Adaptive Boundary Observer for Parabolic PDEs Subject to Domain and Boundary Parameter Uncertainties. Automatica, 72, 115-122. https://doi.org/10.1016/j.automatica.2016.06.006
[26]
Schaum, A., Moreno, J.A., Fridman, E. and Alvarez, J. (2014) Matrix Inequality-Based Observer Design for a Class of Distributed Transport-Reaction Systems. International Journal of Robust and Nonlinear Control, 129, 2213-2230. https://doi.org/10.1002/rnc.2981
[27]
Ioannou, P.A. and Sun, J. (1995) Robust Adaptive Control Prentice-Hall. New Jersey.
[28]
Smyshlyaev, A. and Krstic, M. (1990) Closed-Form Boundary State Feedbacks for a Class of 1-D Partial Integro-Differential Equations. IEEE Transactions on Automatic Control, 49, 2185-2022. https://doi.org/10.1109/TAC.2004.838495
[29]
Krstic, M. and Smyshlyaev, A. (2008) Boundary Control of PDEs: A Course on Backstepping Designs, Philadelphia, Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9780898718607
[30]
Khalil, H.K. (2002) Nonlinear Systems, Prentice Hall. New Jersey.
[31]
Hardy, G.H., Littlewood, J.E. and Pólya, G. (1952) Inequalities Cambridge University Press.
