Khalid Zguaid, Fatima Zahrae El ALAOUİ

On the regional boundary observability of semilinear time-fractional systems with Caputo derivative

This paper considers the regional boundary observability problem for semilinear time-fractional systems. The main objective is to reconstruct the initial state on a subregion of the boundary of the evolution domain of the considered fractional system using the output equation. We proceed by providing a link between the regional boundary observability of the considered semilinear system on the desired boundary subregion, and the regional observability of its linear part, in a well chosen subregion of the evolution domain. By adding some assumptions on the nonlinear term appearing in the considered system, we give the main theorem that allows us to reconstruct the initial state in the well-chosen subregion using the Hilbert uniqueness method (HUM). From it, we recover the initial state on the boundary subregion. Finally, we provide a numerical example that backs up the theoretical results presented in this paper with a satisfying reconstruction error.

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Curtain, R.F., & Zwart, H. (1995). An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York. https://doi.org/10.1007/BFb0006761
El Jai, A. (1997). Capteurs et actionneurs dans l’analyse des systemes distribues. Elsevier Masson, Paris.
Amouroux, M., El Jai A., & Zerrik, E. (1994). Regional observability of distributed systems. International Journal of Systems Science, 25(2), 301- 313. https://doi.org/10.1080/00207729408928961
El Jai, A., Somon, M.C., Zerrik, E. & Pritchard, A.J. (1995). Regional controllability of distributed parameter systems. International Journal of Control, 62(6), 1351-1365.
El Jai, A., Afifi, L. & Zerrik, E. (2012). Systems Theory: Regional Analysis of Infinite Dimensional Linear Systems. Presses Universitaires de Perpignan, Perpignan.
Boutoulout, A., Bourray, H. & El Alaoui, F.Z. (2013). Boundary gradient observability for semi-linear parabolic systems: Sectorial approach. Mathematical Sciences Letters, 2(1), 45-54. https://doi.org/10.12785/msl/020106
Boutoulout, A., Bourray, H., El Alaoui, F.Z., & Benhadid, S. (2014). Regional observability for distributed semi-linear hyperbolic systems. International Journal of Control, 87(5), 898-910. https://doi.org/10.1080/00207179.2013.861929
Zguaid, K., & El Alaoui, F.Z. (2022). Regional boundary observability for Riemann–Liouville linear fractional evolution systems. Mathematics and Computers in Simulation, 199, 272-286. https://doi.org/10.1016/j.matcom.2022.03.023
Baleanu, D., & Lopes, A.M. (2019). Handbook of Fractional Calculus with Applications: Applications in Engineering, Life and Social Sciences, Part A. De Gruyter, Berlin, Boston.
Petras, I. (2019). Handbook of Fractional Calculus with Applications: Applications in Control. De Gruyter, Berlin, Boston.
Tarasov, V.E. (2019). Handbook of Fractional Calculus with Applications: Applications in Physics, Part A. De Gruyter, Berlin, Boston.
Skovranek, T., & Despotovic, V. (2019). Signal prediction using fractional derivative models. In: Handbook of Fractional Calculus with Applications: Applications in Engineering, Life and Social Sciences, Part B. De Gruyter, Berlin, Boston, 179–206.
Sahijwani, N., & Sukavanam, N. (2023). Approximate controllability for systems of fractional non-linear differential equations involving Riemann-Liouville derivatives. An International Journal of Optimization and Control: Theories & Applications, 13(1), 59-67. https://doi.org/10.11121/ijocta.2023.1178
Pandey, R., Shukla, C., Shukla, A., Upadhyay, A., & Singh, A.K. (2023). A new approach on approximate controllability of Sobolev-type Hilfer fractional differential equations. An International Journal of Optimization and Control: Theories & Applications, 13(1), 130–138. https://doi.org/10.11121/ijocta.2023.1256
Zguaid, K., El Alaoui, F.Z., & Torres D.F.M. (2023). Regional gradient observability for fractional differential equations with Caputo time-fractional derivatives. International Journal of Dynamics and Control. https://doi.org/10.1007/s40435-022-01106-0
Zguaid, K., & El Alaoui, F.Z. (2022). Regional boundary observability for linear time-fractional systems. Partial Differential Equations in Applied Mathematics, 6, 100432. https://doi.org/10.1016/j.padiff.2022.100432
Zguaid, K., El Alaoui, F.Z., & Boutoulout, A. (2021). Regional Observability of Linear Fractional Systems Involving Riemann-Liouville Fractional Derivative. In: Z. Hammouch, H. Dutta, S. Melliani, and M. Ruzhansky, eds. Nonlinear Analysis: Problems, Applications and Computational Methods, Springer International Publishing, 164–178.
Zguaid, K., El Alaoui, F.Z., & Boutoulout, A. (2021). Regional observability for linear time fractional systems. Mathematics and Computers in Simulation, 185, 77–87. https://doi.org/10.1016/j.matcom.2020.12.013
Zguaid, K., & El Alaoui, F.Z. (2023). Regional boundary observability for semilinear fractional systems with Riemann-Liouville derivative. Numerical Functional Analysis and Optimization, 44(5), 420–437. https://doi.org/10.1080/01630563.2023.2171055
El Alaoui, F.Z., Boutoulout, A., & Zguaid, K. (2021). Regional reconstruction of semilinear Caputo type time-fractional systems using the analytical approach. Advances in the Theory of Nonlinear Analysis and its Application, 5(4), 580- 599. https://doi.org/10.31197/atnaa.799236
Boutoulout, A., Bourray, H., & El Alaoui, F.Z. (2010). Regional boundary observability for semi-linear systems approach and simulation. International Journal of Mathematical Analysis, 4(24), 1153–1173.
Boutoulout, A., Bourray, H., & El Alaoui, F.Z. (2015). Regional boundary observability of semi-linear hyperbolic systems: sectorial approach. IMA Journal of Mathematical Control and Information, 32(3), 497–513.
Lions, J.L., & Magenes, E. (1972). Non- Homogeneous Boundary Value Problems and Applications Vol. 1. Springer-Verlag, Berlin.
Mu, J., Ahmad, B., & Huang, S. (2017). Existence and regularity of solutions to time-fractional diffusion equations. Computers & Mathematics with Applications, 73(6), 985–996.
Ge, F., Quan, Y.C., & Kou, C. (2018). Regional Analysis of Time-Fractional Diffusion Processes. Springer International Publishing, Switzerland.
Tiomela, R.F., Norouzi, F., Nguerekata, G., & Mophou, G. (2020). On the stability and stabilization of some semilinear fractional differential equations in Banach Spaces. Fractional Differential Calculus, 10(2), 267–290.
Gottlieb, D., & Orszag, S.A. (1977). Numerical Analysis of Spectral Methods. Society for Industrial and Applied Mathematics, Philadelphia.
Garrappa, R. (2018). Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics, 6(2), 16