Some results regarding observability and initial state reconstruction for time-fractional systems
The aim of this study is to present the notion of observability for a specific class of linear time-fractional systems of Riemann-Liouville type with a differentiation order between 1 and 2. To accomplish this goal, we first define the concept of observability and its features, then we extend the Hilbert Uniqueness Method (HUM) to determine the system's initial state. This method converts the reconstruction problem into a solvability one, leading to an algorithm that calculates the initial state. The effectiveness of the proposed algorithm is demonstrated through numerical simulations, which are presented in the final section.
[1] Renardy, M., Hrusa, W.J., & Nohel, J.A. (1987). Mathematical Problems in Viscoelasticity. Long- man Science & Technology, Longman Scientific and Technical, Essex.
[2] Metzler, R., & Klafter, J. (2000). The Ran- dom Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach. Physics Reports, 339(1), 1–77.
[3] Atangana, A. (2014). Convergence and stability analysis of a novel iteration method for fractional biological population equation. Neural Computing and Applications, 25(5), 1021-1030, doi: https: //doi.org/10.1007/s00521-014-1586-0.
[4] Hilfer, R. (2000). Applications Of Fractional Cal- culus In Physics. World Scientific, Singapore.
[5] Sabatier, J., Agrawal, O.P. & Machado, J.A.T.(2007). Advances in Fractional Calculus: Theo- retical Developments and Applications in Physics and Engineering. Springer, Dordrecht.
[6] Wang, P.K.C. (1964). Control of Distributed Pa- rameter Systems. Advances in Control Systems, 1, 75–172, doi:https://doi.org/10.1016/B978 -1-4831-6717-6.50008-5.
[7] Goodson, R., & Klein, R.A. (1970). Definition and Some Results for Distributed System Observ- ability. IEEE Transactions on Automatic Control, 15(2), 165–174, doi: https://doi.org/10.110 9/TAC.1970.1099407.
[8] 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.
[9] 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.
[10] Zouiten, H., Boutoulout, A., & El Alaoui, F.Z.(2017). On the Regional Enlarged Observability for Linear Parabolic Systems. Journal of Mathe- matics and System Science, 7, 79-87, doi: https: //doi.org/10.17265/2159-5291/2017.03.001.
[11] Ge, F., Chen, Y., & Kou, C. (2016). On the Re- gional Gradient Observability of Time Fractional Diffusion Processes. Automatica, 74, 1–9, doi: https://doi.org/10.1016/j.automatica.2 016.07.023.
[12] Zguaid, K., El Alaoui, F.Z., & Boutoulout, A.(2021). Regional Observability for Linear Time Fractional Systems. Mathematics and Computers in Simulation, 185, 77-87, doi: https://doi.or g/10.1016/j.matcom.2020.12.013.
[13] Awais, Y., Javaid, I., & Zehra, A. (2017). On Controllability and Observability of Fractional Continuous-Time Linear Systems with Regular Pencils. Bulletin of the Polish Academy of Sci- ences Technical Sciences, 65(3), 297-304, doi: https://doi.org/10.1515/bpasts-2017-0033.
[14] Cai, R., Ge, F., Chen, Y., & Kou, C.(2019). Regional Observability for Hadamard- Caputo Time Fractional Distributed Parameter Systems. Applied Mathematics and Computation, 360, 190–202, doi: https://doi.org/10.1016/ j.amc.2019.04.081.
[15] Sabatier, J., Farges, C., Merveillaut, M., & Feneteau, L. (2012). On Observability and Pseudo State Estimation of Fractional Order Systems. European Journal of Control, 18(3), 260–271, doi: https://doi.org/10.3166/ejc.18.260-271.
[16] Zguaid, K., & El Alaoui, F.Z. (2023). The Re- gional Observability Problem for a Class of Semi- linear Time-Fractional Systems With Riemann- Liouville Derivative. In: Advanced Mathematical Analysis and its Applications, P. Debnath, D. F. M. Torres, and Y. Je Cho, eds., CRC Press, Boca Raton, 251–264.
[17] Boutoulout, A., Bourray, H., & El Alaoui, F.Z.(2012). Regional Gradient Observability for Dis- tributed Semilinear Parabolic Systems. Journal of Dynamical and Control Systems, 18(2), 159–179, doi: https://doi.org/10.1007/s10883-012-9 138-3.
[18] Boutoulout, A., Bourray, H., El Alaoui, F.Z., & Benhadid, S. (2014). Regional Observability for Distributed Semi-Linear Hyperbolic Systems. In- ternational Journal of Control, 87(5), 898–910, doi: https://doi.org/10.1080/00207179.2 013.861929.
[19] Zguaid, K., & El Alaoui, F.Z. (2022). Regional boundary observability for linear time-fractional systems. Partial Differential Equations in Applied Mathematics, 6, 100432.
[20] Zguaid, K., & El Alaoui, F.Z. (2023). Regional Boundary Observability for Semilinear Fractional Systems with Riemann-Liouville Derivative. Nu- merical Functional Analysis and Optimization, 44(5), 420–437.
[21] 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.
[22] Zerrik, E., Bourray, H., & El Jai, A. (2004). Re- gional Observability for Semilinear Distributed Parabolic Systems. Journal of Dynamical and Control Systems, 10(3), 413–430, doi: https: //doi.org/10.1023/B:JODS.0000034438.72 863.ca.
[23] Zguaid, K., El Alaoui, F.Z., & Torres, D. F. M.(2023). Regional Gradient Observability for Frac- tional Differential Equations with Caputo Time- Fractional Derivatives. International Journal of Dynamics and Control, 11(5), 2423-2437, doi: ht tps://doi.org/10.48550/arXiv.2301.00238.
[24] Zguaid, K., El Alaoui, F.Z., & Boutoulout, A.(2021). Regional Observability of Linear Frac- tional Systems Involving Riemann-Liouville Frac- tional Derivative. In: Z. Hammouch, H. Dutta, S. Melliani, and M. Ruzhansky, eds. Nonlinear Anal- ysis: Problems, Applications and Computational Methods. Springer, Cham, 164-179.
[25] Enrique, C., Jimenez, P., Menendez, J.M., & Conejo, A.J. (2008) The Observability Problem in Traffic Models: Algebraic and Topological Meth- ods. IEEE Transactions on Intelligent Transporta- tion Systems, 9(2), 275-87, doi:https://doi.or g/10.1109/TITS..922929.
[26] Jose, A.L.G., Mar´ıa, N., Enrique, C., & Jose, T. (2013) Application of Observability Techniques to Structural System Identification. Computer- Aided Civil and Infrastructure Engineering, 28(6), 434-450, doi: https://doi.org/10.111 1/mice.12004.
[27] Elbukhari, A.B., Fan, Z., & Li, G. (2023) The Re-gional Enlarged Observability for Hilfer Fractional Differential Equations. Axioms 12(7), 648, doi: https://doi.org/10.3390/axioms12070648.
[28] Viti, F., Rinaldi, M., Corman, F., & Tamp`ere, C.M.J. (2014) Assessing partial observability in network sensor location problems. Transportation Research Part B: Methodological, 70, 65-89, doi: https://doi.org/10.1016/j.trb.2014.08.00 2.
[29] Lions, J.L. (1998). Contrˆolabilit´e Exacte Pertur- bations et Stabilisation de Syst`emes Distribu´es, Tome 1: Contrˆolabilit´e exacte. Dunod, Paris.
[30] Zguaid, K., El Alaoui, F.Z., & Boutoulout, A.(2023). Regional observability of Caputo semilin- ear fractional systems, Asian Journal of Control, doi: https://doi.org/10.1002/asjc.3218.
[31] Lagnese, J. (2006). The Hilbert Uniqueness Method: A Retrospective. In: K.H. Hoffmann, W. Krabs, eds. Optimal Control of Partial Dif- ferential Equations. Springer, Berlin, 158–181.
[32] Zguaid, K., & El Alaoui, F.Z. (2023). On the re- gional boundary observability of semilinear time- fractional systems with Caputo derivative. An International Journal of Optimization and Con- trol: Theories & Applications (IJOCTA) , 13(2), 161-170.
[33] Pedersen, M. (2020). Functional Analysis in Ap- plied Mathematics and Engineering. CRC Press, Boca Raton.
[34] Zguaid, K., & El Alaoui, F.Z. (2022). Regional boundary observability for Riemann–Liouville lin- ear fractional evolution systems. Mathematics and Computers in Simulation, 199, 272-286.
[35] Kilbas, A.A., Srivastava, H.M. & Trujillo, J.J.(2006). Theory And Applications of Fractional Differential Equations. Elsevier, Boston.
[36] Travis, C.C., & Webb, G.F. (1978). Cosine Fami- lies and Abstract Nonlinear Second Order Differ- ential Equations. Acta Mathematica Academiae Scientiarum Hungarica, 32(1), 75–96.
[37] Vasil’ev, V.V., Krein, S., & Sergey, P. (1991). Semigroups of Operators, Cosine Operator Func- tions, and Linear Differential Equations. Journal of Mathematical Sciences, 54, 1042–1129.
[38] Boua, H. (2021). Spectral Theory For Strongly Continuous Cosine. Concrete Operators, 8, 40–47.
[39] Hassani, R.A., Blali, A., Amrani, A.E., & Mous- saouja, K. (2018). Cosine Families of Operators in a Class of Fr´echet Spaces. Proyecciones (Antofa- gasta), 37(1), 103–118.
[40] Ge, F., Chen, Y., & Kou, C. (2018). Regional Analysis of Time-Fractional Diffusion Processes. Springer, Cham.
[41] Gorenflo, R., Kilbas, A.A., Mainardi, F., & Ro- gosin, S. (2020). Mittag-Leffler Functions, Re- lated Topics and Applications. Springer, Berlin.
[42] Brahim, H.B., Zguaid, K., & El Alaoui, F.Z.(2024). A New and specific definition for the mild solution of Riemann-Liouville time-fractional sys- tems with 1 < α < 2,, To appear.
[43] Tucsnak, M., & Weiss, G. (2009). Observation and Control for Operator Semigroups. Birkh¨auser, Basel.
[44] El Jai, A., & Pritchard, A.J. (1986). Capteurs et actionneurs dans l’analyse des syst`emes dis- tribu´es. Elsevier Masson, Paris.
[45] Floridia, G., & Yamamoto, M. (2020). Backward Problems in Time for Fractional Diffusion-Wave Equation. Inverse Problems, 36(12), 125016, doi: https://dx.doi.org/10.1088/1361-6420/ab bc5e.
[46] Almeida, R. (2016). A Caputo Fractional Deriva- tive of a Function with Respect to Another Func- tion. Communications in Nonlinear Science and Numerical Simulation, 44, doi: https://doi.or g/10.1016/j.cnsns.2016.09.006.
[47] Lions, J.L., & Magenes, E. (1972). Non- Homogeneous Boundary Value Problems and Ap- plications. Springer, Berlin.
[48] Zhou, Y., & Wei, H.J. (2020). New Results on Controllability of Fractional Evolution Systems with Order α ∈ (1, 2). Computers & Evolution Equations and Control Theory, 10(3), 491–509, doi: https://doi.org/10.3934/eect.2020077.