Regional Reconstruction of Semilinear Caputo Type Time-Fractional Systems Using the Analytical Approach

Abstract

The aim of this paper is to investigate the concept of regional observability which is a very important notion of systems theory, precisely regional reconstruction of the initial state for a semilinear Caputo type time-fractional diffusion system which is an interesting class of sytems . Then we give some definitions and properties to introduce our notion. The approaches attempted in this work are both based on fixed point techniques that leads to a successful algorithm which is tested by numerical examples which valid the used approach.

Keywords

• Regional Obervability
• Control Theory
• Fractional Calculus
• Semilinear Systems

Kaynakça

1. [1] A. Amara, S. Etemad, S. Rezapour, Approximate solutions for a fractional hybrid initial value problem via the Caputoconformable derivative. Adv. Differ. Equ. 2020(1) (2020) 608.
2. [2] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory And Applications of Fractional Differential Equations, Elsevier (2006).
3. [3] A. Boutoulout, H. Bourray, F.Z. El Alaoui, Boundary gradient observability for semilinear parabolic systems: Sectorial approach, Math. Sci. Lett. 2(1) (2013) 45-54.
4. [4] A. Boutoulout, H. Bourray, F.Z. El Alaoui, S. Benhadid, Regional observability for distributed semi-linear hyperbolic systems, Int. J. Control. 87(5) (2014) 898-910.
5. [5] A. Boutoulout, H. Bourray, F. Z. El Alaoui, Regional Boundary Observability for Semi-Linear Systems Approach and Simulation, Int. J. Math. Anal. 4(24) (2010) 1153-1173.
6. [6] A. Dzielinski, D. Sierociuk, Fractional Order Model of Beam Heating Process and Its Experimental Verification, In New Trends in Nanotechnology and Fractional Calculus Applications, D. Baleanu, Z. B. Guvenc, and J. A. T. Machado, Eds. Dordrecht: Springer Netherlands, (2010) 287-294.
7. [7] A. Dzielinski, D. Sierociuk, G. Sarwas, Some applications of fractional order calculus, Bull. Pol. Acad. Sci. 58(4) (2010) 583-592.
8. [8] A. Dzielinski, G. Sarwas, D. Sierociuk, Time domain validation of ultracapacitor fractional order model, In 49th IEEE Conference on Decision and Control (CDC), (2010) 3730-3735.

9. [9] A. El Jai, Eléments d'analyse et de contrôle des systemes, Perpignan: Presses Universitaires de Perpignan, (2005).
10. [10] A. El Jai, A.J. Pritchard, Capteurs et actionneurs dans l'analyse des systèmes distribués, Elsevier Masson, (1997).
11. [11] A.J. Pritchard, A. Wirth, Unbounded Control and Observation Systems and Their Duality, SIAM J. Control Optim. 16(4) (1978) 535-545.
12. [12] A. L. Alaoui, E. Azroul, A. A. Hamou, Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems, Adv. Theory Nonlinear Anal. Appl. 4(3) (2020) 197-213.
13. [13] A. Salim, M. Benchohra, E. Karapinar, J.E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv. Differ. Equ. 2020(1) (2020) 601.
14. [14] D. Baleanu, S. Etemad, H. Mohammadi, S. Rezapour, A novel modeling of boundary value problems on the glucose graph, Commun. Nonlinear Sci. Numer. Simul. 100 (2021) 105844.
15. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Berlin, Heidelberg: Springer, 840 (1981).
16. [16] D. Salamon, Infinite Dimensional Linear Systems With Unbounded Control and Observation: A Functional Analytic Approach, Trans. Am. Math. Soc. 300 (1987) 383-431.
17. [17] D. Xu, Q. Wang, Y. Li, Controllability and observability of fractional linear systems with multiple different orders. In 2016 31st Youth Academic Annual Conference of Chinese Association of Automation (YAC), Wuhan, Hubei Province, China, (2016) 286-291.
18. [18] E. Zerrik, H. Bourray, A. El Jai, Regional Observability for Semilinear Distributed Parabolic Systems, J. Dyn. Control Syst. 10(3) (2004) 413-430.
19. [19] F. Ge, Y. Chen, C. Kou, Regional Analysis of Time-Fractional Diffusion Processes, Springer International Publishing (2018).
20. [20] F.Z. El Alaoui, Regional observability of semilinear systems, Ph.D thesis, Faculty of Sciences, Moulay Ismail University, Meknes, (2011).
21. [21] I.I. Vrabie, C 0 -semigroups and applications, 1st edition. Amsterdam, Boston: Elsevier Science, (2003).
22. [22] H. Mohammadi, S. Kumar, S. Rezapour, S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Solitons Fractals, 144(2021) 110668.
23. [23] H. Zouiten, F.Z. El Alaoui, A. Boutoulout, Regional Boundary Observability with Constraints: a Numerical Approach, Int. Rev. Autom. Control. 8(5) (2015) 354-361.
24. [24] J.E. Lazreg, S. Abbas, M. Benchohra, E. Karapinar, Impulsive Caputo-Fabrizio fractional di?erential equations in b-metric spaces, Open Math. 19(1) (2021) 363-3752.
25. [25] J. Mu, B. Ahmad, S. Huang, Existence and regularity of solutions to time-fractional difffusion equations, Comput. Math. with Appl. 73(6) (2017) 985-996.
26. [26] K. Fujishiro, M. Yamamoto, Approximate controllability for fractional diffusion equations by interior control, Appl. Anal. 93(9) (2014) 1793-1810.
27. [27] K.G. Magnusson, Observability of Nonlinear Systems, IMA J. Math. Control Inf. 1(4) (1984) 339-358.
28. [28] K. Taira, Analytic semigroups and semilinear initial boundary value problems, Cambridge: Cambridge Univ. Press, (2004).
29. [29] K. Zguaid, F.Z. El Alaoui, A. Boutoulout, Regional Observability of Linear Fractional Systems Involving Riemann-Liouville Fractional Derivative, In Nonlinear Analysis: Problems, Applications and Computational Methods, Z. Hammouch, H. Dutta, S. Melliani, and M. Ruzhansky, Eds. Springer International Publishing, (2021) 164-178.
30. [30] K. Zguaid, F.Z. El Alaoui, A. Boutoulout, Regional observability for linear time fractional systems, Math. Comput. Simul. 185 (2021) 77-87.
31. [31] M. Amouroux, A. El Jai, E. Zerrik, Regional observability of distributed systems, Int. J. Syst. Sci. 25(2) (1994) 301-313.
32. [32] M. Farman, A. Akgul, A. Ahmad, Analysis and Simulation of Fractional-Order Diabetes Model, Adv. Theory Nonlinear Anal. Appl. 4(4) (2020) 483-497.
33. [33] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sci- ences. New York: Springer. 144 (1983).
34. [34] R.F. Curtain, H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, New York: Springer-Verlag, (1995).
35. [35] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the Solutions of Fractional Di?erential Equations via Geraghty Type Hybrid Contractions, Appl. Comput. Math. 20(2) (2021) 313-333.
36. [36] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions, RACSAM. REV. R. ACAD. A. 115(3) (2021) 155.
37. [37] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Methods Appl. Sci. (2020) 1-12.
38. [38] R. Xue, Observability for fractional diffusion equations by interior control, Fract. Calc. Appl. Anal. 20(2) (2017) 537-552.
39. [39] S. Rezapour, S. Etemad, H. Mohammadi, A mathematical analysis of a system of Caputo-Fabrizio fractional differential equations for the anthrax disease model in animals, Adv. Differ. Equ. 2020(1) (2020) 481.
40. [40] S. Rezapour, S. Etemad, B. Tellab, P. Agarwal, J.L. Garcia Guirao, Numerical Solutions Caused by DGJIM and ADM Methods for Multi-Term Fractional BVP Involving the Generalized Ψ-RL-Operators, Symmetry. 13 (2021) 532.
41. [41] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd edition, Basel: Birkhauser (2011)
42. [42] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. with Appl. 59(3) (2010) 1063-1077.
43. [43] Y. Zhou, L. Zhang, X.H. Shen, Existence of mild solutions for fractional evolution equations, J. Integral Equ. Appl. 25(4) (2013) 557-586.