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Regional Observability of Linear Fractional Systems Involving Riemann-Liouville Fractional Derivative

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Nonlinear Analysis: Problems, Applications and Computational Methods (SM2A 2019)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 168))

Abstract

In this paper, we study the concept of regional observability, more precisely the regional reconstruction of the initial state of a linear fractional system on a subregion \(\omega \) of the evolution domain \(\varOmega \). We use the Hilbert uniqueness method in order to reconstruct the initial state of the given system, which consists of transforming the reconstruction problem into a solvability one. After presenting an algorithm that allows us to reconstruct the regional initial state, we give, at the end, two successful numerical results, in order to backup our theoretical work, each with a different type of sensor and with a reasonable value of error.

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References

  1. Afifi, L., El Jai, A.: Strategic sensors and spy sensors. Int. J. Appl. Math. Comput. Sci. 4(4), 553–573 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Amouroux, M., El Jai, A., Zerrik, E.: Regional observability of distributed systems. Int. J. Syst. Sci. 25(2), 301–313 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baleanu, D., Golmankhaneh, A.K., Golmankhaneh, A.K.: On electromagnetic field in fractional space. Nonlinear Anal. Real World Appl. 11(1), 288–292 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boutoulout, A., Bourray, H., El Alaoui, F.Z., Benhadid, S.: Regional boundary observability for semi-linear systems approach and simulation. Int. J. Math. Anal. 4(24), 1153–1173 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Boutoulout, A., Bourray, H., El Alaoui, F.Z.: Regional gradient observability for distributed semilinear parabolic systems. J. Dyn. Control. Syst. 18(2), 159–179 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boutoulout, A., Bourray, H., El Alaoui, F.Z.: Boundary gradient observability for semilinear parabolic systems: sectorial approach. Math. Sci. Lett. 2(1), 45–54 (2013)

    Article  Google Scholar 

  7. Boutoulout, A., Bourray, H., El Alaoui, F.Z., Benhadid, S.: Regional observability for distributed semi-linear hyperbolic systems. Int. J. Control. 87(5), 898–910 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boutoulout, A., Bourray, H., El Alaoui, F.Z.: Regional boundary observability of semilinear hyperbolic systems: sectorial approach. IMA. J. Math. Control. Inform. 32(3), 497–513 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York (1995)

    Book  MATH  Google Scholar 

  10. El Alaoui, F. Z.: Regional observability of semilinear systems. Ph.D thesis. Faculty of Sciences. Moulay Ismail University. Meknes (2011)

    Google Scholar 

  11. El Jai, A.: Eléments d’analyse et de contrôle des systèmes. Presses Universitaires de Perpignan, Perpignan (2005)

    Google Scholar 

  12. El Jai, A., Simon, M.C., Zerrik, E.: Regional observability and sensor structures. Sens. Actuators Phys. 39(2), 95–102 (1993)

    Article  Google Scholar 

  13. Ge, F., Chen, Y., Kou, C.: Regional Analysis of Time-Fractional Diffusion Processes. Springer International Publishing (2018). www.springer.com/gp/book/9783319728957

  14. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta. 45(5), 765–771 (2006)

    Article  Google Scholar 

  15. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Pub. Co., Singapore River Edge N.J (2000)

    Google Scholar 

  16. Kalman, R.E.: On the general theory of control systems. In: IFAC Proceedings Volumes, vol. 1 (1), pp. 491–502 (1960)

    Google Scholar 

  17. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  18. Kumar, D., Singh, J., Al Qurashi, M., Baleanu, D.: A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Adv. Differ. Equ. 2019(1), 278 (2019)

    Article  MathSciNet  Google Scholar 

  19. Li, Y., Liu, F., Turner, I.W., Li, T.: Time-fractional diffusion equation for signal smoothing. Appl. Math. Comput. 326, 108–116 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Lions, J.L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Paris (1997)

    Google Scholar 

  21. Mainardi, F., Mura, A., Pagnini, G.: The M-Wright function in time-fractional diffusion processes: a tutorial survey. Int. J. Differ. Equ. 22(3), 87–99 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Mophou, G.M.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61(1), 68–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1st edn. Academic Press, Cambridge (1998)

    MATH  Google Scholar 

  24. Pritchard, A.J., Wirth, A.: Unbounded control and observation systems and their duality. SIAM. J. Control. Optim. 16(4), 535–545 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  25. Qureshi, S., Yusuf, A., Shaikh, A.A., Inc, M., Baleanu, D.: Fractional modeling of blood ethanol concentration system with real data application. Chaos Interdiscip. J. Nonlinear. Sci. 29(1), 013143 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Salamon, D.: Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach. Trans. Amer. Math. Soc. 300, 383–431 (1987)

    MathSciNet  MATH  Google Scholar 

  27. Weiss, G.: Admissible observation operators for linear semigroups. Israel. J. Math. 65(1), 17–43 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zerrik, E., Bourray, H., El Jai, A.: Regional observability for semilinear distributed parabolic systems. J. Dyn. Control Syst. 10(3), 413–430 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhou, Y., Zhang, L., Shen, X.H.: Existence of mild solutions for fractional evolution equations. J. Integral Equ. Appl. 25(4), 557–586 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zouiten, H., El Alaoui, F.Z., Boutoulout, A.: Regional boundary observability with constraints: a numerical approach. Int. Rev. Automat. Contr. 8(5), 354–361 (2015)

    Google Scholar 

  31. Zouiten, H., Boutoulout, A., Torres, D.: Regional enlarged observability of Caputo fractional differential equations. Discrete. Cont. Dyn-S. 13(3), 1017–1029 (2018)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Sciences, Moulay Ismail University, 11201, Mekens, Morocco

    Khalid Zguaid, Fatima Zahrae El Alaoui & Ali Boutoulout

Authors
  1. Khalid Zguaid
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  2. Fatima Zahrae El Alaoui
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  3. Ali Boutoulout
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Corresponding author

Correspondence to Khalid Zguaid .

Editor information

Editors and Affiliations

  1. FSTE, Moulay Ismail University, Meknes, Morocco

    Zakia Hammouch

  2. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  3. Department of Mathematics, Sultan Moulay Slimane University, Beni Mellal, Morocco

    Said Melliani

  4. Department of Mathematics, Ghent University, Gent, Belgium

    Michael Ruzhansky

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Zguaid, K., El Alaoui, F.Z., Boutoulout, A. (2021). Regional Observability of Linear Fractional Systems Involving Riemann-Liouville Fractional Derivative. In: Hammouch, Z., Dutta, H., Melliani, S., Ruzhansky, M. (eds) Nonlinear Analysis: Problems, Applications and Computational Methods. SM2A 2019. Lecture Notes in Networks and Systems, vol 168. Springer, Cham. https://doi.org/10.1007/978-3-030-62299-2_12

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  • DOI: https://doi.org/10.1007/978-3-030-62299-2_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-62298-5

  • Online ISBN: 978-3-030-62299-2

  • eBook Packages: EngineeringEngineering (R0)

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