March  2020, 13(3): 1017-1029. doi: 10.3934/dcdss.2020060

Regional enlarged observability of Caputo fractional differential equations

1. 

TSI Team, MACS Laboratory, Department of Mathematics and Computer Science, Moulay Ismail University, Faculty of Sciences, 11201 Meknes, Morocco

2. 

Center for Research & Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal

* Corresponding author: delfim@ua.pt

Received  April 2018 Revised  July 2018 Published  March 2019

Fund Project: This research is part of first author's Ph.D. project, which is carried out at Moulay Ismail University, Meknes.

We consider the regional enlarged observability problem for fractional evolution differential equations involving Caputo derivatives. Using the Hilbert Uniqueness Method, we show that it is possible to rebuild the initial state between two prescribed functions only in an internal subregion of the whole domain. Finally, an example is provided to illustrate the theory.

Citation: Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 1017-1029. doi: 10.3934/dcdss.2020060
References:
[1]

S. Abbas, M. Benchohra and G. M. N'Guérékata, Topics in Fractional Differential Equations, Developments in Mathematics, 27, Springer, New York, 2012. doi: 10.1007/978-1-4614-4036-9.  Google Scholar

[2] R. AlmeidaS. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015.  doi: 10.1142/p991.  Google Scholar
[3]

M. AmourouxA. El Jaȉ and E. Zerrik, Regional observability of distributed systems, Internat. J. Systems Sci., 25 (1994), 301-313.  doi: 10.1080/00207729408928961.  Google Scholar

[4]

M. Axtell and M. E. Bise, Fractional calculus applications in control systems, IEEE Conference on Aerospace and Electronics, 2 (1990), 563-566.  doi: 10.1109/NAECON.1990.112826.  Google Scholar

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D. BaleanuA. K. Golmankhaneh and A. K. Golmankhaneh, On electromagnetic field in fractional space, Nonlinear Anal. Real World Appl., 11 (2010), 288-292.  doi: 10.1016/j.nonrwa.2008.10.058.  Google Scholar

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A. Carpinteri and F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer Verlag, 378 (1997), 291-348.  doi: 10.1007/978-3-7091-2664-6_7.  Google Scholar

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R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

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A. Debbouche and D. F. M. Torres, Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions, Fract. Calc. Appl. Anal., 18 (2015), 95-121.  doi: 10.1515/fca-2015-0007.  Google Scholar

[9]

K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, in: Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer Verlag, Heidelberg, (1999), 217–224. Google Scholar

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S. Dolecki and D. L. Russell,, A general theory of observation and control, SIAM J. Control Optimization, 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[11]

A. Dzieliński and D. Sierociuk, Ultracapacitor modelling and control using discrete fractional order state-space model, Acta Montan. Slovaca, 13 (2008), 136-145.   Google Scholar

[12]

A. Dzieliński and D. Sierociuk, Fractional order model of beam heating process and its experimental verification, in New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, (2010), 287–294. doi: 10.1007/978-90-481-3293-5_24.  Google Scholar

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A. DzielińskiD. Sierociuk and G. Sarwas, Some applications of fractional order calculus, Bull. Polish Acad. Sci. Tech. Sci., 58 (2010), 583-592.   Google Scholar

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A. El Jaȉ and A. J. Pritchard, Capteurs et Actionneurs Dans L'analyse des Systémes Distribués, Recherches en Mathématiques Appliquées, 3, Masson, Paris, 1986.  Google Scholar

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A. El JaȉM. C. Simon and E. Zerrik, Regional observability and sensors structures, Sensors and Actuators Journal, 39 (1993), 95-102.   Google Scholar

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F. GeY. Chen and C. Kou, On the regional gradient observability of time fractional diffusion processes, Automatica J. IFAC, 74 (2016), 1-9.  doi: 10.1016/j.automatica.2016.07.023.  Google Scholar

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F. Ge, Y. Q. Chen and C. Kou, Regional Analysis of Time-Fractional Diffusion Processes, Springer, Cham, 2018. doi: 10.1007/978-3-319-72896-4.  Google Scholar

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R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar

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R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

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A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[21]

M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, Czestochowa University of Technology, Czestochowa, 2009. Google Scholar

[22]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2, Recherches en Mathématiques Appliquées, 9, Masson, Paris, 1988.  Google Scholar

[23]

J.-L. Lions, Sur la contrôlabilité exacte élargie, in Partial differential equations and the calculus of variations, Vol. II, 703–727, Progr. Nonlinear Differential Equations Appl., 2, Birkhäuser Boston, Boston, MA, (1989).  Google Scholar

[24]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Inc., 2006. Google Scholar

[25]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, arXiv: 0704.0320 (2007). Google Scholar

[26]

D. Mozyrska and D. F. M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative, Carpathian J. Math., 26 (2010), 210-221.   Google Scholar

[27]

A. Oustaloup, From fractality to non integer derivation: A fundamental idea for a new process control strategy, In: Bensoussan A., Lions J.L. (eds) Analysis and Optimization of Systems, Lecture Notes in Control and Information Sciences, 111, Springer, Berlin, Heidelberg, (1988), 53–64. Google Scholar

[28] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press , Inc., San Diego, CA, 1999.   Google Scholar
[29]

A. J. Pritchard and A. Wirth, Unbounded control and observation systems and their duality, SIAM J. Control Optim., 16 (1978), 535-545.  doi: 10.1137/0316036.  Google Scholar

[30]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[31]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[32]

J. A. Tenreiro MachadoF. MainardiV. Kiryakova and T. Atanacković, Fractional calculus: D'où venons-nous Que sommes-nous? Où allons-nous, Fract. Calc. Appl. Anal., 19 (2016), 1074-1104.  doi: 10.1515/fca-2016-0059.  Google Scholar

[33]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.  Google Scholar

[34]

E. H. Zerrik and and H. Bourray, Gradient observability for diffusion systems, Int. J. Appl. Math. Comput. Sci., 13 (2003), 139-150.   Google Scholar

[35]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.  Google Scholar

show all references

References:
[1]

S. Abbas, M. Benchohra and G. M. N'Guérékata, Topics in Fractional Differential Equations, Developments in Mathematics, 27, Springer, New York, 2012. doi: 10.1007/978-1-4614-4036-9.  Google Scholar

[2] R. AlmeidaS. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations, Imperial College Press, London, 2015.  doi: 10.1142/p991.  Google Scholar
[3]

M. AmourouxA. El Jaȉ and E. Zerrik, Regional observability of distributed systems, Internat. J. Systems Sci., 25 (1994), 301-313.  doi: 10.1080/00207729408928961.  Google Scholar

[4]

M. Axtell and M. E. Bise, Fractional calculus applications in control systems, IEEE Conference on Aerospace and Electronics, 2 (1990), 563-566.  doi: 10.1109/NAECON.1990.112826.  Google Scholar

[5]

D. BaleanuA. K. Golmankhaneh and A. K. Golmankhaneh, On electromagnetic field in fractional space, Nonlinear Anal. Real World Appl., 11 (2010), 288-292.  doi: 10.1016/j.nonrwa.2008.10.058.  Google Scholar

[6]

A. Carpinteri and F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer Verlag, 378 (1997), 291-348.  doi: 10.1007/978-3-7091-2664-6_7.  Google Scholar

[7]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[8]

A. Debbouche and D. F. M. Torres, Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions, Fract. Calc. Appl. Anal., 18 (2015), 95-121.  doi: 10.1515/fca-2015-0007.  Google Scholar

[9]

K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, in: Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer Verlag, Heidelberg, (1999), 217–224. Google Scholar

[10]

S. Dolecki and D. L. Russell,, A general theory of observation and control, SIAM J. Control Optimization, 15 (1977), 185-220.  doi: 10.1137/0315015.  Google Scholar

[11]

A. Dzieliński and D. Sierociuk, Ultracapacitor modelling and control using discrete fractional order state-space model, Acta Montan. Slovaca, 13 (2008), 136-145.   Google Scholar

[12]

A. Dzieliński and D. Sierociuk, Fractional order model of beam heating process and its experimental verification, in New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, (2010), 287–294. doi: 10.1007/978-90-481-3293-5_24.  Google Scholar

[13]

A. DzielińskiD. Sierociuk and G. Sarwas, Some applications of fractional order calculus, Bull. Polish Acad. Sci. Tech. Sci., 58 (2010), 583-592.   Google Scholar

[14]

A. El Jaȉ and A. J. Pritchard, Capteurs et Actionneurs Dans L'analyse des Systémes Distribués, Recherches en Mathématiques Appliquées, 3, Masson, Paris, 1986.  Google Scholar

[15]

A. El JaȉM. C. Simon and E. Zerrik, Regional observability and sensors structures, Sensors and Actuators Journal, 39 (1993), 95-102.   Google Scholar

[16]

F. GeY. Chen and C. Kou, On the regional gradient observability of time fractional diffusion processes, Automatica J. IFAC, 74 (2016), 1-9.  doi: 10.1016/j.automatica.2016.07.023.  Google Scholar

[17]

F. Ge, Y. Q. Chen and C. Kou, Regional Analysis of Time-Fractional Diffusion Processes, Springer, Cham, 2018. doi: 10.1007/978-3-319-72896-4.  Google Scholar

[18]

R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar

[19]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[21]

M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, Czestochowa University of Technology, Czestochowa, 2009. Google Scholar

[22]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2, Recherches en Mathématiques Appliquées, 9, Masson, Paris, 1988.  Google Scholar

[23]

J.-L. Lions, Sur la contrôlabilité exacte élargie, in Partial differential equations and the calculus of variations, Vol. II, 703–727, Progr. Nonlinear Differential Equations Appl., 2, Birkhäuser Boston, Boston, MA, (1989).  Google Scholar

[24]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Inc., 2006. Google Scholar

[25]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, arXiv: 0704.0320 (2007). Google Scholar

[26]

D. Mozyrska and D. F. M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative, Carpathian J. Math., 26 (2010), 210-221.   Google Scholar

[27]

A. Oustaloup, From fractality to non integer derivation: A fundamental idea for a new process control strategy, In: Bensoussan A., Lions J.L. (eds) Analysis and Optimization of Systems, Lecture Notes in Control and Information Sciences, 111, Springer, Berlin, Heidelberg, (1988), 53–64. Google Scholar

[28] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press , Inc., San Diego, CA, 1999.   Google Scholar
[29]

A. J. Pritchard and A. Wirth, Unbounded control and observation systems and their duality, SIAM J. Control Optim., 16 (1978), 535-545.  doi: 10.1137/0316036.  Google Scholar

[30]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[31]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[32]

J. A. Tenreiro MachadoF. MainardiV. Kiryakova and T. Atanacković, Fractional calculus: D'où venons-nous Que sommes-nous? Où allons-nous, Fract. Calc. Appl. Anal., 19 (2016), 1074-1104.  doi: 10.1515/fca-2016-0059.  Google Scholar

[33]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.  Google Scholar

[34]

E. H. Zerrik and and H. Bourray, Gradient observability for diffusion systems, Int. J. Appl. Math. Comput. Sci., 13 (2003), 139-150.   Google Scholar

[35]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.  Google Scholar

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