Relative controllability of linear systems of fractional order with delay

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December 2015, 5(4): 845-858. doi: 10.3934/mcrf.2015.5.845

Relative controllability of linear systems of fractional order with delay

Therese Mur ^1, and Hernan R. Henriquez ^1,

1.	Departmento de Matemática, Universidad de Santiago-USACH, Casilla 307, Correo-2, Santiago, Chile, Chile

Received December 2014 Revised April 2015 Published October 2015

Abstract
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In this paper we are concerned with the controllability of control systems governed by a fractional differential equations with delay. Using the Mittag-Leffler function we define the concept of solution, and applying the properties of the Laplace transform we characterize the relative or pointwise controllability of the system. Our results generalize those of Kirillova and Churakova, which were established for first order systems. Finally, we show that functionally controllable fractional systems are rare.

Keywords: Controllability of systems, retarded functional differential equations, fractional differential equations, Mittag-Leffler function..

Mathematics Subject Classification: Primary: 93B05, 93C23; Secondary: 34A08, 26A3.

Citation: Therese Mur, Hernan R. Henriquez. Relative controllability of linear systems of fractional order with delay. Mathematical Control & Related Fields, 2015, 5 (4) : 845-858. doi: 10.3934/mcrf.2015.5.845

References:

[1]	R. P. Agarwal, A propos d'une note de M. Pierre Humbert,, C. R. Séances Acad. Sci., 236 (1953), 2031.
[2]	K. Balachandran, J. Kokila and J. J. Trujillo, Relative controllability of fractional dynamical systems with multiple delays in control,, Comput. Math. Appl., 64 (2012), 3037. doi: 10.1016/j.camwa.2012.01.071.
[3]	K. Balachandran and J. Kokila, On the controllability of fractional dynamical systems,, Internat. J. Appl. Math. Comput. Sci., 22 (2012), 523.
[4]	K. Balachandran, J. Y. Park and J. J. Trujillo, Controllability of nonlinear fractional dynamical systems,, Nonlinear Anal., 75 (2012), 1919. doi: 10.1016/j.na.2011.09.042.
[5]	D. Baleanu, J. A. Tenreiro Machado and A. C. J. Luo, Fractional Dynamics and Control,, Springer Science, (2012). doi: 10.1007/978-1-4614-0457-6.
[6]	E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces,, Eindhoven University of Technology, (2001).
[7]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, $2^{nd}$ edition, (2007). doi: 10.1007/978-0-8176-4581-6.
[8]	A. A. Chikrii and I. I. Matichin, Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann-Liouville, Caputo and Miller-Ross,, J. of Automat. Inform. Sci., 40 (2008), 1.
[9]	R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4224-6.
[10]	A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,, Comput. Math. Appl., 62 (2011), 1442. doi: 10.1016/j.camwa.2011.03.075.
[11]	K. Diethelm and A. D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoelasticity,, in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, (1999), 217.
[12]	G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation,, Springer-Verlag, (1974).
[13]	M. Feckan, J.-R. Wang and Y. Zhou, Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators,, J. Optim. Theory Appl., 156 (2013), 79. doi: 10.1007/s10957-012-0174-7.
[14]	L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators,, Mech. Syst. Signal Processing, 5 (1991), 81. doi: 10.1016/0888-3270(91)90016-X.
[15]	T. L. Guo, Controllability and observability of impulsive fractional linear time-invariant system,, Comput. Math. Appl., 64 (2012), 3171. doi: 10.1016/j.camwa.2012.02.020.
[16]	J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media,, Comput. Methods Appl. Mech. Eng., 167 (1998), 57. doi: 10.1016/S0045-7825(98)00108-X.
[17]	R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific Publ. Co., (2000). doi: 10.1142/9789812817747.
[18]	T. Kaczorek, Selected Problems of Fractional Systems Theory,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-20502-6.
[19]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006).
[20]	F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect,, Differ. Equ., 3 (1967), 221.
[21]	J. Klamka, Controllability of Dynamical Systems,, Kluwer Academic Publishers, (1991).
[22]	J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus,, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1140. doi: 10.1016/j.cnsns.2010.05.027.
[23]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010). doi: 10.1142/9781848163300.
[24]	M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems,, North-Holland, (1987).
[25]	D. Matignon and B. d'Andréa-Novel, Some results on controllability and observability of finite dimensional fractional differential systems,, in CESA'96 IMACS Multiconference, (1996), 952.
[26]	T. Mur and H. R. Henríquez, Controllability of abstract systems of fractional order,, preprint, (2015).
[27]	J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus,, Springer, (2007). doi: 10.1007/978-1-4020-6042-7.
[28]	D. Salamon, Control and Observation of Neutral Systems,, Research Notes in Mathematics, (1984).
[29]	X. Zhang, Some results of linear fractional order time-delay system,, Appl. Math. Comput., 197 (2008), 407. doi: 10.1016/j.amc.2007.07.069.
[30]	H. Zhang, J. Cao and W. Jiang, Controllability criteria for linear fractional differential systems with state delay and impulses,, J. Appl. Math., 2013 (1460).

show all references

References:

[1]	R. P. Agarwal, A propos d'une note de M. Pierre Humbert,, C. R. Séances Acad. Sci., 236 (1953), 2031.
[2]	K. Balachandran, J. Kokila and J. J. Trujillo, Relative controllability of fractional dynamical systems with multiple delays in control,, Comput. Math. Appl., 64 (2012), 3037. doi: 10.1016/j.camwa.2012.01.071.
[3]	K. Balachandran and J. Kokila, On the controllability of fractional dynamical systems,, Internat. J. Appl. Math. Comput. Sci., 22 (2012), 523.
[4]	K. Balachandran, J. Y. Park and J. J. Trujillo, Controllability of nonlinear fractional dynamical systems,, Nonlinear Anal., 75 (2012), 1919. doi: 10.1016/j.na.2011.09.042.
[5]	D. Baleanu, J. A. Tenreiro Machado and A. C. J. Luo, Fractional Dynamics and Control,, Springer Science, (2012). doi: 10.1007/978-1-4614-0457-6.
[6]	E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces,, Eindhoven University of Technology, (2001).
[7]	A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, $2^{nd}$ edition, (2007). doi: 10.1007/978-0-8176-4581-6.
[8]	A. A. Chikrii and I. I. Matichin, Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann-Liouville, Caputo and Miller-Ross,, J. of Automat. Inform. Sci., 40 (2008), 1.
[9]	R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4224-6.
[10]	A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,, Comput. Math. Appl., 62 (2011), 1442. doi: 10.1016/j.camwa.2011.03.075.
[11]	K. Diethelm and A. D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoelasticity,, in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, (1999), 217.
[12]	G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation,, Springer-Verlag, (1974).
[13]	M. Feckan, J.-R. Wang and Y. Zhou, Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators,, J. Optim. Theory Appl., 156 (2013), 79. doi: 10.1007/s10957-012-0174-7.
[14]	L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators,, Mech. Syst. Signal Processing, 5 (1991), 81. doi: 10.1016/0888-3270(91)90016-X.
[15]	T. L. Guo, Controllability and observability of impulsive fractional linear time-invariant system,, Comput. Math. Appl., 64 (2012), 3171. doi: 10.1016/j.camwa.2012.02.020.
[16]	J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media,, Comput. Methods Appl. Mech. Eng., 167 (1998), 57. doi: 10.1016/S0045-7825(98)00108-X.
[17]	R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific Publ. Co., (2000). doi: 10.1142/9789812817747.
[18]	T. Kaczorek, Selected Problems of Fractional Systems Theory,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-20502-6.
[19]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006).
[20]	F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect,, Differ. Equ., 3 (1967), 221.
[21]	J. Klamka, Controllability of Dynamical Systems,, Kluwer Academic Publishers, (1991).
[22]	J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus,, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1140. doi: 10.1016/j.cnsns.2010.05.027.
[23]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010). doi: 10.1142/9781848163300.
[24]	M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems,, North-Holland, (1987).
[25]	D. Matignon and B. d'Andréa-Novel, Some results on controllability and observability of finite dimensional fractional differential systems,, in CESA'96 IMACS Multiconference, (1996), 952.
[26]	T. Mur and H. R. Henríquez, Controllability of abstract systems of fractional order,, preprint, (2015).
[27]	J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus,, Springer, (2007). doi: 10.1007/978-1-4020-6042-7.
[28]	D. Salamon, Control and Observation of Neutral Systems,, Research Notes in Mathematics, (1984).
[29]	X. Zhang, Some results of linear fractional order time-delay system,, Appl. Math. Comput., 197 (2008), 407. doi: 10.1016/j.amc.2007.07.069.
[30]	H. Zhang, J. Cao and W. Jiang, Controllability criteria for linear fractional differential systems with state delay and impulses,, J. Appl. Math., 2013 (1460).

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