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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

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

Received  December 2014 Revised  April 2015 Published  October 2015

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.
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. Google Scholar [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. Google Scholar [3] K. Balachandran and J. Kokila, On the controllability of fractional dynamical systems,, Internat. J. Appl. Math. Comput. Sci., 22 (2012), 523. Google Scholar [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. Google Scholar [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. Google Scholar [6] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces,, Eindhoven University of Technology, (2001). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation,, Springer-Verlag, (1974). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific Publ. Co., (2000). doi: 10.1142/9789812817747. Google Scholar [18] T. Kaczorek, Selected Problems of Fractional Systems Theory,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-20502-6. Google Scholar [19] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006). Google Scholar [20] F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect,, Differ. Equ., 3 (1967), 221. Google Scholar [21] J. Klamka, Controllability of Dynamical Systems,, Kluwer Academic Publishers, (1991). Google Scholar [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. Google Scholar [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010). doi: 10.1142/9781848163300. Google Scholar [24] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems,, North-Holland, (1987). Google Scholar [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. Google Scholar [26] T. Mur and H. R. Henríquez, Controllability of abstract systems of fractional order,, preprint, (2015). Google Scholar [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. Google Scholar [28] D. Salamon, Control and Observation of Neutral Systems,, Research Notes in Mathematics, (1984). Google Scholar [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. Google Scholar [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). Google Scholar

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. Google Scholar [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. Google Scholar [3] K. Balachandran and J. Kokila, On the controllability of fractional dynamical systems,, Internat. J. Appl. Math. Comput. Sci., 22 (2012), 523. Google Scholar [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. Google Scholar [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. Google Scholar [6] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces,, Eindhoven University of Technology, (2001). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation,, Springer-Verlag, (1974). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific Publ. Co., (2000). doi: 10.1142/9789812817747. Google Scholar [18] T. Kaczorek, Selected Problems of Fractional Systems Theory,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-20502-6. Google Scholar [19] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006). Google Scholar [20] F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect,, Differ. Equ., 3 (1967), 221. Google Scholar [21] J. Klamka, Controllability of Dynamical Systems,, Kluwer Academic Publishers, (1991). Google Scholar [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. Google Scholar [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010). doi: 10.1142/9781848163300. Google Scholar [24] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems,, North-Holland, (1987). Google Scholar [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. Google Scholar [26] T. Mur and H. R. Henríquez, Controllability of abstract systems of fractional order,, preprint, (2015). Google Scholar [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. Google Scholar [28] D. Salamon, Control and Observation of Neutral Systems,, Research Notes in Mathematics, (1984). Google Scholar [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. Google Scholar [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). Google Scholar
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