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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 |
References:
[1] |
R. P. Agarwal, A propos d'une note de M. Pierre Humbert, C. R. Séances Acad. Sci., 236 (1953), 2031-2032. |
[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), 3037-3045.
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), 523-531. |
[4] |
K. Balachandran, J. Y. Park and J. J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Anal., 75 (2012), 1919-1926.
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, New York, 2012.
doi: 10.1007/978-1-4614-0457-6. |
[6] |
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, Eindhoven, 2001. |
[7] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, $2^{nd}$ edition, Birkhäuser, Boston, 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), 1-11. |
[9] |
R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 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), 1442-1450.
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, Reaction Engineering and Molecular Properties (eds. F. Keil, W. Machens, H. Voss and J. Werther), Springer-Verlag, Heidelberg, 1999, 217-224. |
[12] |
G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, Berlin, 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), 79-95.
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), 81-88.
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), 3171-3182.
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), 57-58.
doi: 10.1016/S0045-7825(98)00108-X. |
[17] |
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., New Jersey, 2000.
doi: 10.1142/9789812817747. |
[18] |
T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 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, Vol. 204, Elsevier, Amsterdam, 2006. |
[20] |
F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect, Differ. Equ., 3 (1967), 221-225. |
[21] |
J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 1991. |
[22] |
J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1140-1153.
doi: 10.1016/j.cnsns.2010.05.027. |
[23] |
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, Singapore, 2010.
doi: 10.1142/9781848163300. |
[24] |
M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems, North-Holland, Amsterdam, 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, Computational Engineering in Systems Applications (ed. P. Borne), 1996, 952-956. |
[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, Dordrecht, 2007.
doi: 10.1007/978-1-4020-6042-7. |
[28] |
D. Salamon, Control and Observation of Neutral Systems, Research Notes in Mathematics, 91, Pitman Advanced Publ. Program, Boston, 1984. |
[29] |
X. Zhang, Some results of linear fractional order time-delay system, Appl. Math. Comput., 197 (2008), 407-411.
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, Article ID 146010, 9pp. |
show all references
References:
[1] |
R. P. Agarwal, A propos d'une note de M. Pierre Humbert, C. R. Séances Acad. Sci., 236 (1953), 2031-2032. |
[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), 3037-3045.
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), 523-531. |
[4] |
K. Balachandran, J. Y. Park and J. J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Anal., 75 (2012), 1919-1926.
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, New York, 2012.
doi: 10.1007/978-1-4614-0457-6. |
[6] |
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, Eindhoven, 2001. |
[7] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, $2^{nd}$ edition, Birkhäuser, Boston, 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), 1-11. |
[9] |
R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 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), 1442-1450.
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, Reaction Engineering and Molecular Properties (eds. F. Keil, W. Machens, H. Voss and J. Werther), Springer-Verlag, Heidelberg, 1999, 217-224. |
[12] |
G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, Berlin, 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), 79-95.
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), 81-88.
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), 3171-3182.
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), 57-58.
doi: 10.1016/S0045-7825(98)00108-X. |
[17] |
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., New Jersey, 2000.
doi: 10.1142/9789812817747. |
[18] |
T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 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, Vol. 204, Elsevier, Amsterdam, 2006. |
[20] |
F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect, Differ. Equ., 3 (1967), 221-225. |
[21] |
J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 1991. |
[22] |
J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1140-1153.
doi: 10.1016/j.cnsns.2010.05.027. |
[23] |
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, Singapore, 2010.
doi: 10.1142/9781848163300. |
[24] |
M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems, North-Holland, Amsterdam, 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, Computational Engineering in Systems Applications (ed. P. Borne), 1996, 952-956. |
[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, Dordrecht, 2007.
doi: 10.1007/978-1-4020-6042-7. |
[28] |
D. Salamon, Control and Observation of Neutral Systems, Research Notes in Mathematics, 91, Pitman Advanced Publ. Program, Boston, 1984. |
[29] |
X. Zhang, Some results of linear fractional order time-delay system, Appl. Math. Comput., 197 (2008), 407-411.
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, Article ID 146010, 9pp. |
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