# American Institute of Mathematical Sciences

• Previous Article
Stochastic recursive optimal control problem with time delay and applications
• MCRF Home
• This Issue
• Next Article
Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation
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), 2031-2032.  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), 3037-3045. 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), 523-531.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, Eindhoven, 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, Birkhäuser, Boston, 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), 1-11. Google Scholar [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.  Google Scholar [10] A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450. 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, Reaction Engineering and Molecular Properties (eds. F. Keil, W. Machens, H. Voss and J. Werther), Springer-Verlag, Heidelberg, 1999, 217-224. Google Scholar [12] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, Berlin, 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), 79-95. 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), 81-88. 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), 3171-3182. 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), 57-58. doi: 10.1016/S0045-7825(98)00108-X.  Google Scholar [17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., New Jersey, 2000. doi: 10.1142/9789812817747.  Google Scholar [18] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 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, Vol. 204, Elsevier, Amsterdam, 2006.  Google Scholar [20] F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect, Differ. Equ., 3 (1967), 221-225. Google Scholar [21] J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 1991.  Google Scholar [22] J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1140-1153. doi: 10.1016/j.cnsns.2010.05.027.  Google Scholar [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, Singapore, 2010. doi: 10.1142/9781848163300.  Google Scholar [24] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems, North-Holland, Amsterdam, 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, Computational Engineering in Systems Applications (ed. P. Borne), 1996, 952-956. 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, Dordrecht, 2007. doi: 10.1007/978-1-4020-6042-7.  Google Scholar [28] D. Salamon, Control and Observation of Neutral Systems, Research Notes in Mathematics, 91, Pitman Advanced Publ. Program, Boston, 1984.  Google Scholar [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.  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, Article ID 146010, 9pp.  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), 2031-2032.  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), 3037-3045. 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), 523-531.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, Eindhoven, 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, Birkhäuser, Boston, 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), 1-11. Google Scholar [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.  Google Scholar [10] A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450. 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, Reaction Engineering and Molecular Properties (eds. F. Keil, W. Machens, H. Voss and J. Werther), Springer-Verlag, Heidelberg, 1999, 217-224. Google Scholar [12] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, Berlin, 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), 79-95. 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), 81-88. 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), 3171-3182. 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), 57-58. doi: 10.1016/S0045-7825(98)00108-X.  Google Scholar [17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., New Jersey, 2000. doi: 10.1142/9789812817747.  Google Scholar [18] T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, 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, Vol. 204, Elsevier, Amsterdam, 2006.  Google Scholar [20] F. M. Kirillova and S. V. Churakova, The controllability problem for linear systems with aftereffect, Differ. Equ., 3 (1967), 221-225. Google Scholar [21] J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 1991.  Google Scholar [22] J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1140-1153. doi: 10.1016/j.cnsns.2010.05.027.  Google Scholar [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, Singapore, 2010. doi: 10.1142/9781848163300.  Google Scholar [24] M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems, North-Holland, Amsterdam, 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, Computational Engineering in Systems Applications (ed. P. Borne), 1996, 952-956. 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, Dordrecht, 2007. doi: 10.1007/978-1-4020-6042-7.  Google Scholar [28] D. Salamon, Control and Observation of Neutral Systems, Research Notes in Mathematics, 91, Pitman Advanced Publ. Program, Boston, 1984.  Google Scholar [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.  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, Article ID 146010, 9pp.  Google Scholar
 [1] Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050 [2] Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031 [3] Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27 [4] Burcu Gürbüz. A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021069 [5] Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283 [6] Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929 [7] Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51 [8] Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 519-537. doi: 10.3934/dcdss.2020029 [9] Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 995-1006. doi: 10.3934/dcdss.2020058 [10] Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 609-627. doi: 10.3934/dcdss.2020033 [11] Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171 [12] Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3577-3587. doi: 10.3934/dcdss.2020428 [13] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [14] Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 [15] Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793 [16] Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157 [17] Francesco Mainardi. On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267 [18] Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $1/2$. Evolution Equations & Control Theory, 2021, 10 (4) : 921-935. doi: 10.3934/eect.2020096 [19] Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005 [20] Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016

2020 Impact Factor: 1.284