## Relative controllability of linear systems of fractional order with delay.(English)Zbl 1332.93061

Summary: 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.

### MSC:

 93B05 Controllability 93C23 Control/observation systems governed by functional-differential equations 34A08 Fractional ordinary differential equations 26A33 Fractional derivatives and integrals
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### References:

 [1] R. P. Agarwal, A propos d’une note de M. Pierre Humbert,, C. R. Séances Acad. Sci., 236, 2031, (1953) · Zbl 0051.30801 [2] K. Balachandran, Relative controllability of fractional dynamical systems with multiple delays in control,, Comput. Math. Appl., 64, 3037, (2012) · Zbl 1268.93021 [3] K. Balachandran, On the controllability of fractional dynamical systems,, Internat. J. Appl. Math. Comput. Sci., 22, 523, (2012) · Zbl 1302.93042 [4] K. Balachandran, Controllability of nonlinear fractional dynamical systems,, Nonlinear Anal., 75, 1919, (2012) · Zbl 1277.34006 [5] D. Baleanu, Fractional Dynamics and Control,, Springer Science, (2012) · Zbl 1231.93003 [6] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces,, Eindhoven University of Technology, (2001) · Zbl 0989.34002 [7] A. Bensoussan, Representation and Control of Infinite Dimensional Systems,, $$2^{nd}$$ edition, (2007) [8] A. A. Chikrii, 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, 1, (2008) [9] R. F. Curtain, An Introduction to Infinite-Dimensional Linear Systems Theory,, Springer-Verlag, (1995) · Zbl 0839.93001 [10] A. Debbouche, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,, Comput. Math. Appl., 62, 1442, (2011) · Zbl 1228.45013 [11] K. Diethelm, 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, 217, (1999) · Zbl 0959.41020 [12] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation,, Springer-Verlag, (1974) · Zbl 0278.44001 [13] M. Feckan, Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators,, J. Optim. Theory Appl., 156, 79, (2013) · Zbl 1263.93031 [14] L. Gaul, Damping description involving fractional operators,, Mech. Syst. Signal Processing, 5, 81, (1991) [15] T. L. Guo, Controllability and observability of impulsive fractional linear time-invariant system,, Comput. Math. Appl., 64, 3171, (2012) · Zbl 1268.93023 [16] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media,, Comput. Methods Appl. Mech. Eng., 167, 57, (1998) · Zbl 0942.76077 [17] R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific Publ. Co., (2000) · Zbl 0998.26002 [18] T. Kaczorek, Selected Problems of Fractional Systems Theory,, Springer-Verlag, (2011) · Zbl 1221.93002 [19] A. A. Kilbas, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006) · Zbl 1092.45003 [20] F. M. Kirillova, The controllability problem for linear systems with aftereffect,, Differ. Equ., 3, 221, (1967) · Zbl 0217.57805 [21] J. Klamka, Controllability of Dynamical Systems,, Kluwer Academic Publishers, (1991) · Zbl 0732.93008 [22] J. T. Machado, Recent history of fractional calculus,, Commun. Nonlinear Sci. Numer. Simulat., 16, 1140, (2011) · Zbl 1221.26002 [23] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010) · Zbl 1210.26004 [24] M. Malek-Zavarei, Time-Delay Systems,, North-Holland, (1987) · Zbl 0658.93001 [25] D. Matignon, Some results on controllability and observability of finite dimensional fractional differential systems,, in CESA’96 IMACS Multiconference, 952, (1996) [26] T. Mur, Controllability of abstract systems of fractional order,, preprint, (2015) · Zbl 1328.93059 [27] J. Sabatier, Advances in Fractional Calculus,, Springer, (2007) · Zbl 1116.00014 [28] D. Salamon, Control and Observation of Neutral Systems,, Research Notes in Mathematics, (1984) · Zbl 0546.93041 [29] X. Zhang, Some results of linear fractional order time-delay system,, Appl. Math. Comput., 197, 407, (2008) · Zbl 1138.34328 [30] H. Zhang, Controllability criteria for linear fractional differential systems with state delay and impulses,, J. Appl. Math., 2013, (1460)
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