Relative controllability of fractional dynamical systems with multiple delays in control. (English) Zbl 1268.93021

Summary: We are concerned with the global relative controllability of fractional dynamical systems with multiple delays in control for finite dimensional spaces. Sufficient conditions for controllability results are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. An example is provided to illustrate the theory.


93B05 Controllability
34A08 Fractional ordinary differential equations
Full Text: DOI


[1] Bagley, R.L.; Torvik, P.J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. rheology, 27, 201-210, (1983) · Zbl 0515.76012
[2] Bagley, R.L.; Torvik, P.J., Fractional calculus in the transient analysis of viscoelastically damped structures, Amer. inst. aeronaut. astronaut., 23, 918-925, (1985) · Zbl 0562.73071
[3] Chow, T.S., Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys. lett. A, 342, 148-155, (2005)
[4] J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288-291.
[5] Magin, R.L, Fractional calculus in bioengineering, Critical rev. biomed. eng., 32, 1-377, (2004)
[6] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[7] Ortigueira, M.D., On the initial conditions in continuous time fractional linear systems, Signal process., 83, 2301-2309, (2003) · Zbl 1145.94367
[8] ()
[9] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. mech. eng., 167, 57-68, (1998) · Zbl 0942.76077
[10] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications, (1999), Academic Press USA · Zbl 0924.34008
[11] Balachandran, K.; Trujillo, J.J., The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear anal.: theory, methods appl., 72, 4587-4593, (2010) · Zbl 1196.34007
[12] Balachandran, K.; Kiruthika, S., Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators, Comput. math. appl., 62, 1350-1358, (2011) · Zbl 1228.34013
[13] Balachandran, K.; Kiruthika, S.; Trujillo, J.J., On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Comput. math. appl., 62, 1157-1165, (2011) · Zbl 1228.34014
[14] Das, S., Functional fractional calculus for system identification and controls, (2008), Springer-Verlag Berlin · Zbl 1154.26007
[15] Adams, J.L.; Hartley, T.T., Finite time controllability of fractional order systems, J. comput. nonlinear dyn., 3, 021402-1-021402-5, (2008)
[16] Chen, Y.Q.; Ahn, H.S.; Xue, D., Robust controllability of interval fractional order linear time invariant systems, Signal process., 86, 2794-2802, (2006) · Zbl 1172.94386
[17] Monje, C.A.; Chen, Y.Q.; Vinagre, B.M.; Xue, D.; Feliu, V., Fractional-order systems and controls; fundamentals and applications, (2010), Springer London
[18] Shamardan, A.B.; Moubarak, M.R.A., Controllability and observability for fractional control systems, J. fract. cal., 15, 25-34, (1999) · Zbl 0964.93013
[19] Balachandran, K.; Dauer, J.P., Controllability of nonlinear systems via fixed point theorems, J. optim. th. appl., 53, 345-352, (1987) · Zbl 0596.93010
[20] Dauer, J.P.; Gahl, R.D., Controllability of nonlinear delay systems, J. optim. theory appl., 21, 59-70, (1977) · Zbl 0325.93007
[21] Balachandran, K.; Somasundaram, D., Controllability of nonlinear systems with time varying delays in control, Kybernetika, 21, 65-72, (1985) · Zbl 0558.93008
[22] Balachandran, K., Global relative controllability of nonlinear systems with time varying multiple delays in control, Internat. J. control, 46, 193-200, (1987) · Zbl 0629.93010
[23] Dacka, C., On the controllability of a class of nonlinear systems, IEEE trans. automat. control, 25, 263-266, (1980) · Zbl 0439.93006
[24] Balachandran, K.; Somasundaram, D., Controllability of nonlinear systems consisting of a bilinear mode with time varying delays in control, Automatica, 20, 257-258, (1984) · Zbl 0535.93006
[25] Klamka, J., Relative controllability of nonlinear systems with delay in control, Automatica, 12, 633-634, (1976) · Zbl 0345.93014
[26] Klamka, J., Controllability of nonlinear systems with distributed delay in control, Internat. J. control, 31, 811-819, (1980) · Zbl 0462.93009
[27] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003
[28] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives; theory and applications, (1993), Gordan and Breach Amsterdam · Zbl 0818.26003
[29] Caputo, M., Linear model of dissipation whose \(Q\) is almost frequency independent II, Geophys. J. royal astronom. soc., 13, 529-539, (1967)
[30] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[31] Chikrii, A.A.; Matichin, I.I., Presentation of solutions of linear systems with fractional derivatives in the sense of riemann – liouville, Caputo and Miller-ross, J. automat. informat. sc., 40, 1-11, (2008)
[32] Dauer, J.P., Nonlinear perturbations of quasi-linear control systems, J. math. anal. appl., 54, 717-725, (1976) · Zbl 0339.93004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.