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Relative controllability of fractional dynamical systems with delays in control. (English) Zbl 1248.93022
Summary: This paper is concerned with the controllability of nonlinear fractional dynamical systems with time varying multiple delays and distributed delays in control defined in finite dimensional spaces. Sufficient conditions for controllability results are obtained using Schauder’s fixed-point theorem and the controllability Gramian matrix which is defined by using the Mittag–Leffler matrix function. Examples are provided to illustrate the theory.
MSC:
93B05Controllability
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
References:
[1]Adams, J. L.; Hartley, T. T.: Finite time controllability of fractional order systems, J comput nonlinear dyn 3, 021402-1-021402-5 (2008)
[2]Balachandran, K.: Controllability of nonlinear perturbation of linear systems with distributed delay in control, Robotica 3, 89-91 (1985)
[3]Balachandran, K.: Global relative controllability of nonlinear systems with time varying multiple delays in control, Int J control 46, 193-200 (1987) · Zbl 0629.93010 · doi:10.1080/00207178708933892
[4]Balachandran, K.; Dauer, J. P.: Controllability of nonlinear systems via fixed point theorems, J optim theory appl 53, 345-352 (1987) · Zbl 0596.93010 · doi:10.1007/BF00938943
[5]Balachandran, K.; Dauer, J. P.: Controllability of nonlinear systems in Banach spaces: a survey, J optim theory appl 115, 7-28 (2002) · Zbl 1023.93010 · doi:10.1023/A:1019668728098
[6]Balachandran, K.; Somasundaram, D.: Controllability of a class of nonlinear systems with distributed delays in control, Kybernetica 19, 475-481 (1983) · Zbl 0528.93012
[7]Balachandran, K.; Somasundaram, D.: Controllability of nonlinear systems with time varying delays in control, Kybernetica 21, 65-72 (1985) · Zbl 0558.93008
[8]Balachandran, K.; Trujillo, J. J.: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear anal 72, 4587-4593 (2010) · Zbl 1196.34007 · doi:10.1016/j.na.2010.02.035
[9]Balachandran, K.; Kiruthika, S.; Trujillo, J. J.: Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun nonlinear sci numer simul 16, 1970-1977 (2011) · Zbl 1221.34215 · doi:10.1016/j.cnsns.2010.08.005
[10]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 · doi:10.1016/j.camwa.2011.03.031
[11]Balachandran, K.; Park, Jy.; Trujillo, Jj.: Controllability of nonlinear fractional dynamical systems, Nonlinear analy theory methods appl 75, 1919-1926 (2012)
[12]Cameron, R. H.; Martin, W. T.: An unsymmetric Fubini theorem, Bull amer math soc 47, 121-125 (1941) · Zbl 0025.15201 · doi:10.1090/S0002-9904-1941-07384-2
[13]Caputo, M.: Linear model of dissipation whose Q is almost frequency independent. Part II, Geophys J royal astronom soc 13, 529-539 (1967)
[14]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 · doi:10.1016/j.sigpro.2006.02.021
[15]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 sci 40, 1-11 (2008)
[16]Cole, K. S.: Electrical excitation in nerve, Cold spring harbor symposia quantit biol 1, 131-137 (1933)
[17]Dacka, C.: On the controllability of a class of nonlinear systems, IEEE trans automat control 25, 263-266 (1980) · Zbl 0439.93006 · doi:10.1109/TAC.1980.1102287
[18]Das, S.: Functional fractional calculus for system identification and controls, (2008)
[19]Do, V. N.: Controllability of semilinear systems, J optim theory appl 65, 41-52 (1990)
[20]Kaczorek, T.: Selected problems of fractional system theory, (2011)
[21]Kexue, L.; Jigen, P.: Laplace transform and fractional differential equations, Appl math lett 24, 2019-2023 (2011)
[22]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[23]Klamka, J.: Relative controllability of nonlinear systems with delay in control, Automatica 12, 633-634 (1976) · Zbl 0345.93014 · doi:10.1016/0005-1098(76)90046-7
[24]Klamka, J.: Relative controllability of nonlinear systems with distributed delay in control, Int J control 28, 307-312 (1978) · Zbl 0402.93012 · doi:10.1080/00207177808922456
[25]Klamka, J.: Controllability of nonlinear systems with distributed delay in control, Int J control 31, 811-819 (1980) · Zbl 0462.93009 · doi:10.1080/00207178008961084
[26]Manabe, S.: The non-integer integral and its application to control systems, ETJ jpn 6, 83-87 (1961)
[27]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[28]Monje, C. A.; Chen, Y. Q.; Vinagre, B. M.; Xue, D.; Feliu, V.: Fractional-order systems and controls; fundamentals and applications, (2010)
[29]Oldham, K.; Spanier, J.: The fractional calculus, (1974)
[30]Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications, (1999)
[31], Advances in fractional calculus: theoretical developments and applications in physics and engineering (2007)
[32]Samko, Sg.; Kilbas, Aa.; Marichev, Oi.: Fractional integrals and derivatives; theory and applications, (1993) · Zbl 0818.26003
[33]Shamardan, A. B.; Moubarak, M. R. A.: Controllability and observability for fractional control systems, J fractional calcul 15, 25-34 (1999) · Zbl 0964.93013
[34]Zhou, Y.; Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal real world appl 11, 4465-4475 (2010)