# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. (English) Zbl 1165.34408
Summary: A stability test procedure is proposed for linear nonhomogeneous fractional order systems with a pure time delay. Some basic results from the area of finite time and practical stability are extended to linear, continuous, fractional order time-delay systems given in state-space form. Sufficient conditions of this kind of stability are derived for particular class of fractional time-delay systems. A numerical example is given to illustrate the validity of the proposed procedure.
##### MSC:
 34K20 Stability theory of functional-differential equations 26A33 Fractional derivatives and integrals (real functions)
##### References:
 [1] Zavarei, M.; Jamshidi, M.: Time-delay systems: analysis,Optimization and applications, (1987) [2] Lee, T. N.; Diant, S.: Stability of time delay systems, IEEE trans. Automat. control AC 31, No. 3, 951-953 (1981) [3] Mori, T.: Criteria for asymptotic stability of linear time delay systems, IEEE trans. Automat. control, AC 30, 158-161 (1985) · Zbl 0557.93058 · doi:10.1109/TAC.1985.1103901 [4] Hmamed, A.: On the stability of time delay systems: new results, Internat. J. Control 43, No. 1, 321-324 (1986) · Zbl 0613.34063 · doi:10.1080/00207178608933467 [5] Chen, J.; Xu, D.; Shafai, B.: On sufficient conditions for stability independent of delay, IEEE trans. Automat control AC 40, No. 9, 1675-1680 (1995) · Zbl 0834.93045 · doi:10.1109/9.412644 [6] Weiss, L.; Infante, F.: On the stability of systems defined over finite time interval, Proc. natl. Acad. sci. 54, No. 1, 44-48 (1965) · Zbl 0134.30702 · doi:10.1073/pnas.54.1.44 [7] Grujić, Lj.T.: Non-Lyapunov stability analysis of large-scale systems on time-varying sets, Internat. J. Control 21, No. 3, 401-405 (1975) · Zbl 0303.93010 · doi:10.1080/00207177508921999 [8] Grujić, Lj.T.: Practical stability with settling time on composite systems, Automatika (YU), T.P. 9, No. 1 (1975) [9] Lashirer, A. M.; Story, C.: Final-stability with applications, J. inst. Math. appl. 9, 379-410 (1972) · Zbl 0254.34055 · doi:10.1093/imamat/9.3.397 [10] D.Lj Debeljković, M.P. Lazarević, Dj. Koruga, S. Tomašević, On practical stability of time delay system under perturbing forces, in: AMSE 97, Melbourne, Australia, October 29-31, 1997, pp. 447–450 [11] Lj.D. Debeljković, M.P. Lazarević, S.A. Milinković, M.B. Jovanović, Finite time stability analysis of linear time delay system: Bellman-Gronwall approach, in: IFAC International Workshop Linear Time Delay Systems, Grenoble,France, 1998, pp. 171–176 [12] Lazarević, M. P.; Debeljković, Lj.D.; Nenadić, Z. Lj.; Milinković, S. A.: Finite time stability of time delay systems, IMA J. Math. control. Inform. 17, 101-109 (2000) · Zbl 0979.93095 · doi:10.1093/imamci/17.2.101 [13] Lj.D. Debeljković, M.P. Lazarević, et al. Further results on non-lyapunov stability of the linear nonautonomous systems with delayed state, Journal Facta Uversitatis, Niš, Serbia,Yugoslavia, 2001, vol.3, No 11, pp. 231–241 [14] D. Matignon, Stability result on fractional differential equations with applications to control processing, in: IMACS - SMC Proceeding, July, Lille, France, 1996, pp. 963–968 [15] D. Matignon, Stability properties for generalized fractional differential systems, ESAIM: Proceedings, 5, December,1998, pp. 145–158 · Zbl 0920.34010 · doi:10.1051/proc:1998004 · doi:http://www.edpsciences.org/articles/proc/Vol.5/contents.htm [16] B.M. Vinagre, C.A. Monje, A.J. Calder’on, Fractional order systems and fractional order control actions, in: Lecture 3 of the IEEE CDC02 TW2: Fractional Calculus Applications in Automatic Control and Robotics, 2002 [17] Chen, Q.; Ahn, H.; Podlubny, I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal processing 86, 2611-2618 (2006) · Zbl 1172.94385 · doi:10.1016/j.sigpro.2006.02.011 [18] Chen, Y. Q.; Moore, K. L.: Analytical stability bound for delayed second order systems with repeating poles using Lambert function W, Automatica 38, No. 5, 891-895 (2002) · Zbl 1020.93019 · doi:10.1016/S0005-1098(01)00264-3 [19] Chen, Y. Q.; Moore, K. L.: Analytical stability bound for a class of delayed fractional-order dynamic systems, Nonlinear dynam. 29, 191-200 (2002) · Zbl 1020.34064 · doi:10.1023/A:1016591006562 [20] Bonnet, C.; Partington, J. R.: Stabilization of fractional exponential systems including delays, Kybernetika 37, 345-353 (2001) [21] Bonnet, C.; Partington, J. R.: Analysis of fractional delay systems of retarded and neutral type, Automatica 38, 1133-1138 (2002) · Zbl 1007.93065 · doi:10.1016/S0005-1098(01)00306-5 [22] Hotzel, R.; Fliess, M.: On linear systems with a fractional derivation: introductory theory and examples, Math. comput. Simul. 45, 385-395 (1998) [23] Lazarević, M. P.: Finite time stability analysis of $PD\alpha$ fractional control of robotic time-delay systems, Mech. res. Comm. 33, 269-279 (2006) [24] Ye, H.; Gao, J.; Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation, J. math. Anal. appl. 328, 1075-1081 (2007) [25] Lacroix, S. F.: 2nd ed.traite du calcul differential et du calcul integral, Traite du calcul differential et du calcul integral 3 (1819) [26] Podlubny, I.: Fractional differential equations, (1999) [27] Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos solitons fractals 7, No. 9, 1461-1477 (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5 [28] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) [29] Kilbas, A.; Srivastava, H.; Trujillo, J.: Theory and applications of fractional differential equations, (2006) [30] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Geophys. J. Royal astronom. Soc. 13, 529-539 (1967) [31] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003 [32] C. Lorenzo, T. Hartley, Initialization,conceptualization,and application, NASA_Tp, 1998-208415, December, 1998 [33] Spasić, A.; Lazarević, M.; Krstic, D.: Chapter: theory of electroviscoelasticity, , 371-394 (2006) [34] A.M. Spasić, P.M. Lazarevi’c, Electroviscoelasticity of Liquid–Liquid Interfaces: Fractional-order model (new constitutive models of liquids), in: Lectures in rheology, Department of Mechanics, Faculty of Mathematics, University of Belgrade,Spring, 2004