Opuscula Math. 39, no. 1 (2019), 77-89
The existence of consensus of a leader-following problem with Caputo fractional derivative
Abstract. In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.
Keywords: leader-following problem, Caputo fractional differential equation, consensus, nonlinear system, Schauder fixed point theorem.
Mathematics Subject Classification: 26A33, 34K20, 45D05.
- R.P. Agarwal, M. Bohner, S.R. Grace, D. O'Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, 2005.
- C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solutions of certain integral equations, Nonlinear Anal. 66 (2007), 472-483.
- L.C. Becker, Resolvents and solutions of weakly singular linear Volterra integral equations, Nonlinear Anal. 74 (2011), 1892-1912.
- H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004.
- T.A. Burton, Liapunov Theory for Integral Equations with Singular Kernels and Fractional Differential Equations, CreateSpace Independent Publishing Platform, 2012.
- F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control. 52 (2007), 852-862.
- F. Cucker, S. Smale, On the mathematics of emergences, Japan. J. Math. 2 (2007), 197-227.
- K. Diethelm, The Analysis of Fractional Differential Equations, Springer, New York, 2004.
- E. Girejko, L. Machado, A.B. Malinowska, N. Martins, Krause's model of opinion dynamics on isolated time scales, Math. Methods Appl. Sci. 39 (2016), 5302-5314.
- E. Girejko, A.B. Malinowska, E. Schmeidel, M. Zdanowicz, The emergence on isolated time scales, IEEExplore (2016), 1246-1251.
- M.N. Islam, Bounded, asymptotically stable, and \(L^1\) solutions of Caputo fractional differential equations, Opuscula Math. 35 (2015), 181-190.
- A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
- V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic System, Cambridge Scientific Publishers, Cambridge, 2009.
- A.B. Malinowska, D.F.M Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, 2012.
- A.B. Malinowska, T. Odzijewicz, D.F.M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer, New York, 2015.
- R.K. Miller, Nonlinear Volterra Integral Equations, Benjamin, New York, 1971.
- E.C. Oliveira, J.A.T. Machado, A review of definitions for fractional derivatives and integral, Math. Probl. Eng., Article ID 238459 (2014), 1-6.
- J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993.
- Z. Yu, H. Jiangn, C. Hu, Leader-following consensus of fractional-order multi-agent systems under fixed topology, Neurocomputing 149 (2015), 613-620.