On fractional extensions of Barbalat Lemma

a b s t r a c t This paper presents Barbalat-like lemmas for fractional order integrals, which can be used to conclude about the convergence of a function to zero, based on some conditions upon its fractional integral. Some examples in the context of asymptotic behaviour of solutions of fractional order differential equations, indicate the potential application of these lemmas in control theory.

[1]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[2]  Changpin Li,et al.  Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative , 2011 .

[3]  Xin Yu,et al.  Corrections to “Stochastic Barbalat's Lemma and Its Applications” [Jun 12 1537-1543] , 2014, IEEE Transactions on Automatic Control.

[4]  Manuel A. Duarte-Mermoud,et al.  Lyapunov functions for fractional order systems , 2014, Commun. Nonlinear Sci. Numer. Simul..

[5]  Dazhi Zhang,et al.  Local existence and uniqueness of solutions of a degenerate parabolic system , 2011 .

[6]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[7]  Yuanqing Xia,et al.  Stochastic Barbalat's Lemma and Its Applications , 2012, IEEE Transactions on Automatic Control.

[8]  Changpin Li,et al.  A survey on the stability of fractional differential equations , 2011 .

[9]  Manuel A. Duarte-Mermoud,et al.  Sufficient Condition on the Fractional Integral for the Convergence of a Function , 2013, TheScientificWorldJournal.

[10]  Guang-Ren Duan,et al.  New versions of Barbalat’s lemma with applications , 2010 .

[11]  Anuradha M. Annaswamy,et al.  Stable Adaptive Systems , 1989 .

[12]  Y. Q. Chen,et al.  Using Fractional Order Adjustment Rules and Fractional Order Reference Models in Model-Reference Adaptive Control , 2002 .

[13]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .