Quantitative asymptotic convergence in Caputo fractional-order systems: Explicit bounds, counterexamples, and chaotic synchronisation

Barbălat’s lemma and its variants are basic tools for proving that an error signal tends to zero, and replacing the ordinary integral by a fractional integral is known to make the lemma false when the order lies in ( 0,1 ) . This paper identifies a fractional counterpart of the direct argument of Tao: a family of pointwise inequalities, indexed by a free dwell level, bounds a signal at time t by the fractional integral of its own p th power at the same time, with constant 2 M ( p + 1 ) p + 1 p − p for a bound M on the Caputo derivative, improved to 2 M ( p + 1 ) by a refinement of the same argument. From these we obtain a fractional Barbălat-type lemma: if the signal belongs to L p and its Caputo derivative is bounded, then the signal tends to zero, with a quantitative estimate of its size in terms of the L p tail and with neither uniform continuity nor an argument by contradiction. The homogeneity exponents are forced by scaling, the optimal constant is enclosed between two explicit expressions which coincide at integer order, where it is exactly ( p + 1 ) M , and an explicit signal which does not converge, although its Caputo derivative and its fractional integral are both bounded, shows that the integrability hypothesis cannot be replaced by a fractional one. The results are applied to mixed fractional–integer systems, in which the fractional block supplies the bounded derivative and the integer block the integrability, giving for fractional adaptive control a computable envelope for the tracking error under a fractional tuning law and convergence with an explicit estimate under an integer-order one, and are illustrated numerically together with a grid-refinement study and the adaptive synchronisation of the chaotic fractional-order Chen system.