Finite-Time Synchronization of Fractional-Order Delayed Fuzzy Cellular Neural Networks With Parameter Uncertainties

Abstract

The finite-time synchronization (FTS) is studied for a class of fractional-order delayed fuzzy cellular neural networks (FODFCNNs) with parameter uncertainties. A linear fractional-order finite time inequality (FOFTI) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -a V(t)-b$</tex-math></inline-formula> is extended to the nonlinear case <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -a V^{-\eta }(t)-b$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\eta \ge 1$</tex-math></inline-formula> , which plays a vital role in the FTS of fractional-order systems. However, for the case of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0&lt; \eta &lt; 1$</tex-math></inline-formula> , a theoretical cornerstone justifying its use is still missing. To fill this research gap, on the basis of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C_{p}$</tex-math></inline-formula> inequality and the rule for fractional-order derivative of composite function, a nonlinear FOFTI <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -a V^{-\eta }(t)-b$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0&lt; \eta &lt; 1$</tex-math></inline-formula> is developed. Furthermore, a nonlinear FOFTI <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{c}_{t_{0}}D^{p}_{t} V(t)\le -bV^{-\xi }(t)-a V^{-\eta }(t)$</tex-math></inline-formula> is also established. These two novel inequalities provide the new tools for the research on the finite time stability and synchronization of fractional-order systems and can greatly extend the pioneer ones. Next, on the basis of these novel inequalities, the feedback controller is designed and two novel FTS criteria of FODFCNNs with parameter uncertainties are obtained. Finally, two examples are presented to verify the effectiveness of the derived results.