New Inequality Approaches for Fixed-Time Stability Lemmas and Application to Discontinuous CGNNs With Nondifferentiable Delays

Abstract

This article proposes fixed-time stability lemmas for the Filippov system via some new inequality approaches. The adopted method no longer needs to integrate the Lyapunov function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$</tex-math> </inline-formula> on the two integral intervals, which is quite different from the existing ones. Some new estimations of the settling times are provided. Also, the steepness exponents of the Lyapunov function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$</tex-math> </inline-formula> in the previous fixed-time stability lemmas are improved. In order to further study the nondifferentiable delayed neural networks modeled by the Filippov system, a class of discontinuous uncertain Cohen–Grossberg neural networks (CGNNs) with mixed delays is formulated and the distributed delays are nondifferentiable, which is more general. Due to the existence of nondifferentiable distributed delays, the existence of the periodic solutions is proved by Kakutani’s fixed-point theorem before considering the stability. By virtue of the obtained fixed-time stability lemmas and the constructed delay-product-type Lyapunov–Krasovskii functional, the fixed-time stabilization is obtained via a no-chattering controller. Clearly, the designed controller does not contain integral terms and delay terms for dealing with the time delays in the closed-loop system, which is more simplified and practical. Finally, two examples help examine the correctness of the main results.