In this article, we initiate the study of new concepts of conformable \(q\)-fractional calculus. The conformable fractional \(q\)-derivative and \(q\)-integral are defined and their fundamental theorems are also proved. The uniform asymptotic stability of the \(q\)-deformed conformable fractional system with constant delay is investigated by using the Lyapunov-Razumikhin method. For application, a new asymptotic stability necessary condition for the conformable \(q\)-fractional linear system with constant delay is obtained in term of linear matrix inequality (LMI). A numerical example is demonstrated for the results given to illustrate the effectiveness.

This paper addresses the stability analysis problem for a class of nonlinear neutral systems with two additive time-varying delays, including a neutral delay, two discrete delays, and two distributed delays. By constructing an improved Lyapunov–Krasovskii functional and employing extended integral inequalities together with free-weighting matrices, we derive novel delay-dependent stability criteria in the form of linear matrix inequalities (LMIs). The proposed approach effectively reduces conservatism in stability analysis. A numerical example validates the theoretical results and demonstrates the applicability of the method.

The problem of delay-range-dependent stability analysis for linear systems with distributed time-varying delays and nonlinear perturbations is studied without using the model transformation and delay-decomposition approach. The less conservative stability criteria are obtained for the systems by constructing a new augmented Lyapunov–Krasovskii functional and various inequalities, which are presented in terms of linear matrix inequalities (LMIs). Four numerical examples are demonstrated for the results given to illustrate the effectiveness and improvement over other methods.