A model of fractional-order discontinuous impulsive delayed gene regulatory networks (GRNs) was investigated in this paper. The impulsive perturbations were at fixed moments of time and measured the impulsive control effects which can be controlled appropriately. A fractional-order modeling approach was applied and distributed delays were taken into account for greater model flexibility. In this paper, rather than studying the classical Lyapunov-type stability of an equilibrium point, we addressed the extended Lipschitz stability behavior of the considered GRNs. By applying the impulsive fractional Lyapunov functions technique, new criteria were derived to ensure the global uniform Lipschitz stability for the fractional impulsive delayed GRNs. Furthermore, the effects of considering uncertain parameters were also analyzed. Finally, an illustrating example was given to support the obtained theoretical results.

In this paper a regularization scheme for a family of uncertain fuzzy systems of differential equations with respect to the uncertain parameters is introduced. Important fundamental properties of the solutions are discussed on the basis of the established technique and new results are proposed. More precisely, existence and uniqueness criteria of solutions of the regularized equations are established. In addition, an estimation is proposed for the distance between two solutions. Our results contribute to the progress in the area of the theory of fuzzy systems of differential equations and extend the existing results to the uncertain case. The proposed results also open the horizon for generalizations including equations with delays and some modifications of the Lyapunov theory. In addition, the opportunities for applications of such results to the design of efficient fuzzy controllers are numerous.

In the present work, we study discontinuous impulsive systems of the type of Cohen-Grossberg Neural Networks (CGNNs) with time-varying delays. The impulsive perturbations are realized not at fixed moments of time, and can be considered as control inputs. The hybrid concept of practical exponential stability with respect to specific manifolds defined by a function is introduced and studied analytically. The established results are applied to the case of Bidirectional Associative Memory (BAM) CGNNs. Lyapunov function method and the Razumikhin technique are the base of the proofs. A numerical example is also presented to demonstrate the applicability and effectiveness of the obtained stability conditions. The proposed results extend and complement some existing stability criteria for impulsive CGNNs with time-varying delays.

In this article, for scalar nonlinear equations with uncertain parameters, a new approach has been proposed for studying the integral funnel (antifunnel) of a family of their solutions. Efficient conditions for a family of solutions to remain in an integral funnel (antifunnel) and conditions for narrowing of the integral funnel were established. The proposed approach is based on a comparison scheme and a set of regularized differential equations. The obtained results contribute to the development of the qualitative theory of uncertain systems and to the expansions of the appropriate applications.