In this paper, we study the exponential stability problem of discrete-time switched delay systems. Combining a multiple Lyapunov function method with a mode-dependent average dwell time technique, we develop novel sufficient conditions for exponential stability of the switched delay systems expressed by a set of numerically solvable linear matrix inequalities. Finally, numerical examples are presented to illustrate less conservativeness of the obtained results.

This paper focuses on the state bounding problem for the time-delay impulsive and switching genetic regulatory networks (ISGRNs) with exogenous disturbances. Firstly, a sufficient criterion for the state bounding is obtained such that all the trajectories of ISGRNs under consideration converge exponentially into a sphere on the basis of an average dwell time (ADT) switching. Besides, globally exponential stability conditions for the considered system are further stated when the exogenous disturbance vanishes. As a special case, the equivalent state bounding criteria are established by using the properties of some special matrices when there exist no impulses at the switching instants in ISGRNs. Finally, an illustrating example is given to demonstrate the derived results. Compared with the existing literatures, the considered genetic regulatory networks (GRNs) have more general structure and the approach adopted in the present paper is more simple than Lyapunov-Krasovskii functional (LKF) approach.

Medical research has shown that overeating betel nut can be addictive and can damage health. More serious cases may cause mouth cancer and other diseases. Even worse, people’s behavior habit of chewing betel nut may influence each other through social interaction with direct or indirect ways, such as face-to-face communication, Facebook, Twitter, microblog, and WeChat, which leads to the spreading phenomenon of betel nut addiction. In order to investigate the dynamic spreading characteristics of betel nut addiction, a PMSR (Potential-Mild-Severe-Recovered) betel nut addiction spreading model is presented on scale-free networks. The basic reproductive number R0 and equilibria are derived. Theoretical results indicate that the basic reproductive number R0 is significantly dependent on the topology of the underlying networks, and some influence parameters. The existence of equilibria is determined by the basic reproductive number R0. Furthermore, we prove that if R0<1 the addiction-elimination equilibrium is globally asymptotically stable. If R0>1, the betel nut addiction spreading is permanent, and the addiction-prevailing equilibrium is globally attractive. Finally, numerical simulations confirm the theoretical analysis results.