Controllability of impulsive singularly perturbed systems and its application to a class of multiplex networks

This paper is concerned with the controllability problem of impulsive singularly perturbed systems (ISPSs). A new analytical approach is developed by integrating the merits of the fast-slow decomposition, Chang transformation and Gram-like matrix, and then some ε-independent necessary and sufficient controllable conditions are obtained. In addition, the upper bound of ε is given when deriving the sufficient controllable conditions. Moreover, a new type of heterogeneous multiplex multi-time-scale networks is introduced and can be further modeled by ISPSs. Based on matrix theory and graph theory, some intuitive and easy-to-test criteria are deduced for the controllability of the proposed networks. It is shown that the network topology, the nodal dynamics, the leader selection, and the inner-coupling interconnection are important controllable factors. Several numerical examples are presented to show the effectiveness of the proposed results.