Controllability on impulsive systems with delays in both input and impulse and its applications to multi-agent networks

This paper investigates the controllability of impulsive systems with input delay and impulse delay and its applications in multi -agent networks. We adopt the geometric and algebraic analytical tools to establish some easily verified controllability conditions for the considered system model. First, the analytic solution of the considered system is established on every impulsive interval by using ordinary differential equation theory. Then, according to the solution derived, some sufficient complete controllability criteria are developed to reveal the role of the Gramian matrices in different subintervals. By introducing a row matrix of different kinds of Gramian matrices, a complete controllability condition that is proved to be necessary and sufficient is further obtained. By using the relevant geometric matrix theory, the derived algebraic controllability condition is then converted to a geometric one. On other hand, we introduce a multi -agent network with delayed input and impulse and investigate its controllability. By resorting to graph theory and matrix theory, several factors affecting the controllability of the considered multi -agent networks are investigated, such as the topology structure, the inner coupling matrix, and the dynamics of each agent. Finally, two numerical examples are worked out to verify the derived controllability criteria.