This article explores the average controllability of composite networks generated by factor networks via Cartesian product from energy control perspective, which can be characterized by the controllability Gramian-based metrics. The considered factors are directed weighted networks with diagonalizable / non-diagonalizbable Laplacian dynamics. It is shown that the average controllability of a composite network is determined by the spectral properties of its factors without calculating its own high-dimensional matrix, which reveals how the energy-related controllability of a Cartesian product network (CPN) can be derived from its factors’ features. This will reduce the computational complexity and provide insights to study the energy control of other graph product networks.

This article studies the controllability of a new composite network generated by two smaller scale factor networks via the Corona product with Laplacian dynamics. First, the eigenvalues and corresponding eigenvectors of a new composite network-the -duplication Corona product network-are derived by some properties of its factor networks. Second, a necessary and sufficient algebra-based criterion for the controllability of such network is established based on the Popov-Belevitch-Hautus (PBH) test. Furthermore, the weights on edges between the different factor networks are considered. Finally, several examples are presented to demonstrate the effectiveness of our results applied to the unmanned aerial vehicle (UAV) formation.

This paper investigates the controllability of Cartesian product networks consisting of smaller factor networks via Cartesian product of graphs, reveals the controllability relationship between the Cartesian product network and its factor networks, and presents a theoretic analysis framework for a Cartesian product network by the eigenvalue-eigenvector pairs of factor networks. A sufficient and necessary condition for the controllability of Cartesian product networks is established based on the PBH test conjoining with the properties of Kronecker products, which is an important theoretical analysis tool to analyze the controllability of product networks. This work will help us to understand the cooperative mechanisms of the system controllability, network topology and nodal dynamics, and provide a theory and technical guidance to build a more reasonable large-scale network model and design an effective control scheme for a real networked system.

A necessary and sufficient algebraic condition is obtained for the controllability of a composite signed network consisting of two factor networks by Cartesian product, which reveals how the controllability of the higher-dimensional Cartesian product network is affected by the controllability of its smaller-scale low-dimensional factor networks. Furthermore, the structural balance of the Cartesian product network can be judged by the structural balance of its factor networks, which can greatly reduce amount of calculation to some extent. And the agents’ state evolution trajectories of Cartesian product networks can also be predicted by those of their factor networks. Moreover, the controllability of general signed networks is considered. Algorithms and numerical examples are exhibited to demonstrate the validity of our methods.

We study the coordinated motion of a group of vehicles with a leader based on nearest neighbor rules. The leader may be a special vehicle in the group or an external signal (virtual leader) to steer the vehicle group. The control law consists of two parts: a potential field force that makes the vehicles attract to each other and at the same time avoid collision in the group, and an alignment force that makes all vehicles' headings and speeds to converge respectively to common values. We assume that the interaction patterns are either fixed or switching based on the information topologies of the leader and member vehicles. Our approach is based on the Lyapunov theory and basic graph theory. We also consider the effect of noise on the collective dynamics of the group. Numerical simulations are worked out to illustrate the analytical results

In this paper we study the property of controllability for a class of interconnected systems with a leader in switching networks. We obtain necessary and sufficient conditions for the interconnected systems using neighbor rules, to be controllable by one of them acting as a leader. Here, we take the control laws to be an attractive force, and we assume the topology of the control interconnections is variant, that is, each swarm individual (agent) updates is current state based on the current information from neighboring individuals and the leader. Results show that the interconnected systems can be completely controllable even though every subsystem cannot be controllable by selecting one of the as a leader and neighbor interaction rules. Moreover, the swarm can begin from the given initial configuration and reach a desired final configuration while satisfying certain conditions. The model and results of this paper present a novel method to investigate a class of swarms via switching systems and provide further insight into the effect of the interaction pattern on self-organized motion in a swarm system. Numerical simulations are also worked out to illustrate the analytical results.