In this technical note, cluster consensus in continuous-time networks of multi-agents with time-varying topologies via non-identical inter-cluster inputs is studied. The cluster consensus contains two aspects: intra-cluster synchronization, that the state differences between agents in the same cluster converge to zero, and inter-cluster separation, that the states of the agents in different clusters do not approach. δ-cluster-spanning-tree in continuous-time networks of multi-agent systems plays essential role in analysis of cluster synchronization. Inter-cluster separation can be realized by imposing adaptive inputs that are identical within the same cluster but different in different clusters, under the inter-cluster common influence condition. Simulation examples demonstrate the effectiveness of the derived theoretical results.

Moving Target Defense (MTD) can enhance the resilience of cyber systems against attacks. Although there have been many MTD techniques, there is no systematic understanding and quantitative characterization of the power of MTD. In this paper, we propose to use a cyber epidemic dynamics approach to characterize the power of MTD. We define and investigate two complementary measures that are applicable when the defender aims to deploy MTD to achieve a certain security goal. One measure emphasizes the maximum portion of time during which the system can afford to stay in an undesired configuration (or posture), without considering the cost of deploying MTD. The other measure emphasizes the minimum cost of deploying MTD, while accommodating that the system has to stay in an undesired configuration (or posture) for a given portion of time. Our analytic studies lead to algorithms for optimally deploying MTD.

Abstract-In this paper, we utilize event-triggered coupling configuration to realize synchronization of linearly coupled dynamical systems. Here, the diffusion couplings are set up from the latest observations of the nodes of its neighborhood and the next observation time is triggered by the proposed criteria based on the local neighborhood information as well. Two scenarios are considered: continuous monitoring, that each node can observe its neighborhood's instantaneous states, and discrete monitoring, that each node can only obtain its neighborhood's states at the same time point when the coupling term is triggered. In both cases, we prove that if the system with persistent coupling can synchronize, then these event-trigger coupling strategies can synchronize the system, too.

In this paper, stability of linearly coupled dynamical systems with feedback pinning algorithm is studied. Here, both the coupling matrix and the set of pinned-nodes vary with time, induced by a continuous-time Markov chain with finite states. Event-triggered rules are employed on both diffusion coupling and feedback pinning terms, which can efficiently reduce the computation load, as well as communication load in some cases and be realized by the latest observations of the state information of its local neighborhood and the target trajectory. The next observation is triggered by certain criterion (event) based on these state information as well. Two scenarios are considered: the continuous monitoring, that each node observes the state information of its neighborhood and target (if pinned) in an instantaneous way, to determine the next triggering event time, and the discrete monitoring, that each node needs only to observe the state information at the last event time and predict the next triggering-event time. In both cases, we present several event-triggering rules and prove that if the conditions that the coupled system with persistent coupling and control can be stabilized are satisfied, then these event-trigger strategies can stabilize the system, and Zeno behaviors are excluded in some cases. Numerical examples are presented to illustrate the theoretical results.

In this paper, we study pinning control problem of coupled dynamical systems with stochastically switching couplings and stochastically selected controller-node set. Here, the coupling matrices and the controller-node sets change with time, induced by a continuous-time Markovian chain. By constructing Lyapunov functions, we establish tractable sufficient conditions for exponentially stability of the coupled system. Two scenarios are considered here. First, we prove that if each subsystem in the switching system, i.e. with the fixed coupling, can be stabilized by the fixed pinning controller-node set, and in addition, the Markovian switching is sufficiently slow, then the time-varying dynamical system is stabilized. Second, in particular, for the problem of spatial pinning control of network with mobile agents, we conclude that if the system with the average coupling and pinning gains can be stabilized and the switching is sufficiently fast, the time-varying system is stabilized. Two numerical examples are provided to demonstrate the validity of these theoretical results, including a switching dynamical system between several stable sub-systems, and a dynamical system with mobile nodes and spatial pinning control towards the nodes when these nodes are being in a predesigned region.