On the reachability and controllability of temporal continuous-time linear networks: A generic analysis

Temporal networks are a class of time-varying networks whose topology evolves through a time-ordered sequence of static networks (known as subsystems), which find wide-ranging applications in modeling complex systems such as social networks, epidemic spreading, and brain dynamics. This paper investigates the reachability and controllability of temporal continuous-time linear networks from a generic viewpoint, where only the zero-nonzero patterns of subsystem matrices are known. It is demonstrated that the reachability and controllability of temporal networks on a single temporal sequence are generic properties with respect to the parameters of subsystem matrices and the time durations of subsystems. Explicit expressions are then given for the minimal subspace that contains the reachable set across all possible temporal sequences (called overall reachable set). It is found that verifying the structural reachability/controllability and structural overall reachability is at least as difficult as the structural target controllability verification problem of a single system, implying that finding verifiable conditions for them is hard. Graph-theoretic lower and upper bounds are provided for the generic dimensions of the reachable subspace on a single temporal sequence, and of the minimal subspace that contains the overall reachable set. These bounds extend classical concepts in structured system theory, including dynamic graph and cactus, to temporal networks, and can be calculated using graph-theoretic algorithms. Finally, applications of these results to the structural controllability of switched linear systems are discussed.