# On the Reachability and Controllability of Temporal Continuous-Time Linear Networks: A Generic Analysis

## TL;DR

This paper provides a generic analysis of reachability and controllability in temporal continuous-time linear networks, revealing their structural properties, computational complexity, and graph-theoretic bounds, with applications to switched systems.

## Contribution

It introduces explicit expressions for the overall reachable set, analyzes the complexity of structural properties, and extends classical graph concepts to temporal networks.

## Key findings

- Reachability and controllability are generic properties in temporal networks.
- Verifying structural reachability is computationally hard.
- Graph-theoretic bounds can be efficiently computed.

## Abstract

Temporal networks are a class of time-varying networks, which change their topology according to a given time-ordered sequence of static networks (known as subsystems). 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. We demonstrate that the reachability and controllability on a single temporal sequence are generic properties with respect to the parameters of subsystem matrices and the time durations of subsystems. We then give explicit expressions 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 are at least as hard 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 the dynamic graph and the cactus, to temporal networks, and can be efficiently calculated using graph-theoretic algorithms. Finally, applications of the results to the structural controllability of switched linear systems are discussed.