This thesis aims to bridge classical geometric control theory with the theory of structured systems within a network-based framework. The proposed methodology integrates structural and geometric control techniques and leverages the key observation that the subspaces of interest contain or are contained in a subspace generated by canonical basis vectors. Since these canonical vectors correspond to individual network nodes, the algebraic analysis of such subspaces can be directly translated into an analysis over node sets.

The first paper included in this thesis investigates the controllability of temporal networks, modeled as linear, piecewise-constant, time-varying dynamical systems. In particular, an upper bound on the number of snapshots, i.e., time intervals during which the system matrices remain constant, necessary to achieve controllability, is derived. The proof exploits structural properties by decomposing the controllability subspace associated with each snapshot into a component which varies depending on the specific entries of the system matrix, and a fixed component, independent of those entries.

The second paper introduces the minimal node cardinality disturbance decoupling problem, studied under three feedback paradigms: state feedback, output feedback, and dynamic feedback. The formulation and analysis of this problem rely on classical concepts from geometric control theory, such as controlled and conditioned invariant subspaces. By exploiting the fixed structure of these subspaces, they can be reformulated as sets of nodes, providing a network-based interpretation that enhances both intuition and visual interpretability.

The third paper applies the developed theoretical framework to the minimal input cardinality disturbance decoupling problem via output feedback in a system of coupled oscillators, linearized around a stable phase-locked synchronized state, and applies it to the disturbance decoupling of power grids. This application illustrates the practical relevance and versatility of the proposed approach.

Overall, the thesis shows that, under mild assumptions, complex geometric control structures can be represented in an intuitive and visually meaningful manner when analyzed through the lens of networks. This perspective not only facilitates the study of large-scale systems, but also opens new research directions at the intersection of geometric control and network theory.