ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

Contributed talks on Dynamical Systems and Control

Mon 14:30–15:00, Auditorium Max Weber, Chair: Maria Isabel Garcia-Planas
Structural and Exact Controllability for Linear Dynamical Systems. A Geometric Approach
Maria Isabel Garcia-Planas (Universitat Politècnica de Catalunya)

In recent years there has been growing interest in the descriptive analysis of complex systems, permeating many aspects of daily life, obtaining considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Between different aspects in which we can study the controllability we have the notion of structural controllability that has been proposed by Lin [1] as a framework for studying the controllability properties of directed complex networks where the dynamics of the system is governed by a linear system: $\dot x(t) = Ax(t) + Bu(t)$ usually the matrix $A$ of the system is linked to the adjacency matrix of the network, x(t) is a time dependent vector of the state variables of the nodes, $u(t)$ is the vector of input signals, and $B$ which defines how the input signals are connected to the nodes of the network and it is the called input matrix. Structurally controllable means that there exists a matrix $\bar{A}$ in which is not allowed to contain a non-zero entry when the corresponding entry in A is zero such that the network can be driven from any initial state to any final state by appropriately choosing the input signals $u(t)$. For complex systems with linear dynamics it is also of interest to study the exact controllability, this measure is defined as the minimum set of controls that are needed in order to steer the whole system toward any desired state. In this paper, focusing the study on the obtention of the set of all $B$ making the system $(A,B)$ exact controllable.

1. C. Lin. Structural controllability. IEEE Trans. Automat. Contr., 19, 201-8, (1974).