On the Structural Target Controllability of Undirected Networks
Jingqi Li, Ximing Chen, Sérgio Pequito, George J. Pappas, Víctor M. Preciado
Abstract
In this article, we study the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">target controllability problem</i> of networked dynamical systems,in which we are tasked to steer a subset of network nodes toward a desired objective. More specifically, we derive necessary and sufficient conditions for the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">structural target controllability</i> of linear time-invariant (LTI) systems with symmetric state matrices, such as those representing undirected dynamical networks with unknown link weights. To achieve our goal, we first characterize the generic rank of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">symmetrically structured matrices</i> , as well as the modes of any numerical realization. Subsequently, we provide graph-theoretic necessary and sufficient conditions for the structural target controllability of undirected networks with multiple control nodes. In addition, we show that these results can be extended and lead to a necessary and sufficient condition of the structural output controllability. However, different from structural target controllability, we prove that verifying the proposed conditions on structural output controllability in undirected networks is NP-hard.