Graphs and controllability completion problems (Q5946188)

Suppose \(A=[a_{ij}]\) is an \(n\times n\) partially specified matrix (some entries are unspecified) and \(b\) is an \(n\times 1\) vector. In this instance, the controllability completion problem concerns the ability to choose values for the unspecified entries of \(A\), thus obtaining a completion \(A_c\) of \(A\), such that \((A_c,b)\) is a completely controllable pair. Let \(G_A\) be the graph on \(n\) vertices having edges \((i,j)\) if and only if \(a_{ij}\) is specified. The authors find sufficient conditions for the existence of \(A_c\) when the specified entries of \(A\) are symmetrically placed and \(G_A\) is a path, cycle or, more generally, a graph whose vertices have degree at most \(2\). Analogous results are obtained in the non-symmetric case of specified entries.