Department of Mathematics2024-11-0920120895-479810.1137/1007879942-s2.0-84867285601https://hdl.handle.net/20.500.14288/1492We consider the 2-norm distance tau(r)(A, B) from a linear time-invariant dynamical system (A, B) of order n to the nearest system (A + Delta A(*), B + Delta B-*) whose reachable subspace is of dimension r < n. We first present a characterization to test whether the reachable subspace of the system has dimension r, which resembles and can be considered as a generalization of the Popov-Belevitch-Hautus test for controllability. Then, by exploiting this generalized Popov-Belevitch-Hautus characterization, we derive the main result of this paper, which is a singular value optimization characterization for tau(r)(A, B). A numerical technique to solve the derived singular value optimization problems is described. The numerical results on a few examples illustrate the significance of the derived singular value characterization for computational purposes.pdfApplied mathematicsNearest linear systems with highly deficient reachable subspacesJournal Article1095-7162https://doi.org/10.1137/100787994310150300015Q2NOIR00067