Qualitative stability of linear systems

Abstract

This paper provides a finitely computable graph-theoretic answer to the following question concerning linear dynamical systems: When, given only the signs of entries (+, -, or 0) in a real square matrix A, can one be certain that all positive trajectories of the system ẋ = Ax are bounded? Matrices having such sign-patterns are called sign-quasistable. With "bounded" replaced by "convergent to the origin," the matrices are called sign-stable and were fully described in earlier papers. However, when A's digraph has several strong components, so that the system is actually a hierarchy of subsystems, and when some of those subsystems fail to be sign-stable, the recognition of sign-quasistability is a very delicate matter. By means of certain graph color tests, it is possible to identify the system variables that are capable (for some choice of matrix-entry magnitudes and initial conditions) of emitting nonzero constant. © 1987.