Abstract:We study the continuous-time Lyapunov equation under the assumption that the state matrix is a Hurwitz companion matrix. The standard Lyapunov theory implies that the unique solution $X$ is positive semidefinite. Motivated by positive systems, we investigate the question of whether $X$ is entrywise nonnegative. We prove that this is the case when the companion matrix has only real eigenvalues. The proof reduces each entry of $X$ to a quadratic form associated with a class of Cauchy-like matrices whose entries are expressed in terms of elementary symmetric polynomials. The required nonnegativity then follows from the positive semidefiniteness of these Cauchy-like matrices. We also discuss a stronger total-positivity property: total nonnegativity does not hold in general, but it is recovered under an additional sign condition on the expansion of the forcing vector in the eigenbasis of $A^\top$.
From: Augusto Ferrante [view email]
[v1]
Mon, 15 Jun 2026 09:57:19 UTC (10 KB)
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