# Proof of marginal stability

Given a homogeneous continuous linear time invariant system:

$$\frac{dx(t)}{dt} = Ax(t) \;,\; A \in \mathbb{R}^{n\times n},\; x(t) \in \mathbb{R}^{n},\; t \ge 0$$

Is there any reference (book or web page) to the proof that this system is marginally stable if and only if the real part of every eigenvalue (of the Jordan matrix associated to A) is non-positive such that one or more eigenvalues have zero real part, and all eigenvalues with zero real part are simple roots?

• Are you more interested in the proof, or in having a reference? – Kwin van der Veen Sep 9 '18 at 9:46