110 Stability cases

Consider $dX/dt=AX$ with $X(0)=X_0$. This has solution $X(t)=\exp (tA)X_0$, and we distinguish the following cases.

(i) Unstable: $X(t)$ $(t>0)$ is unbounded for some $X_0$, which occurs when either $\mathrm{\Re }{\lambda }_j>0$ for some eigenvalue ${\lambda }_j$ of $A$, or $\mathrm{\Re }{\lambda }_j=0$ for some ${\lambda }_j$ that has a Jordan block of size $\ge 2$.

(ii) Marginally stable: $X(t)$ is bounded for $t>0$ for all $X_0$, which occurs when , or $\mathrm{\Re }{\lambda }_j=0$ and the corresponding Jordan blocks are all of size $1\times 1$. Later we will regard this marginal case as BIBO unstable.

(iii) Exponentially stable: there exist $M,\delta >0$ such that $\parallel X(t)\parallel \le M{e}^{-\delta t}$ for all $t>0$ and all $X_0$. This occurs when for all eigenvalues ${\lambda }_j$.