115 Undamped harmonic oscillator

Example

The undamped harmonic oscillator

{{d^{2}x}\over{dt^{2}}}+\nu^{2}x=U_{0}\cos\omega t

with $U_{0}$ real and $\omega,\nu>0$ is marginally stable, but not BIBO stable. We introduce

X=\begin{bmatrix}x\\ v\end{bmatrix},A=\begin{bmatrix}0&1\\ -\nu^{2}&0\end{bmatrix},B=\begin{bmatrix}0\\ 1\end{bmatrix},

{{d}\over{dt}}X=AX+BU_{0}\cos\omega t.

Note that $A$ has eigenvalues $\pm i\nu$ on the imaginary axis.

$\bullet$ For $U=0$ the system oscillates at its natural angular frequency $\nu$