Systems stability with zero poles
The transfer function of a stable (LTI) system needs to have all its poles in the left half-plane, i.e. any pole \$s_{\infty}\$ must satisfy

$$\text{Re}(s_{\infty})&lt;0\tag{1}$$

If this condition is satisfied, then any bounded input signal \$|x(t)|\le K\$ will result in a bounded output signal \$|y(t)|\le L\$ with some positive constants \$K\$ and \$L\$. This concept is called BIBO-stability. Poles on the imaginary axis, i.e. poles with \$\text{Re}(s_{\infty})=0\$ do not satisfy (1), and, consequently, systems with such poles are not stable in the BIBO sense.

In some contexts, systems with poles on the imaginary axis are called marginally stable, but such systems will generally produce unbounded outputs for bounded input signals.