Why non repeated poles at imaginary axis makes LTI system marginally stable?

Question

I understand that stability for an LTI system is defined with respect to Bounded input bounded output condition. However I'm not clear on why non repeated poles on the imaginary axis makes the system marginally stable. For a unit step input, a single pole at origin produces an unbounded ramp output [Unbounded Response] and non repeated conjugate poles on the imaginary axis produces a bounded sinusoidal output [Bounded response]. Then why are both these Marginally stable systems?

Answer 1

Complex conjugate poles on the \$j\omega\$ axis can also produce an unbounded output, just like a pole at \$s=0\$. It just depends on the input signal. If you excite a system with a single pole at \$s=0\$ with an impulse, the output is bounded (it's a step). If you excite if with a step, the output is unbounded (it's a ramp). If you excite a system with complex conjugate poles at \$\pm j\omega_0\$ with a sinusoidal input signal with frequency \$\omega_0\$, then you'll get a sinusoidal output signal with linearly increasing amplitude (a ramped sinusoid), i.e., an unbounded signal.