= Zero-input response
The zero-input response is the output due to a system's initial internal state when its externally applied input is set to zero. It describes memory left by earlier excitation. For a <linear time-invariant system>, <linearity> separates this from the <zero-state response> due to the subsequent input.
For constant matrices in $\dot x=Ax+Bu$, $y=Cx+Du$, setting $u=0$ and $x(0)=x_0$ gives
$$
x_{\rm zi}(t)=e^{At}x_0,\qquad y_{\rm zi}(t)=Ce^{At}x_0.
$$
The <matrix exponential> solves the homogeneous state equation. With an arbitrary input, the complete solution adds the <zero-state response>:
$$
y(t)=Ce^{At}x_0+C\int_0^t e^{A(t-s)}Bu(s)\,ds+Du(t).
$$
A zero initial output $Cx_0=0$ alone does not establish that the whole state is zero.
For an <RC circuit> with <capacitor> difference $v(0)=V_0$ and no applied source, $RC\,\dot v+v=0$, so $v(t)=V_0e^{-t/(RC)}$. The <capacitor energy> falls as $CV_0^2e^{-2t/(RC)}/2$; the resistor receives the dissipated energy. The primary MIT lecture explicitly distinguishes the capacitor's initial state, zero external input and zero initial state.<State and Memory by Anant Agarwal and Jeffrey Lang>[PDF pages 2–6, RC memory and zero-input/zero-state decomposition]{ref}
At a <time constant> $RC=2\,\mathrm s$ and $V_0=3\,\mathrm V$, the modeled difference after two seconds is $3/e\approx1.104\,\mathrm V$. Zero input has not meant zero output or zero stored energy. A direct experiment charges the same <capacitor> to different known initial differences, removes the charging source and records the decay across a fixed <resistor>. The amplitudes should scale with $V_0$ while the normalized decay remains the same in the linear model.
A zero-input response need not decay in every system: an ideal undamped oscillator retains its initial motion, and an unstable homogeneous system can amplify a nonzero initial state. The <transfer function> describes the zero-state input-output map, so its product with a zero input cannot represent a separate nonzero initial-state contribution.
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