Input-output analysis has been widely used to predict the transition to turbulence in wall-bounded shear flows, but it typically does not capture the full nonlinear effects involved. This work employs the small-signal finite-gain (SSFG) ${\mathcal{L}}_p$ stability theorem to assess nonlinear input-output stability. This SSFG ${\mathcal{L}}_p$ stability theorem can predict permissible forcing amplitudes below which a finite nonlinear input-output gain can be maintained. Our analysis employs linear matrix inequalities and sum-of-squares as the primary tools to search for a quadratic Lyapunov function of an unforced nonlinear system. The resulting Lyapunov function can certify the SSFG ${\mathcal{L}}_p$ stability of a nonlinear input-output system. We demonstrate the applicability of the SSFG ${\mathcal{L}}_p$ stability theorem using a nine-mode shear flow model with a random body force. The predicted nonlinear input-output ${\mathcal{L}}_p$ gain (amplification) is consistent with numerical simulations; the ${\mathcal{L}}_p$ norm of the output from numerical simulations remains bounded by the theoretical prediction from the SSFG ${\mathcal{L}}_p$ stability theorem, with the gap between simulated and theoretical bounds narrowing as $p\to \infty$. The input-output gain obtained from the nonlinear SSFG ${\mathcal{L}}_p$ stability theorem is higher than the linear ${\mathcal{L}}_p$ gain, which suggests that nonlinearity can significantly amplify small disturbances. Both nonlinear ${\mathcal{L}}_p$ gain and linear ${\mathcal{L}}_p$ gain are valid for each $p\in [1,\infty ]$, and such generalizability leads to much higher upper bounds on input-output gain than those predicted by linear ${\mathcal{L}}_2$ gain (${\mathcal{H}}_{\infty }$ norm of transfer function). The SSFG ${\mathcal{L}}_p$ stability theorem requires the input forcing to be smaller than a permissible forcing amplitude to maintain finite input-output gain, which is an inherently nonlinear behavior that cannot be predicted by linear input-output analysis. We also identify such permissible forcing amplitude using numerical simulations and bisection search, where below such forcing amplitude, the output norm at any time will be lower than a given threshold value. The permissible forcing amplitude identified from the SSFG ${\mathcal{L}}_p$ stability theorem is conservative but also consistent with that obtained by numerical simulations and bisection search. This study demonstrates that the nonlinear SSFG ${\mathcal{L}}_p$ stability theorem is a powerful tool for analyzing nonlinear input-output systems such as transitional shear flows with external disturbance.

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• Flow control
• Transition to turbulence