Nonlinear input-output analysis of transitional shear flows using small-signal finite-gain Lp stability

This work employs the Small-Signal Finite-Gain (SSFG) Lp stability theorem to analyze nonlinear input-output amplification of a nine-mode shear flow model with random input forcing. Using linear matrix inequalities and sum-of-squares techniques, we search for a quadratic Lyapunov function of unforced system to certify SSFG Lp stability of nonlinear input-output system. The predicted nonlinear input-output Lp gain (amplification) is consistent with numerical simulations; the Lp norm of the output from numerical simulations remains bounded by the theoretical prediction from SSFG Lp stability theorem, with the gap between simulated and theoretical bounds narrowing as p approaches infinity. The nonlinear Lp gain exceeds the linear Lp gain and linear L2 gain, indicating that nonlinearity can significantly amplify small disturbances. The SSFG Lp 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. The permissible forcing amplitude identified from the SSFG Lp stability theorem is also consistent with that obtained by numerical simulations and bisection search. This study demonstrates that the nonlinear SSFG Lp stability theorem is a powerful tool for analyzing transitional shear flows with external disturbance, which can predict permissible forcing amplitudes below which a finite nonlinear input-output gain can be maintained.