Nonlinear input-output analysis of transitional shear flows using small-signal finite-gain $\mathcal{L}_p$ stability

Abstract

This SSFG Lp stability theorem can predict permissible forcing amplitudes below which a finite nonlinear input-output gain can be maintained. Our analysis employs Linear Matrix Inequalities (LMI) and Sum-of-Squares (SOS) as the primary tools to search for a quadratic Lyapunov function of an unforced nonlinear system. The resulting Lyapunov function can certify the SSFG Lp stability of a nonlinear input-output system. We demonstrate the applicability of the SSFG Lp stability theorem using a nine-mode shear flow model with a random body force. The predicted nonlinear input-output Lp gain 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 $p \rightarrow \infty$. The input-output gain obtained from the nonlinear SSFG Lp stability theorem is higher than the linear Lp gain. Both nonlinear Lp gain and linear Lp 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 L2 gain. 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 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 Lp stability theorem is conservative but also consistent with that obtained by numerical simulations and bisection search.