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

TL;DR

This paper introduces a nonlinear stability theorem using LMI and SOS techniques to predict input-output gains in shear flows, validated through simulations and providing higher bounds than linear analysis.

Contribution

The paper develops a novel SSFG Lp stability theorem for nonlinear shear flows, enabling prediction of permissible forcing amplitudes and input-output gains with improved accuracy over linear methods.

Findings

The nonlinear Lp gain exceeds the linear Lp gain.

Predicted bounds are consistent with numerical simulations.

Permissible forcing amplitude is conservatively estimated.

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.