  The precise set of parameters governing the transition to turbulence in
wall-bounded shear flows remains an open question; many theoretical bounds have
been obtained, but there is not yet a consensus between these bounds and
experimental/simulation results. In this work, we focus on a method to provide
a provable Reynolds number dependent bound on the amplitude of perturbations a
flow can sustain while maintaining the laminar state. Our analysis relies on an
input--output approach that partitions the dynamics into a feedback
interconnection of the linear and nonlinear dynamics (i.e., a Lur\'e system
that represents the nonlinearity as static feedback). We then construct
quadratic constraints of the nonlinear term that is restricted by system
physics to be energy-conserving (lossless) and to have bounded input--output
energy. Computing the region of attraction of the laminar state (set of safe
perturbations) and permissible perturbation amplitude are then reformulated as
Linear Matrix Inequalities (LMI), which provides a more computationally
efficient solution than prevailing nonlinear approaches based on the sum of
squares programming. The proposed framework can also be used for energy method
computations and linear stability analysis. We apply our approach to low
dimensional nonlinear shear flow models for a range of Reynolds numbers. The
results from our analytically derived bounds are consistent with the bounds
identified through exhaustive simulations. However, they have the added benefit
of being achieved at a much lower computational cost and providing a provable
guarantee that a certain level of perturbation is permissible.
