  The Navier Stokes equations (NSEs) are partial differential equations (PDEs)
to describe the nonlinear convective motion of fluids and they are
computationally expensive to simulate because of their high nonlinearity and
variables being fully coupled. Reduced-order models (ROMs) are simpler models
for evolving the flows by capturing only the dominant behaviors of a system and
can be used to design controllers for high-dimensional systems. However it is
challenging to guarantee the stability of these models either globally or
locally. Ensuring the stability of ROMs can improve the interpretability of the
behavior of the dynamics and help develop effective system control strategies.
For quadratically nonlinear systems that represent many fluid flows, the
Schlegel and Noack trapping theorem (JFM, 2015) can be used to check if ROMs
are globally stable (long-term bounded). This theorem was subsequently
incorporated into system identification techniques that determine models
directly from data. In this work, we relax the quadratically energy-preserving
constraints in this theorem, and then promote local stability in data-driven
models of quadratically nonlinear dynamics. First, we prove a theorem outlining
sufficient conditions to ensure local stability in linear-quadratic systems and
provide an estimate of the stability radius. Second, we incorporate this
theorem into system identification methods and produce a-priori locally stable
data-driven models. Several examples are presented to demonstrate the
effectiveness and accuracy of the proposed algorithm.
