Stability Domains for Quadratic-Bilinear Reduced-Order Models

We propose a computational approach to estimate the stability domain of quadratic-bilinear reduced-order models (ROMs), which are low-dimensional approximations of large-scale dynamical systems. For nonlinear ROMs, it is not only important to show that the origin is locally asymptotically stable, but also to quantify if the operative range of the ROM is included in the region of convergence. While accuracy and structure preservation remain the main focus of development for nonlinear ROMs, computational methods that go beyond the existing highly conservative analytical results have been lacking thus far. In this work, for a given quadratic Lyapunov function, we first derive an analytical estimate of the stability domain, which is rather conservative but can be evaluated efficiently. With the goal to enlarge this estimate, we provide an optimal ellipsoidal estimate of the stability domain by solving a convex optimization problem. This provides us with valuable information about stability properties of the ROM, an important aspect of predictive simulation. We do not assume a specific ROM method, so a particular appeal is that the approach is applicable to quadratic-bilinear models obtained via data-driven approaches, where ROM stability properties cannot - per definition - be derived from the full-order model. Numerical results for a LQG-balanced ROM of Burgers' equation, a proper orthogonal decomposition ROM of FitzHugh-Nagumo, and a non-intrusive ROM of Burgers' equation demonstrate the scalability and quantitative advantages of the proposed approach. The optimization-based estimates of the stability domain are found to be up to four orders of magnitude less conservative than analytical estimates.