Trapping Regions for Quadratic Systems with Generalized Lossless Nonlinearities

We consider a class of quadratic systems, primarily motivated by incompressible fluid flows, where the nonlinearities are generalized lossless: they do not produce or dissipate energy, as measured by a generalized quadratic metric. Our goal is to compute trapping regions, which are forward invariant sets that certify ultimate boundedness. The key contribution is a novel parameterization of the generalized lossless condition that enables optimization of trapping regions for a broader class of quadratic systems. We also formulate the conditions for ellipsoidal trapping regions, whereas spherical regions have been the focus of prior works. We provide three numerical examples, which demonstrate the improvements offered by the proposed approach relative to existing methods.