A Convex Optimization Approach to Compute Trapping Regions for Lossless Quadratic Systems

Abstract: Quadratic systems with lossless quadratic terms arise in many applications, including models of atmosphere and incompressible fluid flows. Such systems have a trapping region if all trajectories eventually converge to and stay within a bounded set. Conditions for the existence and characterization of trapping regions have been established in prior works for boundedness analysis. However, prior solutions have used non-convex optimization methods, resulting in conservative estimates. In this paper, we build on this prior work and provide a convex semidefinite programming condition for the existence of a trapping region. The condition allows precise verification or falsification of the existence of a trapping region. If a trapping region exists, then we provide a second semidefinite program to compute the least conservative trapping region in the form of a ball. Two low-dimensional systems are provided as examples to illustrate the results. A third high-dimensional example is also included to demonstrate that the computation required for the analysis can be scaled to systems of up to $\sim O(100)$ states. The proposed method provides a precise and computationally efficient numerical approach for computing trapping regions. We anticipate this work will benefit future studies on modeling and control of lossless quadratic dynamical systems.