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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 work 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 for 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 radius of the spherical trapping region. 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 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. 

Abstract The structured singular value (SSV), or , is used to assess the robust stability and performance of an uncertain linear time‐invariant system. Existing algorithms compute upper and lower bounds on the SSV for structured uncertainties that contain repeated (real or complex) scalars and/or nonrepeated complex full‐blocks. This paper presents algorithms to compute bounds on the SSV for the case of repeated complex full‐blocks. This specific class of uncertainty is relevant for the input‐output analysis of many convective systems, such as fluid flows. Specifically, we present a power iteration to compute the SSV lower bound for the case of repeated complex full‐blocks. This generalizes existing power iterations for repeated complex scalars and nonrepeated complex full‐blocks. The upper bound can be formulated as a semi‐definite program (SDP), which we solve using a standard interior‐point method to compute optimal scaling matrices associated with the repeated full‐blocks. Our implementation of the method only requires gradient information, which improves the computational efficiency of the method. Finally, we test our proposed algorithms on an example model of incompressible fluid flow. The proposed methods provide less conservative bounds as compared to prior results, which ignore the repeated full‐block structure.