Quadratization refers to a transformation of an arbitrary system of polynomial ordinary differential equations to a system with at most quadratic right-hand side. Such a transformation unveils new variables and model structures that facilitate model analysis, simulation, and control and offers a convenient parameterization for data-driven approaches. Quadratization techniques have found applications in diverse fields, including systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis. In this study, we focus on quadratizations that preserve the stability properties of the original model, specifically dissipativity at given equilibria. This preservation is desirable in many applications of quadratization including reachability analysis and synthetic biology. We establish the existence of dissipativity-preserving quadratizations, develop an algorithm for their computation, and demonstrate it in several case studies.

翻译：二次化是指将任意多项式常微分方程组转化为至多二次右端项系统的变换。这种变换能揭示新变量和模型结构，促进模型分析、仿真与控制，并为数据驱动方法提供便捷的参数化表示。二次化技术已广泛应用于系统理论、流体力学、化学反应建模及数学分析等多个领域。本研究聚焦于保持原始模型稳定性性质的二次化方法，特别是给定平衡点的耗散性。这种保持特性在可达性分析和合成生物学等二次化应用场景中具有重要价值。我们证明了耗散保持型二次化的存在性，开发了相应的计算算法，并通过多个案例研究进行了验证。