Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws such as the well-known policy iteration. In this manuscript, the focus is on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman or composition operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup, in a specially weighted $\mathrm{L}^p$-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay which justifies finite rank approximations with numerical methods. The potential benefit for numerical computations will be demonstrated with two short examples.

翻译：李雅普诺夫函数在非线性动力学系统的控制理论中发挥着关键作用。除了其在稳定性分析中的经典用途外，李雅普诺夫函数还出现在计算最优反馈律的迭代方案中，例如著名的策略迭代算法。本文重点关注非线性自治有限维动力学系统的李雅普诺夫函数，该函数将利用Koopman算子或复合算子重构为无限维线性系统。由于该无限维系统具有弱*连续半群的结构，在特定加权的$\mathrm{L}^p$空间中，可以建立算子李雅普诺夫方程解与所需李雅普诺夫函数之间的联系。研究表明，该算子方程的解具有快速特征值衰减特性，这保证了使用数值方法进行有限秩近似的合理性。最后，通过两个简短示例展示了该方法在数值计算中的潜在优势。