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系统控制理论中发挥着关键作用。除了稳定性分析这一经典用途外，李雅普诺夫函数还出现在迭代方案中，用于计算最优反馈律，例如著名的策略迭代。本文关注的是非线性自治有限维动力系统的李雅普诺夫函数，该函数将利用Koopman算子或复合算子重写为无限维线性系统。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Lyapunov方程的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，这验证了利用数值方法进行有限秩近似的合理性。通过两个简短示例，将展示其对数值计算的潜在益处。