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：Lyapunov 函数在非线性动态系统的控制理论方面发挥着关键作用。 Lyapunov 函数除了在稳定分析中古典使用外, 在计算最佳反馈法的迭接方案中也产生 Lyapunov 函数, 如众所周知的政策迭代。 在本手稿中, 重点是非线性自主的有限维动态系统的Lyapunov 函数, 这个系统将使用 Koopman 或构成操作员重写成一个无限线性线性系统。 由于这个无线性系统有一个弱* 连续的半组结构, 在一个特别加权的 $\ mathrm{L ⁇ ⁇ p$- space one 中, 一个可以在操作者 Lyapunov 方程式的解决方案和想要的 Lyapunov 函数之间建立连接。 将会显示, 这个操作者方程式的解决方案会达到快速的天价衰变, 从而可以用数字方法来证明定级对准。 数字计算的潜在好处将用两个简短的例子来证明 。