This paper aims at developing a functional approach to investigate the Morse structures of attractors for infinite-dimensional dynamical systems generated by PDEs. Let $S\left(t\right)$ be a semiflow on a complete metric space $X$, and $\mathcal{A}$ an attractor of $S\left(t\right)$ with a Morse decomposition $\mathcal{M}.$ We first construct some good Morse–Lyapunov functions for $\mathcal{M}$ which seem to be of independent interest in their own right. Then we establish some fundamental deformation lemmas. Based on these results, we finally introduce the concept of critical groups for Morse sets and prove Morse inequalities and equations. The problem of how to calculate Morse equations of attractors by using a natural Morse–Lyapunov function which may be defined only on a dense subspace of $X$ is also addressed, and an illustrating example of parabolic equation is presented.

本文旨在开发一种功能方法来研究由偏微分方程产生的无限维动力系统的吸引子的莫尔斯结构。让$秒\left(吨\right)$ 是完全度量空间上的半流 $X$， 和 $\mathcal{一种}$ 的吸引子 $秒\left(吨\right)$ 用莫尔斯分解 $\mathcal{米}.$ 我们首先构造一些好的 Morse-Lyapunov 函数 $\mathcal{米}$这似乎是他们自己的独立利益。然后我们建立一些基本的变形引理。基于这些结果，我们最终引入了 Morse 集的临界群的概念，并证明了 Morse 不等式和方程。如何使用自然的 Morse-Lyapunov 函数计算吸引子的 Morse 方程，该函数只能定义在$X$ 还提出了一个抛物线方程的说明性示例。