I have been studying about gradient flow in Hilbert space of a Morse function $f$. Specifically, let $X$ be a Hilbert space and $f : X\to \mathbb R$ be $C^3$ function. The gradient flow here is defined by $x'(t) = \text{grad} \ f(x(t))$. I know that if $f$ is a Morse function (i.e, all critical points are non- degenerate) and if $f$ satisfies the Palais-Smale condition, then all flow trajectory have limits, i.e, $\lim_{t \to \infty} x(t)$ exists. Jürgen Jost's Riemannian Geometry and Geometric Analysis has a very nice discussion (in chapter 6 Morse Theory and Floer Homology) on this topic if $X$ is of finite dimensional. He does mention that the result can be generalized to (infinite dimensional) Hilbert manifolds. However, he doesn't write down the full details of the infinitely dimensional space case.
So suppose X is a Hilbert space. Given $f$ satisfying Palais-Smale condition condition and $f$ being a Morse function, I am wondering whether flow trajectories converge or not. And I would really appreciate if you can point me any reference about this subject. Thanks a head.