Fix a Riemannian metric on a manifold $M$. Suppose that we fix two points $x,y \in M$. We start with the space

$C^{\infty}_{\searrow}(x,y) = \left\{\gamma: \mathbb{R}\to M:\,\lim_{t\to-\infty}\gamma(t) = x, \lim_{t\to\infty}\gamma(t)=y, \exists\,C,\delta > 0 \text{ such that} \vert\gamma'(t)\vert \leq Ce^{-\delta\vert t\vert}\right\}$

We can complete this space to a Banach manifold, $\mathcal{P}^{1,p}(x,y)$. The charts are given by pairs $(U_{\gamma},\gamma^{*}\exp)$, where $\gamma \in C^{\infty}_{\searrow}(x,y)$, $\exp: TM \to M$ is the exponential map coming from our metric, and $U_{\gamma} \subseteq W^{1,p}(\gamma^{*}TM)$ is chosen small enough so that for any $X \in U_{\gamma}$, $\exp_{\gamma(t)}{X(t)}$ stays within an injectivity radius of $\gamma(t)$.

I'm interested in a generalization of this construction when the endpoints of my curves lie in submanifolds, $S_{1},S_{2}\subseteq M$. I could try and do the same thing as above, varying over all pairs $(x,y) \in S_{1}\times S_{2}$:

$\mathcal{P}^{1,p}(S_1,S_2) := \coprod_{(x,y) \in S_1\times S_2} \mathcal{P}^{1,p}(x,y)$

But this gives me a space with the "wrong" topology. For example, any two paths with different endpoints are not in the same path component of $\mathcal{P}^{1,p}(S_1,S_2)$ as I've defined it here. I want paths with nearby endpoints to be considered close together.

To fix the previous problem, I could instead complete to a space where my local model is:

$C^{k}_{S_1,S_2}(\gamma^{*}TM) = \left\{X \in C^{k}(\gamma^{*}TM):\, \lim_{t\to-\infty} X(t) \in TS_1, \lim_{t\to\infty} X(t) \in TS_2\right\}$

This local model allows for variations of the end points; however, the drawback is that I can't use the same charts as I did before. The problem is that $S_1$ and $S_2$ might not be totally geodesic manifolds, so my endpoints will detach from $S_1$ and $S_2$ when I try and deform through some $X \in C^{k}_{S_1,S_2}(\gamma^{*}TM)$ using the exponential map. For the applications I have in mind, assuming that $S_1$ and $S_2$ are totally geodesic submanifolds is too restrictive.

Here are my questions:

Is there already in the literature a construction of a Banach manifold of paths whose endpoints are on fixed submanifolds? If it helps, I'm interested in this space in the context of Morse-Bott theory.

Is there any other construction which allows me to produce a 1-parameter family of curves $\gamma_s(t)$ such that $\partial_s\gamma = X$, and so that the endpoints of $\gamma_s$ are always on $S_1$ and $S_2$?