I am an engineering student who works in the field of optimal control. Problems in this field are typically framed in infinite dimensional Sobolev space, $W^{k,p}(\Omega)$, as:
\begin{equation} \label{optimalControl} \begin{aligned} &J[u(t)] = \underset{u(t)}{\mathrm{minimize}} \int_{t_{0}}^{t_{f}} L(t,x(t),u(t))\hspace{0.1cm} dt \\ &s.t. \hspace{0.8cm} \dot{x}(t) = f(t,x(t),u(t)) \\ & \hspace{1.5cm}x(0) = x_{start} \\ & \hspace{1.5cm}x(t_f) = free \end{aligned} \end{equation}
where,
$\hspace{4.6cm} x(t)\in C^1[t_o,t_f]$ is state vector,
$\hspace{4.6cm}u(t)\in C^1[t_o,t_f]$ is control vector,
$\hspace{4.6cm}f[t,x(t),u(t)]$ is system dynamics,
$\hspace{4.6cm}L[t,x(t),u(t)]$ is Lagrangian, and
$\hspace{4.6cm}J[u(t)]$ is a scalar cost function
One of the methods to solve such a problem is to discretize it directly and recast it as a nonlinear program (NLP) in finite dimensional real space, $\mathbb{R}^{n}$. Here is my question:
What mathematical justification do we have to solve an infinite dimensional problem by converting it to a finite dimensional problem and expect to get a correct solution? In addition to looking for some relevant theorems that allow us to do that, it will be very helpful if somebody can also help me understand this intuitively.
Note: After digging a bit myself, I found a theorem which may be relevant here and is called Lax Equivalence Theorem. Is this it?