I heard that the manifold consisting of solutions of a given PDEs is of infinite dimension.
Is this statement true for all PDEs ?
Could we write a PDE as infinite ODEs and in that case, the statement makes sense ?
Here is a precise context.
We define an optimal control problem as $$ \underset{u(.)}{ \min } \; C(u),\quad \dot q = f(q,u), \quad q(t) \in M $$ in that case $M$ is of finite dimension because we have an ODE but if we replace this ODE to a PDE, $M$ is of infinite dimension.