Quantum Mechanics and bilinear optimal control theory

I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.

So my question is something like this:

Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t) + u(t)H_1(x)\psi(x,t)$ be a Schrödinger equation. $H_0,H_1$ are nice operators (as nice as there is theory available, so for instance: $H_0$ has compact resolvent and $H_1$ is bounded, both on a finite interval with dirichlet conditions).

now we want to solve the problem whether we can move from a state $\psi_0$ to a state $\psi_1$ and want to minimize the function $u$ with respect to some norm.

Is there any theory available?

I already read that in an infinite dimensional space, we can only expect approximate controllability in the sense that $\psi_0$ can be only moved to $\psi_1$ up to an error of $\epsilon.$

So all this is fine, but I would really like to know if there is any literature on when this problem has a solution?

This is not quite what you are asking, but I addressed time optimality in my paper "Time-optimal control of finite quantum systems", J. Math. Phys. 41 (2000), 5262–5269. We have results like: if any control function reaches a desired target then there is one which does so in minimal time; a target is reached by a unique trajectory at time $t$ if it is an extreme point of the set of reachable targets at time $t$; any target reachable at time $t$ is reachable by a bang-bang control. The important qualifiers are that we work in the Born approximation, so the results are only valid over small times or energies, and we require the Hilbert space to be finite dimensional.