= Solution
For a normalized state $|\psi(t_F;f)\rangle$ driven by the admissible control $f$, maximize
$$
\boxed{J_A[f]=\langle\psi(t_F;f)|A|\psi(t_F;f)\rangle.}
$$
Here $A$ is the specified <Hermitian operator>, $t_F$ the final time, and the dependence on $f$ is through the <Schrödinger equation>. The objective is real and is exactly the desired <expectation value>, so increasing it increases the requested observable. A control-resource penalty may be included when the admissible set does not already constrain field strength or fluence.
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