= Solution
For a concrete variational method, maximize the terminal-observable objective with a fluence penalty,
$$
J[f]=\langle\psi(T)|A|\psi(T)\rangle-\frac\lambda2\int_0^Tf(t)^2dt,\qquad
i\hbar\dot\psi=(H_0+fH_1)\psi,\quad\psi(0)=\psi_0,
$$
where $\lambda>0$ weights the control cost. Enforce the dynamics with a complex <costate> $|\chi(t)\rangle$ and the real augmented functional
$$
\mathcal J=J-2\operatorname{Re}\int_0^T\left\langle\chi\left|\dot\psi+\frac i\hbar(H_0+fH_1)\psi\right.\right\rangle dt.
$$
Variation of $\chi$ restores the state equation. Integrate the $\dot\psi$ variation by parts; because $\delta\psi(0)=0$, the terminal coefficient gives $\chi(T)=A\psi(T)$, and the interior coefficient gives
$$
i\hbar\dot\chi=(H_0+fH_1)\chi.
$$
This equation has final data, so it is integrated backward. Variation of the real control gives the <variational costate gradient for Hamiltonian quantum control>
$$
\boxed{\frac{\delta J}{\delta f(t)}=\frac2\hbar\operatorname{Im}\langle\chi(t)|H_1|\psi(t)\rangle-\lambda f(t).}
$$
In particular an unconstrained stationary pulse satisfies $f(t)=2\operatorname{Im}\langle\chi|H_1|\psi\rangle/(\lambda\hbar)$. The state, adjoint and control equations form a coupled two-boundary-value problem, so the stationarity formula is not an independent explicit solution for $f$.
Use <direct-adjoint looping>: start from an admissible pulse, propagate the state forward, set the terminal <costate>, propagate it backward, compute the <gradient> on the time grid, and update the pulse with an ascent step and line search. Project amplitude bounds and impose bandwidth constraints within the parametrization or update. Repeat until the projected <gradient> is small. For gate objectives the forward variable is the propagator and the <costate> is matrix-valued; for mixed-state objectives it is the <density operator>. <Gradient ascent pulse engineering> applies the same efficient forward/backward differentiation to time-slice amplitudes. The variational conditions identify local stationary controls and require numerical convergence checks; they do not prove global optimality of a nonconvex control landscape.
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