 Solution

For a concrete variational method, maximize the terminal-observable objective with a fluence penalty,where $\lambda >0$ weights the control cost. Enforce the dynamics with a complex costate $|\chi (t)⟩$ and the real augmented functionalVariation 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 givesThis equation has final data, so it is integrated backward. Variation of the real control gives the variational costate gradient for Hamiltonian quantum controlIn particular an unconstrained stationary pulse satisfies $f(t)=2\mathrm{Im}⟨\chi |H_1|\psi ⟩/(\lambda \hslash )$. 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.