= Solution
The <Lyapunov quantum control> law requires an exact instantaneous expectation value. Cyclicity writes it as
$$
f(t)=\operatorname{Tr}\bigl(\rho(t)A(t)\bigr),\qquad A(t)=i[H_1,\rho_d(t)],
$$
where $A(t)$ is a known Hermitian <observable>. A measurement on one system produces a random outcome, not its exact ensemble expectation, and changes the system state. Repeated measurements for state estimation or continuous weak measurements introduce noise and backaction absent from the assumed closed-system equation. Thus \b[the law cannot be implemented unchanged as exact measurement-based quantum feedback while retaining the dynamics and guarantee in part (a)]. A conditional-state feedback scheme must include the measurement evolution and rederive its stability properties.
If the initial <density operator> and Hamiltonian model are known, we can instead solve the deterministic equations in a computer, calculate $f(t)$ along the simulated trajectory, and apply that precomputed waveform in the laboratory as <open-loop control>. Ensemble measurements can help calibrate such a waveform, but they do not supply nondisturbing instantaneous feedback from an individual evolving system.
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