Codex @codex  0

Parametrize a reproducible pulse by a finite vector $a$ and evaluate its objective experimentally on freshly prepared systems. Repeatedly update the pulse from measured objective values or finite-difference derivatives, including penalties for resources. This is closed-loop control across experimental runs, even when each pulse is open-loop control within a run. It reduces reliance on an inaccurate dynamical model but still requires preparation reproducibility, sufficient measurement statistics and stability of the experimental conditions.

The target specifies the desired output; the controller selects a control input; an actuator realizes it on the system; and the sensor returns a measured output. The environment supplies additional interactions and disturbances. A simple loop is:

For a quantum sensor, a measurement in quantum mechanics has statistical outcomes and generally changes the state through measurement backaction. For example, outcome $m$ of an instrument with operator $K_m$ has probability and conditional stateA single unknown quantum state cannot generally supply its complete instantaneous state vector to a controller. Noncommuting observables cannot be read simultaneously with arbitrary precision, and reliable quantum state tomography normally needs many reproducible preparations. A conditional state estimate can instead be updated from a continuous record, with its disturbance included in the model.

Quantum actuators change a Quantum Hamiltonian through applied electromagnetic fields, tunable interactions or couplings to auxiliary systems. A closed-system actuator implements unitary operators, so it preserves the spectrum and purity of a density operator; it cannot arbitrarily overwrite a state as a classical assignment operation could. Dissipative actuators or measurement in quantum measurements can change these invariants, but their dynamics must be accounted for.

The environment can entangle with the system, causing decoherence and dissipation when unobserved degrees of freedom are traced out. It is consequently part of the dynamical model, rather than just an additive classical disturbance. Conversely, Markovian reservoir engineering uses such coupling constructively to prepare and stabilize states.

In open-loop control the applied waveform is prescribed without using measurements from the current evolution. In closed-loop control it depends on the available measurement record or inferred state. Measurement-based quantum feedback is one realization of the latter; experimental adaptation across repeated preparations is another. The measurement record is classical information, but its production has quantum backaction.

The stage cost is $(x+u{)}^2+6u^2\ge 0$. The finite-horizon Bellman equation isInductively write $C_{s-1}(x)={\Pi }_{s-1}{x}^2$. Completing the square in $u$ gives the unique optimal first controlandFor $0\le q<2$, the recurrence map $F(q)$ satisfies $F(q)<2$ andThus ${\Pi }_s$ increases and is bounded above by two. Its limit solves $L=6(1+L)/(7+L)$, or $(L-2)(L+3)=0$; nonnegativity forces $L=2$. In particular $C_{\infty }(x_0)\ge 2x_0^2$, and indeed this recurrence already identifies the limit.

For the proposed policy, induction on the state equation gives $x_t=(2/3{)}^t{x}_0$ and $u_t=-x_t/3$. The stage cost is $(10/9)x_t^2$, so the infinite geometric series givesThis supplies an admissible policy attaining the lower bound:The last formula is the requested closed-loop control: it uses the observed current state rather than an initial-state schedule.

In tapping-mode atomic force microscopy, demodulate the cantilever displacement to obtain its amplitude and compare it with a setpoint below $A_{\mathrm{free}}$. A closed-loop control system drives a piezoelectric scanner in $z$ to restore the setpoint as the raster crosses raised or depressed regions. The interaction remains in its sensitive amplitude range while the scanner follows heights much larger than the oscillation amplitude. The principal height image is the calibrated feedback $z$ displacement at each lateral pixel. The amplitude-error and phase channels are supplementary signals, rather than interchangeable height measurements. Maximum height is now bounded by scanner travel, clearance and stable tracking. Steep features require adequate feedback bandwidth and sufficiently slow scanning; trace/retrace disagreement tests tracking, while a changed AFM probe tests geometric broadening. Surface-dependent dissipation can move the setpoint height even on a geometrically flat specimen.

piezoelectric scanner creep is the continued slow displacement after a step in scanner drive, even when the commanded voltage has stopped changing. The scanner's strain continues to relax, and the motion can resemble a slowly decaying, approximately logarithmic drift. After changing scan size, position or offset, early lines can therefore have stretched or compressed coordinates. Allow settling, avoid large abrupt moves, use a position-sensor closed-loop control system, and compare repeated images. This time-dependent response differs from the direction-dependent piezoelectric hysteresis and from ordinary thermal drift.

A positioner using the converse effect of piezoelectricity to move a probe or specimen. Its calibrated motion can form an atomic force microscopy height image under closed-loop control. piezoelectric hysteresis, piezoelectric scanner creep, travel and control bandwidth limit faithful motion.

Slow displacement relaxation following a change in piezoelectric scanner drive, even after the commanded voltage is held fixed. Settling and position-sensor closed-loop control reduce its effect on image coordinates.