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Controllability asks whether the available control inputs can move a dynamical system between every pair of states in its declared state space. Fixing a duration $T>0$ gives the precise condition ${\mathcal{R}}_T(x_0)=X$ for every $x_0\in X$, where $X$ is the state space and ${\mathcal{R}}_T(x_0)$ its reachable set. With freely chosen duration, every pair must instead be connectable in some permitted finite time, which may depend on the pair. Always specify admissible control inputs, deadlines and any constraints along the path. ^[ref]

The surprising part is that one control input can control several state variables by acting at different times. For the double integrator, acceleration changes velocity directly, while the accumulated velocity changes position. Starting from rest, accelerate during the first half of the journey and brake during the second: the final velocity can vanish although the final position has changed. The figure is an analytic mathematical example in dimensionless quantities, with no experimental measurements.

Controllability is useful before trying to design a controller: a requested endpoint outside the reachable set cannot be attained by clever choice of feedback control. The Kalman rank criterion makes this an exact linear algebra calculation for the stated linear time-invariant system model. It concerns whole states, so attaining one plant output is not enough to establish controllability. In quantum control, the declared state space may be an orbit constrained by conservation laws; see quantum controllability and unitary orbit of a density operator.

For $T>0$ and the constant matrices over the real numbers in the Kalman rank criterion, the finite-time Controllability Gramian accumulates the influence of all command times. Its column space equals that of the controllability matrix, and $W_T$ is a symmetric positive-definite matrix exactly when the controllability matrix has rank of a matrix $n$. If so, put $\delta =x_1-e^{AT}{x}_0$. Among square-integrable functions steering $x_0$ to $x_1$ at $T$, the commandminimizes ${\int }_0^T\Vert u(t){\Vert }^{2}dt$, with value ${\delta }^T{W}_T^{-1}\delta$. Here $T$ in a matrix superscript denotes transpose. This cost is a squared input norm, not necessarily physical energy, and the optimizing command need not satisfy an amplitude bound. ^[ref]

For the double integrator, $e^{As}B=(s,1{)}^T$, so direct integration givesFor a rest-to-rest distance $d$, $u_{\ast }(t)=6d/T^2-12dt/T^3$ and the minimum squared input norm is $12d^2/T^3$. Compare the two constant commands in the double integrator example, whose cost is $16d^2/T^3$: both attain the target, but the smooth command costs less. At $T=2$, $d=1$, however, $u_{\ast }$ peaks at $3/2$, so it violates $|u|\le 1$ while the two constant commands remain admissible. Controllability, optimal control, and constrained reachability in control theory answer different questions.

The double integrator is the linear time-invariant system in which one control input is integrated into velocity, and velocity into position. It is a mathematical idealization of one-dimensional motion under commanded acceleration; for a constant mass $m$ and net force $F$, Newton's second law gives $u=F/m$. The worked examples here use normalized dimensionless quantities and assume no control disturbances, rather than describing a laboratory record. ^[ref]

Writing $x=(p,v{)}^T$ givesThe Kalman rank criterion therefore gives controllability in every $T>0$ for unrestricted real control inputs. More concretely, choose constants $a,b$ on the first and second halves. Starting at $x_0=(p_0,v_0{)}^T$, direct integration givesThe endpoint map has determinant $T^3/8\ne 0$, so these two amplitudes can supply any desired endpoint displacement. For a rest-to-rest move of distance $d$, take $a=4d/T^2$, $b=-4d/T^2$. At $T=2$ and $d=1$ these are the two commands in the first figure. No impulse is needed; shorter deadlines require larger finite commands.

For a finite-dimensional continuous-time linear time-invariant system $\dot{x}=Ax+Bu$ on ${\mathbb{R}}^n$, with constant matrices $A,B$ over the real numbers, no state constraints, and every piecewise continuous function $u:[0,T]\to {\mathbb{R}}^m$ allowed as a control input without an amplitude bound, controllability in any fixed $T>0$ is equivalent to the controllability matrix having rank of a matrix $n$. The same equivalence holds with all square-integrable functions as inputs. A stronger useful statement iswhere $e^{AT}$ is a matrix exponential and $\mathrm{im}\mathcal{C}$ is the column space. Thus a rank of a matrix $r<n$ leaves only an $r$-dimensional affine subspace reachable at that time. ^[ref]

The Kalman rank criterion allows arbitrarily large commands as $T$ shrinks. It does not assert controllability with bounded control inputs, nor test a general nonlinear dynamical system just by counting its inputs. For the double integrator, $B=(0,1{)}^T$ and $AB=(1,0{)}^T$ give two independent directions, despite there being only one control input.

Set ${\omega }_1={\omega }_2=\omega$. Define the quantum rotating frame byThe isotropic Heisenberg model coupling commutes with the total $z$ spin angular momentum, so $[H_Z,C]=0$. The transformed Hamiltonian operator is thereforeHere removal of the common Zeeman energy is exact; the exchange is resonant and cannot be discarded using the weak coupling condition for two NMR spins.

Put $a=Jt$, $c=\cos a$, and $s=\sin a$. Exponentiating the parallel-spin entries and the middle $2\times 2$ matrix gives the time-evolution operatorFor example, at $a=\pi /2$ this equals the SWAP gate times $e^{-i\pi /4}$, which checks both the exchange angle and the overall phase.

The printed network reads from left to right as $U$, a Pauli Z gate on the upper spin only, $U$, and another upper-spin Pauli Z gate. LetIts total time-evolution operator, with the first operation on the right, is $W=PUPU$. For this Heisenberg exchange echo,The three paired Pauli operators $X\otimes X$, $Y\otimes Y$, $Z\otimes Z$ commute pairwise: for example both products $(X\otimes X)(Y\otimes Y)$ and $(Y\otimes Y)(X\otimes X)$ equal $-Z\otimes Z$. Thus $P{H}_{\mathrm{r}}P$ commutes with $H_{\mathrm{r}}$, andThe spin echo cancels the $XX+YY$ exchange terms and retains the Ising coupling. Equivalently, over the total free-evolution time $2t$ it produces the effective Hamiltonian operator $JZ\otimes Z/2$ exactly for ideal instantaneous gates. It is generally not the identity. A physical $\pi$ rotation about $z$ is $e^{-i\pi Z/2}=-iZ$; using two such pulses instead of the ideal $Z$ gates changes $W$ only by a physically irrelevant global phase of minus one.

The implementation requires frequency-selective spin control. Under the usual common-radio frequency NMR assumptions, equal resonance frequencies give no way to apply a pulse to the first spin while leaving the second untouched. For equally coupled spins in a spatially uniform control field, the transverse control Hamiltonian operator has the formBoth this control and the drift Hamiltonian operator commute with the SWAP gate $S$. Every reachable time-evolution operator consequently commutes with $S$. But $SPS=I\otimes Z\ne P$: the individual pulse required by the diagram fails this symmetry test. This is the exchange symmetry obstruction to individual spin control. Applying simultaneous $Z$ gates gives $Q=Z\otimes Z$, which commutes with $H_{\mathrm{r}}$ and instead yields $QUQU=U^2$; it does not refocus the exchange. A common change of the RF phase reference also supplies only a collective rotation.

This obstruction assumes the usual common-RF control with no extra handle distinguishing the spins. Equal resonance frequencies do not universally forbid individual addressing: a spatially selective field, unequal control couplings, or another symmetry-breaking interaction could alter controllability. Such resources are absent from the specified equivalent-spin NMR model.

Learning theory explains pathological behaviour through representations, action selection, controllability and retrieval as well as through prediction errors; mathematical description becomes explanatory only when those quantities are connected to people, environments and interventions. A useful computational psychiatry account identifies the problem a person faces, the information available, the internal representation and the rule selecting an action. A fitted learning rate alone does not establish the cause of a mental disorder.

First, distinguish goal-directed action from habit. A goal-directed controller evaluates outcomes using their current value, whereas a habit can repeat a previously acquired stimulus–response policy despite a changed outcome. An illustrative mixture isIf washing formerly relieved distress, lowering the presently expected benefit need not immediately eliminate washing when the habitual component remains influential. This formulation distinguishes an erroneous threat appraisal from failure to use an accurately represented outcome. Either can produce repetitive behaviour, so persistence alone does not distinguish the mechanisms.

An outcome devaluation experiment makes the distinction testable. In Claire M. Gillan's study, people learned button responses to pictured stimuli for outcomes; instructions then made selected outcomes worthless, and a speeded slips-of-action test required withholding their associated responses. Participants with OCD showed poorer outcome knowledge and greater devaluation-insensitive responding despite acquiring the trained responses. This supports deficient goal-directed control rather than demonstrating exceptionally strong habit acquisition. Medication, task understanding and the difference between laboratory responses and actual compulsions limit the inference.^[ref]

Second, action-outcome contingency separates frequent adversity from absent control. Define $\Delta P=P(O\mid A)-P(O\mid \neg A)$ for an aversive outcome $O$. With experimental action assignment, values $0.2-0.8=-0.6$ indicate that acting protects against the outcome, whereas $0.8-0.8=0$ supplies no such control. A learner who generalises zero control to a new situation may stop trying even when a useful action has become available. This is a route to helpless behaviour without merely asserting a large negative prediction error. Observational conditional probabilities require caution because an action may be chosen precisely when danger is greatest.

The motivating yoked-control logic holds exposure constant while changing whether a response terminates it. In rodent experiments, Jose Amat and colleagues linked the protective effect of control to medial prefrontal cortex regulation of the dorsal raphe nucleus: prefrontal inactivation removed that protection. The circuit findings constrain a purely abstract account of learned helplessness, but neither the rodent procedure nor reduced escape defines human depression.^[ref]

Third, negative reinforcement can make avoidance locally successful and globally disabling. Leaving a feared situation removes distress, strengthening escape, while preventing experience that the situation would have been safe. The crucial informational question becomes whether safety is attributed to the environment or to the escape response. In a human conditioning experiment, retaining an avoidance button during otherwise unreinforced cue presentations preserved threat expectancy and skin conductance responses when the button was subsequently unavailable. This supports a mechanism by which safety behavior can obstruct corrective learning.^[ref]

Fourth, retrieval is context dependent. An old fear association and subsequent extinction learning can coexist; the action observed depends on which learned association guides responding. In rat conditioned-suppression experiments, extinguishing a cue outside its acquisition context allowed fear renewal on return. Thus relapse can reflect a retrieval change rather than a failure to acquire anything during treatment. The account motivates practising safety across settings and testing retention outside the treatment room.^[ref]

Mathematics also exposes ambiguity. For softmax action choice,The choice-value scale non-identifiability follows because rescaling every value by $c>0$ and replacing $\beta$ by $\beta /c$ leaves all choice probabilities unchanged. Choice data alone cannot identify both scales. What appears to be low reward valuation may instead be noisy choice, a representation difference or a poorly specified task. Parameter recovery, independent measurements, alternative-model comparison and predictions for new conditions are therefore necessary.

Finally, psychopathology involves suffering, impairment, bodily state, development and social circumstances. The same observed withdrawal can be adaptive in a dangerous environment and disabling elsewhere. A mathematical objective does not decide whose goals matter or whether a preference warrants clinical concern. Accounts must incorporate the person's history and meaning, ascertain distress and functional impact, and test feasible interventions. The strongest explanation combines formal constraints with biological and behavioural evidence and a contextual clinical formulation; a compact equation is a tool, not a diagnosis.

Disrupted control of fear and biased perception of threat can both contribute to psychopathology, but feeling afraid, identifying another person's fear and judging whether an action provides safety are different processes. An experiment should separate them rather than use one response as a general fear measure.

Avoidance learning provides one control mechanism. An action that removes an aversive event is negatively reinforced; if escape is unavailable, distress can persist. Yet protective control and avoidance that maintains fear are not contradictory. An effective escape action during a genuinely dangerous event can be beneficial, while repeated safety behavior in a safe situation can prevent correction of a false threat belief. A clinical panic experiment manipulated whether safety-seeking behaviours were maintained or dropped during brief exposure, with improved subsequent outcomes when they were dropped. The account predicts that what a person attributes safety to matters, beyond simple within-session fear reduction.^[ref]

Circuit interventions offer a different level of evidence. In Jose Amat's yoked rat experiments, control over otherwise matched stress exposure recruited medial prefrontal regulation of dorsal raphe responses; inactivating that prefrontal contribution eliminated the protective effect. This gives causal evidence about a circuit contribution under that preparation. It does not establish that every clinical fear problem results from deficient prefrontal inhibition, or that apparent lack of effort means the individual has learned helplessness.^[ref]

Perception provides another route. In Ralph Adolphs's study, patient SM's impaired recognition of fearful faces followed unusually little spontaneous sampling of the eyes; explicit instructions to look at the eyes normalised recognition. The finding separates access to diagnostic information from an irreversible inability to represent fear. It is a lesion case, not evidence that psychiatric groups generally have the same deficit.^[ref]

Further, Justin S. Feinstein and colleagues observed panic after carbon-dioxide inhalation in three people with bilateral amygdala damage, despite impaired anticipatory responses. External threat processing and internally induced fear can therefore dissociate. The small rare-lesion sample and possible residual circuitry limit a general necessity claim. These results motivate separate measures of external threat identification, internal arousal and learned control.^[ref]

A proposed study would recruit adults with a diagnosed anxiety disorder and a matched comparison group, measuring medication, symptom severity and relevant comorbidity. Diagnosis is observed, not randomised. Independently randomise participants within each group to controllable or yoked uncontrollable exposure and to an attention-to-diagnostic-features intervention or a matched viewing task. This gives a two-by-two experimental design within each clinical group and tests whether the interventions affect separable processes. Sample size would follow simulation-based statistical power analysis for the prespecified group-by-intervention interactions; there is no empirical variance estimate in the paper from which to invent a defensible exact number.

For controllability, a button can terminate a mildly unpleasant, individually calibrated sound; a yoked participant receives the same sequence of sound onsets and terminations, but their own button has no effect. Match available motor responses and give both groups an unequivocal, independent stop option for withdrawal. Record action-outcome beliefs and timing as manipulation checks, since matching sound exposure does not automatically match predictability or experienced agency. Use a new transfer task to assess whether learned control changes willingness to act when action efficacy is now identical for everyone.

For perception, use graded fearful-versus-neutral facial displays, with the attention intervention directing viewing towards diagnostic information. Eye tracking tests whether sampling changed. Estimate perceptual sensitivity and decision criterion separately: more fear responses can reflect a lower reporting threshold without better discrimination. Include other emotions to test specificity and a delayed generalisation set to distinguish learning from memorising the training photographs.

Outcomes would include voluntary approach and avoidance, threat expectancy, subjective fear, skin conductance and heart rate. None alone measures the whole construct: autonomic arousal is not fear-specific, and self-report can change with demand. Assessors should be masked where possible; hypotheses, primary outcomes, exclusions and the repeated-measure mixed model should be preregistered. Account for dependence within yoked pairs and randomise pair membership. Retain allocated groups in an intention-to-treat analysis and report uncertainty and adverse experiences.

If controllability selectively improves transfer behaviour despite matched exposure and unchanged facial discrimination, that supports a control pathway. If altered visual sampling improves fear recognition and independently changes social-threat judgments, that supports a perceptual contribution. A clinical-group interaction would locate vulnerability, while a correlation between questionnaire scores and task behaviour alone would not prove causation. A combined effect could indicate interacting pathways. Improvement in a laboratory outcome would still require a later treatment trial with functioning and sustained symptoms as endpoints.

Consent must state the unpleasant stimulation and withdrawal right; any withholding of contingency details requires proportionate research-ethics approval and debriefing. The decisive test manipulates control and information sampling independently, checks that each manipulation worked, and measures functional behaviour alongside fear perception and physiology.

An open-loop control is computed from a dynamical model and the initial preparation, then applied without changing it in response to measurements during that run. Controllability determines which transformations are possible in principle; field design must also account for duration, available control generators, amplitude and bandwidth limits, and calibration errors.

Three useful model-based strategies are the following. Resonant quantum control chooses carrier frequencies and phases to select transitions, and uses quantum pulse area or elementary rotation gates to compile the desired transformation. Adiabatic quantum control chooses a slowly varying path of Quantum Hamiltonians whose relevant eigenstate connects the initial and target states, maintaining a sufficiently large spectral gap. Quantum optimal control parametrizes or varies the fields and maximizes a fidelity or observable objective, while penalizing resources and enforcing the dynamical equations. These can all produce predetermined laboratory waveforms; experimental feedback is a separate design choice.

For a detailed variational construction, letChoose a positive target observable $O$, for example $O=|{\psi }_d⟩⟨{\psi }_d|$ for pure-state transfer. MaximizeThe terminal term is the target probability or observable expectation, and the quadratic integral penalizes field energy. A costate $\chi (t)$ enforces the Schrödinger equation through the real augmented functionalVariation in $\chi$ recovers the state equation. Integrating the variation in $\dot{\psi }$ by parts gives the terminal boundary term $2\mathrm{Re}⟨O\psi (T)-\chi (T)|\delta \psi (T)⟩$ and the interior adjoint equation. Thus the Euler-Lagrange conditions areThe costate obeys the same Hamiltonian equation but is propagated backward from its terminal value. The field variation isHence the variational costate gradient for Hamiltonian quantum control isAt an unconstrained stationary field,These equations couple forward and backward boundary data, so this is not a single initial-value integration, nor a direct feedback law for an unknown experimental state.

A practical direct-adjoint looping algorithm starts from an admissible trial field. Propagate $\psi$ forward and store its trajectory; set $\chi (T)=O\psi (T)$ and propagate $\chi$ backward; compute $g_k$ along the two trajectories; then update $u_k^{\mathrm{new}}=u_k+\eta g_k$ for a positive step size $\eta$. A line search evaluates the actual objective and decreases $\eta$ until the ascent condition is met. Project into the admissible amplitude range or optimize a finite Fourier or pulse basis to enforce bandwidth constraints. Repeat until the projected gradient and change in objective are small. Repeated initializations help explore different local optima; the stationarity equations alone do not guarantee a global optimum.

For a piecewise constant discretization, define $H_j=H_0+{\sum }_k{u}_{k,j}{H}_k$ and $U_j=e^{-i\Delta t{H}_j/\hslash }$. Forward multiplication gives the state trajectory; backward multiplication gives the costate. The exact matrix-exponential derivative isThis supplies accurate discrete gradients even when $H_j$ and $H_k$ do not commute. The small-step expression $\partial U_j/\partial u_{k,j}\simeq -i\Delta t{H}_k{U}_j/\hslash$ is only an approximation. Forward/backward factorization is the basis of gradient ascent pulse engineering. For a full-gate objective, propagate all basis columns or $U(t)$ itself and use a terminal objective such as the phase-insensitive unitary gate error; optimizing one state alone does not implement a prescribed operation on every input.

To implement the result, convert the optimized envelope and quadratures into the calibrated electric or magnetic drive fields. An arbitrary-waveform source can set time-domain quadratures; optical spectral pulse shaping applies a complex mask to the available pulse spectrum,using, for example, a 4f pulse shaper and spatial light modulator. Finite spectral support, mask resolution and actuator limits must be included in the optimization: smoothing an unconstrained optimum afterward may spoil it. Simulating the calibrated final pulse, optionally over an ensemble of detunings or calibration errors, checks its predicted fidelity and robustness. The computed fields are then executed as open-loop control, with all measurements used for later calibration rather than an assumed instantaneous state readout.

A reachable set contains all states attainable from a specified initial state by admissible control inputs in the allowed times, and controllability means that every target in the declared state space is reachable from every initial state. For a closed quantum control system, distinguish pure-state controllability on state rays, density operator controllability on fixed-spectrum unitary orbits, and unitary operator controllability for entire propagators. State engineering asks for a particular state transfer; process engineering asks for an entire quantum gate, which must act correctly on every input. A closed Hamiltonian evolution preserves density-operator eigenvalues, so arbitrary mixed states of different spectra cannot be connected even by a fully operator-controllable system.

Assume piecewise-constant real controls with freely chosen duration, as in the usual finite-dimensional quantum controllability theorem. We can establish the strongest notion here: the Dynamical Lie algebra is $\mathfrak{u}(3)$, so the system has unitary operator controllability, and consequently both pure-state and fixed-spectrum density operator controllability.

To verify the hypotheses explicitly, use matrix units $E_{jk}=|j⟩⟨k|$ and put $K_0=-i{H}_0$, $X_{jk}=-i(E_{jk}+E_{kj})$, $Y_{jk}=E_{jk}-E_{kj}$. Then $K_1=-i{H}_1=d_1{X}_{12}+d_2{X}_{23}$ andSince $d_1{d}_2\ne 0$ and ${\omega }_1^2\ne {\omega }_2^2$, linear combinations of these two elements isolate $X_{12}$ and $X_{23}$. Their commutators with $K_0$ give ${\omega }_1{Y}_{12}$ and $-{\omega }_2{Y}_{23}$, respectively. Further commutators give two independent traceless diagonal generators, since $[X_{jk},Y_{jk}]=2i(E_{jj}-E_{kk})$, as well as the two $1$–$3$ generators. These eight generators span the special unitary Lie algebra $\mathfrak{su}(3)$.

Finally $\mathrm{Tr}{K}_0=i({\omega }_1+{\omega }_2)\ne 0$. Subtract its traceless component, already in the generated algebra, to obtain a nonzero multiple of $iI$. This proves the Lie algebra of the unitary group is all of $\mathfrak{u}(3)$, not merely $\mathfrak{su}(3)$. The compact-group controllability theorem then gives access to every unitary with a suitable control and some duration. This does not assert controllability at every predetermined short time or under additional bandwidth constraints. This proves that two adjacent transitions generate the qutrit unitary Lie algebra.

For $|D⟩=(|1⟩-|3⟩)/\sqrt{2}$, direct multiplication gives $H_1|D⟩=0$ and $H_0|D⟩=-\omega |D⟩$. It is therefore a dark state of a driven Hamiltonian for every real control, andStarting from this state, the reachable physical ray is a singleton, so it cannot be transferred to $|2⟩$ or any distinct ray. More generally $P_D=|D⟩⟨D|$ commutes with both generators; hence $\mathrm{Tr}(P_D\rho )$ is conserved. The system thus fails full-qutrit pure-state controllability, density operator controllability, and unitary operator controllability.

This is the appropriate full-space meaning of the PDF's phrase about all senses of controllability. It does not preclude control within a restricted invariant subspace: the bright state $|B⟩=(|1⟩+|3⟩)/\sqrt{2}$ couples to $|2⟩$ with strength $\sqrt{2}df(t)$, while the dark line is isolated. The bright and dark state reduction of a star-coupled Hamiltonian makes that invariant decomposition explicit.

A reachable set consists of states obtainable from one specified initial state using admissible inputs and allowed durations. Controllability means that every target in the declared state space is reachable from every initial state there; the declared space and time convention are essential.

For a closed quantum system, let $U_f$ solve the controlled Schrödinger equation with $U_f(0)=I$. The propagator and state reachable sets areA pure-state version acts on normalized vectors up to global phase. Pure-state controllability makes every pure target accessible; unitary operator controllability permits every desired coherent process, with global phase treated according to the definition.

For mixed states, density operator controllability means transitivity on each unitary orbit of a density operator, not access to all density operators regardless of spectrum. The Dynamical Lie algebra generated by $-i{H}_0,-i{H}_1,\dots$ describes the accessible connected unitary transformations under the usual unconstrained-time control assumptions. Conserved quantities or restricted controls can shrink the reachable set. Thus successful preparation of one state is weaker than implementation of an arbitrary gate on all inputs, and coherent controllability does not imply purification.

For reachability in control theory, a state $x_1$ is reachable from $x_0$ if an admissible input and a permitted duration drive the dynamical system from $x_0$ to $x_1$. Controllability requires this for every initial-target pair in the specified state space. Restrictions on duration or field amplitudes change the reachable set, so the admissible controls are part of the definition.

For a closed quantum system, $\dot{U}=-iH[f(t)]U$, $U(0)=I$, and states evolve as $\psi (t)=U(t)\psi (0)$ or $\rho (t)=U(t)\rho (0)U(t{)}^{†}$. This gives three notions of quantum controllability. Unitary operator controllability requires every desired $U_d\in U(N)$ to be reachable from the identity; when a global phase is immaterial, one instead asks for every special unitary operation. Density operator controllability requires every pair of density operators with the same spectrum to be connected. The spectral qualification is necessary because unitary time evolution cannot change eigenvalues; the natural state space is a unitary orbit of a density operator. Pure-state controllability requires any normalized initial state to be transferred to any target ray, $U(T){\psi }_0=e^{i\chi }{\psi }_d$.

Thus $\boxed{\text{operator controllability}\Rightarrow \text{density operator controllability}\Rightarrow \text{pure-state controllability}}$, with the operator implication also valid if global phase is discarded. The converse implications need not hold. A pure-state task concerns one vector, whereas an operator task must act correctly on every input vector at once.

Reachability in control theory asks about a particular initial state $x_0$ and target $x_1$: does some admissible control input make $x(T)=x_1$ at the declared time $T$, or at some allowed duration? Controllability requires this for every pair in the declared state space. This is a property of state evolution, distinct from graph reachability.

For a linear time-invariant system, solving the ordinary differential equation by a matrix exponential givesThe first term is the unforced endpoint. Reachability in control theory asks whether the control input can supply exactly the displacement $x_1-e^{AT}{x}_0$. The timing matters because the same command is carried by different amounts of subsequent dynamics. ^[ref]

Here is a contrasting mathematical example with one shared control input:The controllability matrix has rank of a matrix one. Since $\frac{d}{dt}(z_1-z_2)=0$, the difference is invariant under every allowed control input. From $(0,0)$ the reachable set is the diagonal $z_1=z_2$ when input amplitudes are unrestricted: $(1,1)$ is reachable at $T=2$ using $u=1/2$, while $(1,0)$ is impossible at every duration. Having a control input appear in both state equations did not give independent control of both states. This reachable set is pictured alongside the bounded double integrator below.