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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 the linear time-invariant system $\dot{x}=Ax+Bu$, with $A\in {\mathbb{R}}^{n\times n}$ and $B\in {\mathbb{R}}^{n\times m}$, the controllability matrix has $n$ rows and $nm$ columns. The columns of $B$ are directions influenced directly by the control inputs; $AB,A^{2}B,\dots$ account for how the dynamics carry their influence into further directions. The column space of the controllability matrix is the zero-state reachable set for any $T>0$ when inputs are unrestricted. ^[ref]

Work with finite graphs, as in the usual matching theorem. Hall's condition is $|N(A)|\ge |A|$ for every $A\subseteq X$. It is necessary because a matching injects $A$ into its neighbours.

For sufficiency, induct on $|X|$. If a nonempty proper subset $A$ has $|N(A)|=|A|$, induction matches $A$ to $N(A)$. For $B\subseteq X\setminus A$, Hall's inequality for $A\cup B$ gives $|N(B)\setminus N(A)|\ge |B|$, so induction matches the complement to unused neighbours. Combining the matchings finishes this case. Otherwise every nonempty proper $A$ has $|N(A)|\ge |A|+1$. Choose an edge $xy$, delete its endpoints, and observe that every subset of the remaining $X$ still has at least its own number of neighbours. Induction matches the remainder, and adjoining $xy$ finishes. The one-vertex case and the empty case are immediate.

For $0\le d\le |X|$, the defect version isNecessity follows because at most $d$ vertices of $X$ can be unmatched. For sufficiency add $d$ new vertices to $Y$, each joined to every vertex of $X$. The displayed inequality makes Hall's condition hold in the enlarged graph. Its full matching uses at most $d$ new vertices, giving the required original edges.

To prove the matching-cover equality, let $M$ be a maximum matching and $U$ its unmatched vertices in $X$. Follow alternating paths from $U$, using nonmatching edges from $X$ to $Y$ and matching edges back. Let $Z_X,Z_Y$ be the reachable sets. No reachable $Y$ is unmatched, since that would give an augmenting path and a larger matching. Matching edges give a bijection $Z_Y\leftrightarrow Z_X\setminus U$. The set $C=(X\setminus Z_X)\cup Z_Y$ covers every edge: an edge from a reachable $X$ to an unreachable $Y$ cannot be a nonmatching edge, and its matching edge, if present, is already reachable. Its size isEvery vertex cover has at least $|M|$ vertices, because the matching edges are disjoint. Therefore maximum matching size equals minimum vertex-cover size, as required.

In units $\hslash =1$, a bilinear Hamiltonian control system haswhere the Hermitian matrices $H_0,H_k$ are fixed, and the real functions $u_k$ are admissible control inputs. The corresponding vector or density operator dynamics are $\dot{\psi }=-iH(u)\psi$ or $\dot{\rho }=-i[H(u),\rho ]$. The dynamics are linear in the state at fixed input and affine in the inputs at fixed state; this is the bilinear terminology.

The reachable set from an initial state $x_0$ at time $T$ is ${\mathcal{R}}_T(x_0)=\{x(T;x_0,u):u\text{ admissible}\}$. With freely chosen duration, take $\mathcal{R}(x_0)={\bigcup }_{T\ge 0}{\mathcal{R}}_T(x_0)$. Controllability means every allowed target is reachable from every allowed initial state. The allowed space and phase convention must be specified. For closed quantum control, density operators remain on their initial unitary orbit of a density operator, rather than becoming arbitrary density operators with different eigenvalues.

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.

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.

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.

The reachable set ${\mathcal{R}}_T(x_0)$ collects all endpoints attainable from $x_0$ at exactly $T$ by admissible control inputs. If arrival any time up to $T$ is allowed, use ${\mathcal{R}}_{\le T}(x_0)={\bigcup }_{0\le t\le T}{\mathcal{R}}_t(x_0)$, with ${\mathcal{R}}_0(x_0)=\{x_0\}$. These definitions differ: a general dynamical system may drift away after early arrival, so waiting need not preserve the target. The reachable set is always relative to a declared input class and any constraints on the entire path.

For the double integrator starting from $(0,0)$, now require measurable functions as control inputs with $|u(t)|\le 1$ almost everywhere. Write $(p,v)$ for its endpoint at $T>0$. Two integrals determine the endpoint:Our mathematical example has the exact reachable setTo see the upper boundary, fix $v$: put $u=1$ first and $u=-1$ last, switching at $s=(T+v)/2$. This maximizes the weighted integral for position because early commands have the largest weight $T-t$; the required net velocity fixes $s$. Formally, $(T-t)-(T-s)=s-t$ has the same sign as this maximizing command, so changing to any other admissible command with the same integral cannot increase $p$. Reversing the order gives the lower boundary. Pointwise convex combinations of these two commands preserve $v$ and attain every intermediate $p$, so the formula describes the whole region, not merely a bound. The extreme commands are bang-bang control.

In particular, the bounded double integrator can move from rest to position $d$ and rest again exactly when $|d|\le T^2/4$. Its minimum duration is therefore $2\sqrt{|d|}$ in these normalized units. With freely chosen time every rest-to-rest distance is attainable; with a fixed deadline many are excluded. Changing the amplitude bound to $|u|\le a$ with $a>0$ scales this condition to $|d|\le a{T}^2/4$. This shows why specifying the controls matters even when the Kalman rank criterion gives full rank of a matrix.

In closed quantum control, a density operator reachable set lies inside its unitary orbit of a density operator, since coherent evolution preserves the spectrum. That orbit need not be fully reachable under the available control inputs.