State Controllability

• State controllability is defined as the ability to drive a system from any initial state to a target state in finite time using admissible control inputs.
• Applications span classical, structural, network, and quantum systems, with techniques like rank tests, graph theory, and optimization ensuring controllability under diverse constraints.
• Emerging methods such as ε-controllability, supervisory frameworks, and positive state constraints extend classic theories to address challenges in complex dynamical systems.

State controllability is the property that a dynamical system can be driven from an initial state to a desired final state in finite time by suitable inputs. In the classical state-space setting, this idea is expressed through equations such as

$$\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t), \qquad \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{u}(t),$$

and is tied to reachability conditions, attainable sets, and rank tests. Recent arXiv literature extends the notion well beyond finite-dimensional linear systems, covering structural and graph-theoretic controllability, target and regional objectives, positivity and state constraints, state covariance controllability, data-driven relaxations such as $\epsilon$-controllability, and quantum or machine-learning realizations (Hamdan et al., 2024, Montanari et al., 2023, Liu et al., 2024).

1. Classical formulations and foundational definitions

In the control-theoretic sense used for structured state space models, a system is controllable if one can drive the internal state from any initial state to any final state in finite time with suitable inputs, and the paper on Sparse Mamba states that this holds when the reachability matrix has full rank. The same paper emphasizes controllable canonical form, in which the pair $(\mathbf{A},\mathbf{B})$ is arranged so the system is explicitly controllable (Hamdan et al., 2024).

For discrete-time nonlinear systems,

$x(t+1)=f(x(t),u(t)),$

the paper on nonlinear state-feedback stabilizability distinguishes several notions. A system is $N$-step controllable to the origin if for every initial state $x_0\in\mathbb{R}^n$, there exists an input sequence $u[0,N-1]$ such that $\phi(N,0,x_0;u)=0$. It is asymptotically controllable to the origin if for every $x_0$, there exists an input sequence $u[0,\infty)$ such that $\epsilon$0. The same work uses a rank controllability condition at the origin,

$\epsilon$1

as a nonlinear analogue of a controllability rank condition (Hanba, 2015).

A geometric formulation appears in the note proving that local controllability implies controllability. There, for a control system on a connected finite-dimensional smooth manifold $\epsilon$2, the attainable set from $\epsilon$3 is

$\epsilon$4

Local controllability means $\epsilon$5 for every $\epsilon$6, while controllability means $\epsilon$7 for every $\epsilon$8. Under the standing assumptions of that paper, local controllability implies controllability (Boscain et al., 2021).

2. Structural, graph-theoretic, and network controllability

For structured systems, the numerical values of $\epsilon$9 and $(\mathbf{A},\mathbf{B})$0 are not fixed; only their sparsity patterns are known. A structured system $(\mathbf{A},\mathbf{B})$1 is structurally controllable if there exists a realization $(\mathbf{A},\mathbf{B})$2 that is controllable, and this property is generic in the sense that if one realization is controllable, then almost all realizations with the same sparsity pattern are controllable. The graph-theoretic characterization used in constrained input selection states that structural controllability is equivalent to two conditions: every non-top linked SCC is reachable from some input, and the system bipartite graph $(\mathbf{A},\mathbf{B})$3 has a perfect matching, equivalently the system digraph has no dilations (Moothedath et al., 2017).

The same paper gives a flow-network characterization. If $(\mathbf{A},\mathbf{B})$4 is the number of non-top linked SCCs in $(\mathbf{A},\mathbf{B})$5, $(\mathbf{A},\mathbf{B})$6 is the number of states, and $(\mathbf{A},\mathbf{B})$7 is the maximum flow value in the constructed network $(\mathbf{A},\mathbf{B})$8, then

$(\mathbf{A},\mathbf{B})$9

This permits a reduction of minimum constrained input selection and minimum cost constrained input selection to the minimum cost fixed flow problem. The optimization problems are NP-hard, but the paper proves that approximation schemes of MCFF directly apply, and proposes a polynomial approximation algorithm based on minimum weight bipartite matching and a greedy selection scheme with complexity $x(t+1)=f(x(t),u(t)),$0 and approximation ratio $x(t+1)=f(x(t),u(t)),$1, where $x(t+1)=f(x(t),u(t)),$2 denotes the maximum in-degree of input vertices in the flow network of the structured system (Moothedath et al., 2017).

For structured network systems with a target subset $x(t+1)=f(x(t),u(t)),$3, target controllability and target observability are formulated as target-restricted versions of output controllability and functional observability. The exact rank test for output controllability is

$x(t+1)=f(x(t),u(t)),$4

while target controllability admits a graph-theoretic characterization: each target node must be reachable from some driver node, and no subset of target nodes may have a dilation in the relevant subgraph. The same paper proves a weak duality,

$x(t+1)=f(x(t),u(t)),$5

and a strong duality with an additional exclusion of minimal dilation sets in the dual graph. It also stresses that target controllability and target observability are not trivially dual when $x(t+1)=f(x(t),u(t)),$6 (Montanari et al., 2023).

In random directed networks, structural controllability is linked to the density of low-degree nodes. The paper on network controllability states that the density of nodes with in-degree and out-degree equal to $x(t+1)=f(x(t),u(t)),$7, $x(t+1)=f(x(t),u(t)),$8, and $x(t+1)=f(x(t),u(t)),$9 determines the number of driver nodes, and that networks with minimum in-degree and out-degree greater than $N$0 are always fully controllable by an infinitesimal fraction of driver nodes, regardless on the other properties of the degree distribution (Menichetti et al., 2014). For large structured networks with high-dimensional node systems, a separate strong structural controllability result shows that controllability can be tested after replacing every original node system by an auxiliary node system of dimension $N$1 or $N$2, so that the required rank test has state space dimension at most twice the number of node systems (Jia et al., 2020).

3. Target, regional, and symbolic-state variants

Regional controllability restricts the objective to a prescribed part of the state. In Boolean cellular automata, the 2019 graph-theoretic treatment defines regional controllability of a target region $N$3 as the ability to steer the state of $N$4 from an initial configuration to a desired target configuration within a finite horizon $N$5 using boundary controls $N$6. The controlled evolution is encoded in a transition graph $N$7 with $N$8 vertices. Two exact criteria are given: a Hamiltonian-circuit criterion through powers of the adjacency matrix, and a practical SCC criterion,

$N$9

Once controllability is established, a preimage method constructs an explicit control sequence (Dridi et al., 2019).

The 2025 SAT-based study of finite one-dimensional Boolean cellular automata reformulates the boundary-control problem for a finite region of length $x_0\in\mathbb{R}^n$0 as a Boolean satisfiability problem. For a horizon $x_0\in\mathbb{R}^n$1, the resulting formula has

$x_0\in\mathbb{R}^n$2

clauses, and

$x_0\in\mathbb{R}^n$3

That paper identifies the peripherally-linear elementary cellular automata rules

$x_0\in\mathbb{R}^n$4

as fully controllable, and states that for other rules the reachability ratio, that is, the fraction of controllable pairs of initial and final configurations, is vanishing when the system size grows (Bagnoli et al., 23 Mar 2025).

A different symbolic formalism appears in reaction systems. There, controllability is defined as the ability of transitioning between any two states through a suitable choice of context sequences. Because unrestricted controllability is trivial, the meaningful notions are restricted controllability, such as $x_0\in\mathbb{R}^n$5-controllability, where contexts have size at most $x_0\in\mathbb{R}^n$6, and $x_0\in\mathbb{R}^n$7-controllability, where contexts are restricted to subsets of $x_0\in\mathbb{R}^n$8. The same framework also defines $x_0\in\mathbb{R}^n$9-target controllability and $u[0,N-1]$0-target controllability for target species $u[0,N-1]$1. The paper proves that the associated decision problems are $u[0,N-1]$2-hard (Ivanov et al., 2020).

In supervisory control theory, “state controllability” is not a single universally accepted notion. The 2025 overview distinguishes language controllability, automata controllability, Flordal–Malik controllability, Kushi–Takai $u[0,N-1]$3-admissibility, partial bisimulation, and Zhou et al.’s state controllability. For a deterministic supervisor $u[0,N-1]$4 and a potentially nondeterministic plant $u[0,N-1]$5, Zhou et al.’s notion requires an embedding $u[0,N-1]$6 such that supervisor transitions are mirrored in the plant and

$u[0,N-1]$7

The paper also states that state controllability does not imply language controllability in general, while Flordal–Malik controllability and Kushi–Takai uncontrollable event admissibility are equivalent and are the notions that imply the traditional notion of language controllability in the general nondeterministic setting (Keiren et al., 7 Aug 2025).

4. State constraints, positivity, and boundary control

State controllability under unilateral constraints is prominent in PDE control. For the coupled linear parabolic system

$u[0,N-1]$8

the reaction-diffusion paper studies controllability to trajectories under nonnegativity constraints on the state. When $u[0,N-1]$9, $\phi(N,0,x_0;u)=0$0, the eigenvalues of $\phi(N,0,x_0;u)=0$1 have nonnegative real part, and the initial data are nonnegative with no component a.e. zero, the paper proves exact nonnegative controllability to trajectories: $\phi(N,0,x_0;u)=0$2 For diagonal $\phi(N,0,x_0;u)=0$3, quasipositive $\phi(N,0,x_0;u)=0$4, ellipticity, and the Kalman-type condition

$\phi(N,0,x_0;u)=0$5

it proves approximate nonnegative controllability,

$\phi(N,0,x_0;u)=0$6

The key proof device is a staircase method, and the paper further shows that state-constrained controllability admits a strictly positive minimal time, even for the weaker unilateral bound $\phi(N,0,x_0;u)=0$7 (Lissy et al., 2020).

Positivity constraints also appear in boundary control systems. For the abstract system

$\phi(N,0,x_0;u)=0$8

the 2022 paper develops frequency-domain controllability criteria for positive controls and for simultaneous positivity constraints on control and state. For positive controls only, boundary approximate controllability is equivalent to the existence of $\phi(N,0,x_0;u)=0$9 such that for all $x_0$0,

$x_0$1

For the stronger positive-state version, the criterion adds $x_0$2 and density of the closed convex hull of $x_0$3 in $x_0$4. In transportation networks, provided that the underlying graph is strongly connected, controllability under positivity constraints on the control/state is fully characterized by a Kalman-type rank condition. In a path-like heat network with Robin boundary conditions, the paper establishes approximate controllability under positivity state-constraint with a single positive input through the starting node, but proves the lack of controllability under unilateral control-constraint (Gantouh, 2022).

A semigroup formulation of exact and positive controllability for boundary control systems is given in the earlier 2016 paper. With controllability map

$x_0$5

the exact reachability space is

$x_0$6

The paper further defines a maximal reachability space $x_0$7 and calls the system maximally controllable if $x_0$8. Under positive controls, it studies the cone

$x_0$9

and, in transport and network-flow examples, obtains exact reachability criteria in terms of cyclic spans such as

$u[0,\infty)$0

or

$u[0,\infty)$1

which the paper describes as Kalman-type controllability conditions (Engel et al., 2016).

5. Data-driven, stochastic, and global geometric reformulations

When a system model is unavailable and only sampled transitions

$u[0,\infty)$2

are known, exact point-to-point controllability is replaced by $u[0,\infty)$3-controllability. A state $u[0,\infty)$4 is $u[0,\infty)$5-controllable with respect to target $u[0,\infty)$6 if

$u[0,\infty)$7

and the $u[0,\infty)$8-controllable set is

$u[0,\infty)$9

Under a Lipschitz continuity assumption on $\epsilon$00, the one-step backpropagation theorem propagates controllability from a known controllable ball to predecessor balls. The paper implements this through the maximum expansion of controllable subset (MECS), which iterates selection, expansion, evaluation, and pruning on a tree of controllable balls, and through the simplified Floyd expansion with radius fixed (FERF), which reduces the test to a shortest-path problem. It evaluates the degree of controllability,

$\epsilon$01

on a mass-spring system, a Van der Pol oscillator, and a tunnel-diode circuit (Yang et al., 2024).

For linear stochastic systems with additive noise, the controlled object may be the state covariance rather than the state itself. In discrete time,

$\epsilon$02

the paper characterizes the reachable set of terminal covariances exactly by projection conditions and a decomposition into contributions from the initial covariance and each noise injection time. In continuous time, it gives an upper bound on the reachable covariance set. For finite-horizon covariance controllability, the decisive conditions are Gramian and range conditions. In discrete time,

$\epsilon$03

are equivalent to covariance controllability; in continuous time, the analogous condition is

$\epsilon$04

This is a second-moment analogue of classical reachability and controllability (Liu et al., 2024).

A global geometric analysis of bounded-control linear systems is developed by compactifying $\epsilon$05 using the Poincaré sphere. The paper distinguishes reachable sets $\epsilon$06, controllable sets $\epsilon$07, control sets, and chain control sets, and studies the induced control flow on the compactification. The Lyapunov decomposition of the uncontrolled linear flow organizes the dynamics at infinity, while the linearization of the induced control flow yields Selgrade decompositions and stable manifolds on the sphere. In the original state space, these manifolds describe trajectories whose norms diverge while directions converge, and the resulting picture shows how bounded control range interacts with asymptotic structure to shape global controllability properties (Colonius et al., 2 May 2025).

6. Architectural and quantum realizations

In structured state space models for sequence modeling, state controllability has been imported directly into neural architecture design. The Sparse Mamba paper argues that vanilla Mamba and Mamba2 do not explicitly enforce control-theoretic structure in their $\epsilon$08 parameterization. Its SC-Mamba variant replaces the unconstrained state matrix by the controllable canonical form

$\epsilon$09

so that the full $\epsilon$10 matrix is determined by only $\epsilon$11 free parameters. The paper reports a reduction of parameters from $\epsilon$12 in Mamba to $\epsilon$13 in SC-Mamba, about a $\epsilon$14 improvement in perplexity, and about a $\epsilon$15 decrease in training time. For Mamba2, it also enforces stability by constraining diagonal parameters to remain negative, and reports improved perplexity, for example CodeParrot 1M $\epsilon$16 (Hamdan et al., 2024).

Quantum controllability introduces a distinct finite-time versus asymptotic dichotomy. For discrete-time Markovian quantum dynamics,

$\epsilon$17

the feedback-control paper distinguishes pure-state to pure-state controllability, density-operator to density-operator controllability, and related notions. Its main message is that generic measurement-based feedback does not imply finite-time exact steering between arbitrary states, but does allow arbitrary asymptotic state-to-state transitions. Under a special measurement structure with two diagonal singular Kraus operators, finite-time controllability is recovered, and the required control horizon scales linearly with the Hilbert-space dimension (Albertini et al., 2010).

For infinite-dimensional quantum systems, the controllability problem is often resolved by symmetry methods rather than finite-dimensional Lie-rank arguments alone. The Jaynes-Cummings-Hubbard paper proves both pure-state controllability and strong operator controllability for the full infinite-dimensional model, provided the cavity graph is connected and one has individual control of every qubit through $\epsilon$18 and $\epsilon$19 together with a single collective control on the photon-hopping term (Heinze et al., 2018).

A related but more specialized result concerns rotational states of chiral molecules. The paper on enantiomer-specific state transfer proves complete controllability for rotational states of an asymmetric top molecule despite degeneracy in the orientational quantum number $\epsilon$20, and introduces enantio-selective controllability for the simultaneous control of both enantiomers. In the finite-dimensional truncations used there, complete controllability corresponds to generation of $\epsilon$21, while enantio-selective controllability corresponds to $\epsilon$22. The work then constructs pulse sequences for complete enantiomer-specific population transfer, overcoming the $\epsilon$23-degeneracy obstacle that limits earlier three-wave-mixing schemes (Leibscher et al., 2020).

State controllability thus functions less as a single theorem than as a unifying control-theoretic principle. Across the cited literature, it appears as exact state transfer, dense reachability, target or regional steering, structural genericity, positivity- or state-constrained evolution, covariance assignment, or asymptotic feedback-mediated preparation. What remains invariant is the underlying question: which portions of state space, or which state descriptors, can be made dynamically accessible under the admissible input architecture and system constraints.