Control-Invariant Sets

• Control-invariant sets are defined as subsets of the state or configuration space where a controlled dynamical system can remain indefinitely within prescribed constraints using suitable inputs.
• The concept extends to maximal, robust, probabilistic, and joint state-control variants, with fixed-point and predecessor operator methods ensuring systematic set computation.
• Applications span safety-critical control, MPC design, and data-driven methods, highlighting a balance between computational tractability and model fidelity.

Control-invariant sets are subsets of the state space, or more generally of a constrained configuration space, from which a controlled dynamical system can be kept indefinitely inside prescribed constraints by suitable admissible inputs. In discrete time, for $x^{+}=f(x,u)$ with state constraints $X$ and input constraints $U(x)$, a set $X\subseteq \mathbb{R}^n$ is control invariant if $\forall x\in X,\ \exists u\in U(x)$ such that $f(x,u)\in X$ (Amine et al., 8 Apr 2026). In continuous time, the same idea appears in controlled invariant sets and control sets, and in constrained control it is closely tied to viability, recursive feasibility, and safety-critical control (Colonius et al., 2018, Fiacchini et al., 2018, Vahs et al., 2024).

1. Foundational definitions and variants

The basic discrete-time notion is accompanied by several standard variants. The maximal control invariant set $X_{\mathrm{MCI}}\subseteq X$ is the largest control-invariant subset of the constraint set $X$: it is control invariant, and every other control-invariant subset of $X$ is contained in it (Amine et al., 8 Apr 2026). Closely related notions differ only in the quantifiers or in the underlying dynamics. Robust control invariant sets require $f(x,u,w)\in X$ for all disturbances $X$0 in a disturbance set. Positive invariance removes the existential control choice and concerns autonomous dynamics. Viability kernels characterize states from which constraints can be kept indefinitely and often coincide with the maximal control invariant set under deterministic control constraints (Amine et al., 8 Apr 2026).

In continuous-time nonlinear control systems $X$1, the literature distinguishes controlled invariant sets from control sets. A control set is a maximal controlled invariant region with approximate controllability, and for a control set with nonvoid interior both the control set and its interior satisfy the no-return property: if a trajectory returns to the set at time $X$2, then it must have remained in the set throughout $X$3 (Colonius et al., 2018). This embeds invariance into a broader geometric theory of reachable regions.

Several domain-specific extensions redefine invariance around the timing or uncertainty structure of the control problem. For stochastic systems, probabilistic controlled invariant sets (PCISs) replace worst-case guarantees by trajectory-level probability thresholds over finite or infinite horizons; the infinite-horizon $X$4 case reduces to robust controlled invariance in the kernel formulation (Gao et al., 2019). For impulsively controlled systems, where the state evolves freely between impulse times and control acts only at $X$5, the paper on impulsive systems defines impulsive controlled invariant sets relative to a target zone $X$6: every state in the set must evolve within $X$7 over the whole dwell interval and admit an admissible impulse that returns the sampled state to the set (Sanchez et al., 2021).

A recurrent misconception is to treat these notions as interchangeable. They are not. Control invariance uses an existential input quantifier, robust invariance changes it to a universal disturbance quantifier, positive invariance removes the control choice, and probabilistic invariance replaces hard set inclusion by probability thresholds. The distinctions are structural, not terminological (Amine et al., 8 Apr 2026, Gao et al., 2019).

2. Predecessor operators and fixed-point characterizations

The standard computational object behind control invariance is the predecessor operator. For any $X$8,

$X$9

The classical fixed-point recursion

$U(x)$0

converges to $U(x)$1 under compactness and continuity assumptions, and for LTI systems with polytopic constraints it terminates in finitely many steps (Amine et al., 8 Apr 2026). This places maximal invariance in the broader class of greatest fixed-point problems.

For switched nonlinear systems, the same structure appears in mode-indexed form. With

$U(x)$2

or equivalently $U(x)$3, a set $U(x)$4 is controlled invariant if and only if $U(x)$5. The maximal controlled invariant set inside $U(x)$6 is the fixed point of $U(x)$7, and interval-analysis algorithms compute outer and inner approximations of this fixed point in finitely many iterations (Li et al., 2016).

For grid-based nonlinear methods, the predecessor concept can also be encoded graph-theoretically. In the graph algorithm for nonlinear cascade systems, a directed graph on state-space cells is built from the set-valued image $U(x)$8, and strongly connected components together with paths to them identify non-leaving cells. The resulting outer approximations converge to the largest control-invariant set as the cell diameter goes to zero (Decardi-Nelson et al., 2022).

The fixed-point viewpoint is important because it separates the mathematical definition of invariance from any particular set representation. Polyhedra, interval unions, graph cell complexes, and algebraic templates all approximate the same predecessor-based object.

3. Joint state-control invariance and maximal state-control invariant sets

A major refinement of the classical state-space view is to work directly in the joint state-control space. Let

$U(x)$9

A set $X\subseteq \mathbb{R}^n$0 is state-control invariant if

$X\subseteq \mathbb{R}^n$1

The corresponding maximal state-control invariant set $X\subseteq \mathbb{R}^n$2 is the largest state-control invariant subset of $X\subseteq \mathbb{R}^n$3 (Amine et al., 8 Apr 2026).

This construction strictly extends the classical maximal control invariant set. If $X\subseteq \mathbb{R}^n$4 denotes projection onto the state coordinates and $X\subseteq \mathbb{R}^n$5 denotes the state-dependent section, then

$X\subseteq \mathbb{R}^n$6

and, for every $X\subseteq \mathbb{R}^n$7,

$X\subseteq \mathbb{R}^n$8

Accordingly, the MSCI simultaneously encodes the maximal safe state set and the admissible invariance-preserving inputs at each safe state (Amine et al., 8 Apr 2026).

The joint-space predecessor operator is

$X\subseteq \mathbb{R}^n$9

For LTI systems $\forall x\in X,\ \exists u\in U(x)$0 and a polytope $\forall x\in X,\ \exists u\in U(x)$1 with state projection $\forall x\in X,\ \exists u\in U(x)$2, this specializes to

$\forall x\in X,\ \exists u\in U(x)$3

The MSCI is then the fixed point of

$\forall x\in X,\ \exists u\in U(x)$4

and for polytopic LTI systems the recursion terminates in finitely many steps (Amine et al., 8 Apr 2026).

The conceptual significance is that classical control invariance answers only where the state may remain, whereas state-control invariance also specifies which control values preserve that property. For constrained synthesis and safety filters, this extra information is operational rather than merely descriptive.

4. Computational frameworks

Model-based computation of control-invariant sets spans several distinct geometric paradigms. For polyhedral LTI systems, classical algorithms iterate predecessor operators in the state space, often through repeated polyhedral projection and intersection. High-dimensional alternatives replace explicit Minkowski sums, projections, and vertex enumeration by LP certificates for $\forall x\in X,\ \exists u\in U(x)$5-step invariance; these LP conditions scale to large systems, including examples with state and input dimensions $\forall x\in X,\ \exists u\in U(x)$6 and $\forall x\in X,\ \exists u\in U(x)$7, and in an earlier formulation produced a $\forall x\in X,\ \exists u\in U(x)$8-step approximation for dimensions $\forall x\in X,\ \exists u\in U(x)$9 and $f(x,u)\in X$0 in less than two seconds (Fiacchini et al., 2018, Fiacchini et al., 2017).

For discrete-time switched nonlinear systems, interval-analysis methods represent sets as finite unions of boxes and use convergent inclusion functions to compute outer and inner approximations of maximal controlled invariant sets in finitely many iterations. Under a robustly controlled invariance condition, they also produce a partition-based invariance controller directly from the computation (Li et al., 2016). For discrete-time nonlinear control-affine systems, a related interval-and-polytope refinement method approximates the maximal controlled invariant set with guaranteed precision and computes sets of control inputs that enforce invariance; an accelerated version updates individual components of set unions rather than all at once (Brown et al., 2023).

For nonlinear cascade systems, graph-based methods discretize the state space into cells, construct a directed graph from set-valued images, compute non-leaving cells through strongly connected components, then decompose the system into overlapping subsystems for distributed computation. The global set is reconstructed and validated from subsystem solutions, and the approximation converges to the centralized graph-based result as the grid is refined (Decardi-Nelson et al., 2022).

Continuous-time geometric methods replace predecessor iteration by inequalities on support functions. For algebraic systems obtained from projected control systems, invariance of a convex set can be expressed as a support-function inequality valid for any convex set. This yields LMIs for ellipsoids and SOS or SDP conditions for polynomial and piecewise quadratic support functions; the same support-function framework extends to hybrid systems with mode-dependent flows and resets (Legat et al., 2021, Legat et al., 2021). Piecewise semi-ellipsoidal sets constitute a convex-programming compromise between ellipsoids and polyhedra: they are quadratic on sectors, more expressive than ellipsoids, and less facet-intensive than fine polyhedra (Legat et al., 2020).

Robust and lifted formulations add further geometric structure. Automaton-based methods for discrete-time linear systems with measurable disturbances encode disturbance-reactive controllers as finite automata and compute robust positively invariant sets as linear inequalities in a lifted space of plant states and controller parameters, then project back to obtain robust controlled invariant sets (Liu et al., 2021). For neural-network dynamical systems, set recursion over unions of hyperboxes together with mixed-integer encodings of ReLU networks yields control-invariant subsets with finite termination guarantees and supports mixed-integer MPC with recursive feasibility (Li et al., 15 May 2025).

Across these frameworks, the main trade-off is between representational richness and tractability. Ellipsoids and LMIs are compact but can be conservative; polyhedra can be exact but may suffer from vertex or facet explosion; interval, graph, and lifted representations trade algebraic simplicity for scalable certification structures.

5. Data-driven and learning-based computation

A separate line of work computes invariant sets directly from data, without first identifying a complete model. One approach characterizes the maximum positively invariant and maximum controlled invariant sets as the zero level set of an optimizer $f(x,u)\in X$1 of an infinite-dimensional LP over bounded functions. Restricting $f(x,u)\in X$2 to a finite-dimensional basis and imposing the Bellman-type inequality only on sampled one-step transitions yields a single finite-dimensional LP; the approximation is

$f(x,u)\in X$3

and the method provides convergence rates, sample-complexity bounds, and hard outer guarantees when additional modeling information is available (Korda, 2019).

For disturbed LTI systems, a direct data-driven RCI method uses a single state-input trajectory and disturbance bounds to construct a feasible model set $f(x,u)\in X$4, then solves LMIs in a semidefinite program to maximize $f(x,u)\in X$5, where $f(x,u)\in X$6 parameterizes a polytopic invariant set. The resulting polytopic RCI sets are of comparable size to model-based sets computed with exact system matrices, while also returning a state-feedback controller (Mejari et al., 2023).

Failure-driven learning offers a different mechanism. The FAIL algorithm operates on deterministic LTI systems with unknown $f(x,u)\in X$7, joint polytopic state-input constraints, noiseless state and input measurements, and sufficient excitation. Starting from the full joint constraint polytope $f(x,u)\in X$8, it collects one-step failing state-input pairs, identifies the violated projected state halfspace, regresses the corresponding predecessor halfspace directly in state-control space, and updates

$f(x,u)\in X$9

The sequence is monotone,

$X_{\mathrm{MCI}}\subseteq X$0

and under the stated learning condition it reaches the MSCI in finitely many iterations (Amine et al., 8 Apr 2026). In the reported double-integrator experiment, the algorithm learned the $X_{\mathrm{MCI}}\subseteq X$1 predecessor halfspaces from $X_{\mathrm{MCI}}\subseteq X$2 failing trajectories and recovered all $X_{\mathrm{MCI}}\subseteq X$3 MSCI halfspaces in $X_{\mathrm{MCI}}\subseteq X$4 iterations (Amine et al., 8 Apr 2026).

Control-oriented identification frameworks incorporate invariant sets into model learning itself. In quasi-LPV identification, one can regularize the model parameters by the size of a robust control invariant set of an associated uncertain linear model, so that model learning and invariant-set synthesis are coupled rather than decoupled. The regularization is the optimal value of a convex quadratic program in one formulation and of a differentiable robust optimization problem in a reduced-conservativeness formulation using tightened multiplicative uncertainty bounds (Mulagaleti et al., 2024, Mulagaleti et al., 12 May 2025). A related neural-network line synthesizes invariant sets offline for ReLU models and then enforces them online through mixed-integer MPC, again making recursive feasibility a learned-model design objective rather than a post hoc verification step (Li et al., 15 May 2025).

The main limitations are method-specific. The LP-on-transitions method may need rapidly growing basis size and sample density in high dimension (Korda, 2019). The single-trajectory RCI construction requires a bounded feasible model set, which in turn demands rank conditions on the data (Mejari et al., 2023). FAIL assumes deterministic noiseless data and explicitly requires failures to reveal missing predecessor constraints (Amine et al., 8 Apr 2026).

6. Quantitative extensions, applications, and interpretive issues

Control invariance also admits quantitative generalizations. Invariance pressure extends invariance entropy by introducing a continuous potential $X_{\mathrm{MCI}}\subseteq X$5. For an admissible pair $X_{\mathrm{MCI}}\subseteq X$6, the pressure-generating functional

$X_{\mathrm{MCI}}\subseteq X$7

induces the invariance pressure

$X_{\mathrm{MCI}}\subseteq X$8

When $X_{\mathrm{MCI}}\subseteq X$9, invariance pressure reduces to invariance entropy, and for linear systems with hyperbolic $X$0 the theory yields explicit upper bounds and exact formulas involving unstable eigenvalues and minimal long-term control cost (Colonius et al., 2018).

Stochastic extensions replace worst-case invariance by probability thresholds. Finite- and infinite-horizon PCISs are computed through dynamic programming, LPs in discrete finite-horizon problems, MILPs in discrete infinite-horizon problems, and discretization schemes with convergence guarantees in continuous-space finite-horizon problems. An infinite-horizon PCIS can also be constructed as a stochastic backward reachable set from an RCIS contained in it (Gao et al., 2019). This positions PCISs as a probabilistic complement to RCISs rather than a replacement.

Impulsive systems require yet another modification. For

$X$1

the admissible set

$X$2

captures states whose uncontrolled inter-impulse evolution stays inside the target zone $X$3. An impulsive controlled invariant set must be contained in $X$4, and a valid target zone must contain both an admissible invariant set and an equilibrium set. This leads naturally to zone MPC formulations that track invariant sets rather than points (Sanchez et al., 2021).

Several interpretive issues recur across the literature. First, maximality does not by itself provide a useful input description; this is why joint state-control constructions and methods that compute invariance-preserving input sets are consequential (Amine et al., 8 Apr 2026, Brown et al., 2023). Second, safe-state geometry depends strongly on modeling assumptions: deterministic, robust, probabilistic, impulsive, and hybrid notions are mathematically distinct, even when all are described informally as “staying safe.” Third, computational ease depends on the chosen representation. Polyhedra, support functions, interval unions, lifted automata, and neural-network mixed-integer encodings all define valid invariant-set computations, but they optimize different trade-offs between exactness, scalability, and feedback recoverability (Legat et al., 2020, Li et al., 15 May 2025).

Control-invariant sets therefore function both as a geometric object and as an overview interface. They delimit the states from which indefinite constraint satisfaction is possible, organize admissible control choices, and, in many modern formulations, are learned, approximated, or regularized jointly with the dynamical model itself.