Controlled Invariant Subspace¶

A controlled invariant subspace is a linear-system state subspace whose trajectories can be kept inside it by suitable control input, equivalently one satisfying AV ⊆ V + im B or made invariant by some static state feedback.

Version: v1 · 2026-08-30 · History
Domain-specific #: 1558
Origin domain: control theory
Subdomain: geometric control of linear systems
Aliases: (A,B)-invariant subspace, Controlled-invariant subspace

Core Idea¶

Consider a finite-dimensional linear time-invariant system over the real or complex numbers,

\[ \dot x(t)=Ax(t)+Bu(t) \]

in continuous time, or $x_{k+1}=Ax_k+Bu_k$ in discrete time. A linear state subspace $V\subseteq X$ is controlled invariant for the pair $(A,B)$ when every initial state $x_0\in V$ admits an input that keeps the resulting state trajectory in $V$ for all future time. The input is allowed to counteract the component of $Ax$ that would otherwise leave $V$.

Three descriptions are equivalent in the standard finite-dimensional, unrestricted-input setting:^[1]

\[ \begin{aligned} &\forall x_0\in V\;\exists u(\cdot)\text{ such that }x(t)\in V\text{ for all }t\geq 0,\\ &AV\subseteq V+\operatorname{im}B,\\ &\exists F:X\to U\text{ linear such that }(A+BF)V\subseteq V. \end{aligned} \]

The feedback convention here is $u=Fx$; authors using $u=-Kx$ write $A-BK$ instead. A matrix $F$ satisfying the last inclusion is commonly called a friend of $V$. The equivalence is the abstraction's center: an open-loop existential ability to remain in a subspace, a local algebraic inclusion, and a closed-loop feedback synthesis condition encode the same property.

The construction was introduced as a generalization of ordinary invariant subspaces in the foundational work of Giuseppe Basile and Giovanni Marro.^[2] It became a basic object of geometric control because it converts feedback-design existence questions into subspace construction and inclusion tests.^[3]

Structural Signature¶

Role structure: finite-dimensional state space $X$ + system map $A:X\to X$ + input map $B:U\to X$ + candidate subspace $V\subseteq X$ + input-selection authority + containment objective + optional feedback friend $F$.

Defining invariant: every drift image $Av$ for $v\in V$ has an input-generated correction modulo which it lies in $V$:

\[ AV\subseteq V+\operatorname{im}B. \]

Equivalently, for each $v\in V$ there are $w\in V$ and $u\in U$ such that $Av=w+Bu$. Linearity permits these pointwise choices to be assembled into a linear map on $V$ and extended to a full-state feedback $F$ with $(A+BF)V\subseteq V$.

Recognition test: specify $A$, $B$, and a linear subspace $V$; compute a basis for $AV$, $V$, and $\operatorname{im}B$; then test whether every column of $AV$ lies in the span of $V$ and $B$. In matrix form, if the columns of $Q$ span $V$, the condition is solvability of

\[ AQ=QH+BG \]

for some matrices $H$ and $G$. Setting $FQ=-G$ under the stated sign convention makes $(A+BF)Q=QH$.

Preserved features: $V$ remains a linear subspace; the original pair $(A,B)$ fixes which subspaces qualify; the existence of at least one correcting input or friend is required; and the full closed-loop trajectory, not merely one sampled state, remains in $V$.

Identity-breaking changes: replacing “there exists an input” by “for every input,” allowing escape and later return, dropping linear-subspace closure, changing the input map without re-evaluation, or adding stability, output-zero, disturbance, or magnitude constraints without naming the stronger variant.

What It Is Not¶

Controlled invariance is not ordinary $A$-invariance. Ordinary invariance requires $AV\subseteq V$, so zero input already preserves the subspace. Controlled invariance permits $Av$ to leave $V$ provided its offending component lies in the directions available through $B$. Every $A$-invariant subspace is controlled invariant, but the converse generally fails.

It is not global controllability or reachability. Controllability asks whether arbitrary states can be steered across the state space; controlled invariance asks whether a specified subspace can be made confining. The whole state space $X$ is always controlled invariant even if $(A,B)$ is uncontrollable. Conversely, a controllable pair can have a particular subspace that is not controlled invariant.

It is not stability. A friend $F$ preserves $V$, but the eigenvalues of $(A+BF)|_V$ may be unstable. Requiring a preserving friend whose restricted dynamics lie in a chosen stability region gives the stronger notion of a stabilizability subspace.^[1]

It is not output nulling, disturbance decoupling, conditioned invariance, strong invariance, or a constrained control-invariant set. Each adds or changes a quantifier, map, or constraint. Those distinctions are part of the node's recognition boundary rather than optional terminology.

Scope of Application¶

The nucleus is finite-dimensional LTI geometric control with unrestricted real- or complex-valued inputs and ordinary state-space dynamics. It applies in continuous time and, through the same algebraic inclusion, in discrete time. The closed-loop characterization presumes that full state is available for static feedback; if only outputs are measured, observer or dynamic-compensator questions must be solved separately.

Within this scope, controlled invariant subspaces organize disturbance decoupling, output-nulling problems, constrained pole assignment, regulation, system decomposition, and computation of the largest subspace contained in a safety or output kernel constraint.^[1]^[3] They are used both as analysis objects and as intermediate certificates from which feedback laws can be constructed.

Extensions exist for descriptor systems, infinite-dimensional systems, nonlinear distributions, switched and hybrid systems, robust invariance, positive systems, and state/input-constrained invariant sets. Those extensions may replace $A$, $B$, subspaces, or unconstrained inputs with different objects and may require topological closure, tangency, disturbance quantifiers, or admissibility assumptions. They should not be silently folded into this node's theorem package.

Clarity¶

A useful diagnostic is: what causes the next-state or velocity component normal to $V$ to vanish?

If $A$ alone maps every $v\in V$ back into $V$, the subspace is ordinarily invariant and therefore controlled invariant trivially.
If the normal component of $Av$ can be canceled by some vector in $\operatorname{im}B$ for every $v\in V$, it is controlled invariant nontrivially.
If even one $v\in V$ has a normal drift component outside the projected input image, the candidate fails.
If containment must hold for every admissible input, the relevant property is strong invariance, not existential controlled invariance.

This test separates the object from the controller used to witness it. A subspace can have many friends; a particular feedback can fail to preserve it even though some other feedback succeeds. Controlled invariance is a property of $(A,B,V)$, invariant under state feedback and invertible changes of input coordinates, not a claim that all controllers are equivalent.^[1]

The terms “controlled invariant” and “$(A,B)$-invariant” are exact aliases in this finite-dimensional setting. “Control invariant set” is broader and should not be used as an unqualified alias: it often denotes nonlinear, nonconvex, state-constrained, or input-bounded sets.

Manages Complexity¶

The abstraction replaces an infinite family of trajectory-existence questions—one for every initial state in $V$ and every future time—with a single finite-dimensional inclusion. Instead of directly searching over time functions $u(\cdot)$, one checks a span or rank relation. Instead of searching blindly over all feedback matrices, one first constructs a subspace with the right containment geometry and then solves linear equations for a friend.

It also gives a finite algorithm for the largest controlled invariant subspace contained in a prescribed subspace $K$. Begin with $V_0=K$ and iterate

\[ V_{j+1}=K\cap A^{-1}(V_j+\operatorname{im}B). \]

The sequence is descending. In an $n$-dimensional state space, every strict step decreases dimension, so it stabilizes after finitely many steps; its fixed point is the supremal controlled invariant subspace $V^*(K)$.^[1] This compresses a synthesis question into linear-algebra operations and exposes failure constructively: each removed direction is one from which no one-step correction keeps the remaining target viable.

For disturbance decoupling with disturbance map $E$ and regulated-output map $H$, the geometric condition $\operatorname{im}E\subseteq V\subseteq\ker H$ for some controlled invariant $V$ turns a transfer-function cancellation question into subspace containment. The largest candidate $V^*(\ker H)$ yields the compact test $\operatorname{im}E\subseteq V^*(\ker H)$ in the standard full-state-feedback formulation.^[1]

Abstract Reasoning¶

The inclusion licenses several deductions.

Feedback existence. If $AV\subseteq V+\operatorname{im}B$, choose for each basis vector $q_i$ of $V$ an input correction so that $Aq_i+Bf_i\in V$. Linear extension gives a single feedback $F$ working for every state in $V$. Thus pointwise correctability does not require a different nonlinear policy for each initial state.

Continuous-time persistence. Once $(A+BF)V\subseteq V$, the matrix exponential also preserves $V$: $e^{(A+BF)t}V\subseteq V$. Starting in $V$ under $u=Fx$ therefore keeps the entire trajectory inside, not merely its initial derivative.

Coordinate invariance. Under an invertible state change $x=Tz$, the triple becomes $(T^{-1}AT,T^{-1}B,T^{-1}V)$, and the inclusion is preserved. Qualification is geometric rather than dependent on the matrix representation.

Closure under sums. If $V_1$ and $V_2$ are controlled invariant, then $A(V_1+V_2)\subseteq V_1+V_2+\operatorname{im}B$. Hence their sum is controlled invariant, which guarantees a unique largest controlled invariant subspace inside any fixed $K$. Intersections need not preserve the property, because corrections available separately for the two subspaces may not admit a common correction in their intersection.

No stability inference. The existence of a friend constrains off-subspace leakage but may leave internal modes fixed or unstable. Pole assignment inside $V$ depends on the controllability structure within $V$, not on controlled invariance alone.

Knowledge Transfer¶

Knowledge transfers exactly across finite-dimensional continuous- and discrete-time LTI models because the operative test is the same pair/subspace inclusion. A proof or algorithm expressed in coordinates can be transported under similarity transformations, changes of input basis, or state feedback, then recovered as a geometric statement about the corresponding subspaces.

Transfer among geometric-control problems proceeds by changing the containing constraint $K$. For output nulling, one searches within an output kernel. For disturbance decoupling, one additionally asks that disturbance directions lie inside the selected subspace. For constrained pole assignment, one first selects a friend and then determines which internal and external modes remain assignable. The same subspace certificate organizes these tasks without making them identical.

Transfer to robust or constrained invariance requires new quantifiers. With additive disturbance $Ed$, keeping $V$ invariant for every disturbance generally strengthens the algebraic test to include disturbance directions, such as $AV+\operatorname{im}E\subseteq V+\operatorname{im}B$ in the standard linear unconstrained formulation. With bounded controls or state constraints, the corrective input must also satisfy those bounds, and a subspace may cease to be viable even when the unconstrained inclusion holds.

Cross-domain transfer is only analogical. The portable skeleton—an intervention authority can cancel departures from a desired region—is covered by broader abstractions such as Invariance, Feedback, and Controllability. The named object requires vector spaces, linear maps, input images, and state trajectories, so it remains domain-specific.

Examples¶

Canonical nontrivial example¶

Let

\[ A=\begin{bmatrix}0&0\\1&0\end{bmatrix},\qquad B=\begin{bmatrix}0\\1\end{bmatrix},\qquad V=\operatorname{span}\!\left\{\begin{bmatrix}1\\0\end{bmatrix}\right\}. \]

Writing $e_1=(1,0)^T$ and $e_2=(0,1)^T$, we have $Ae_1=e_2$, so $V$ is not $A$-invariant. But $\operatorname{im}B=\operatorname{span}\{e_2\}$, so

\[ AV=\operatorname{span}\{e_2\}\subseteq V+\operatorname{im}B. \]

Choose $F=[-1\;0]$. Then $BFe_1=-e_2$ and $(A+BF)e_1=0$, so $V$ is invariant under the closed-loop dynamics. In state equations, $\dot x_1=0$ and $\dot x_2=x_1+u$; on $V$, the feedback $u=-x_1$ cancels the normal drift and keeps $x_2=0$.

Controllable system, failing candidate subspace¶

Keep the same $A$ and $V$ but take $B=e_1$. Then $V+\operatorname{im}B=V$, whereas $Ae_1=e_2\notin V$. No input can cancel the $x_2$ velocity, so $V$ is not controlled invariant. Nevertheless, the controllability matrix $[B\;AB]=[e_1\;e_2]$ has full rank. This gives a sharp counterexample to equating controlled invariance of a selected $V$ with global controllability.

Supremal-subspace computation¶

For the first pair and constraint $K=\operatorname{span}\{e_1\}$, $K+\operatorname{im}B=\mathbb R^2$, so the iteration returns $V_1=K$ and stabilizes: $V^*(K)=K$. For the second pair, $K+\operatorname{im}B=K$, and only $x=0$ in $K$ has $Ax\in K$; hence $V_1=\{0\}$ and the largest controlled invariant subspace in $K$ is trivial.

The subspaces $\{0\}$ and $X$ are always controlled invariant. These trivial examples are useful checks but supply no evidence of global controllability, stabilizability, or a meaningful containment objective.

Structural Tensions¶

Open-loop existence vs. closed-loop uniformity. The definition permits a different input function for each initial state, yet finite-dimensional linearity produces one static feedback friend for all of $V$. Diagnostic: verify the algebraic inclusion before claiming a common feedback law.
Containment vs. internal performance. A friend stops leakage from $V$ but does not guarantee decay, speed, robustness, or acceptable control effort. Diagnostic: inspect the spectrum, robustness margins, and input constraints separately.
Subspace geometry vs. physical constraints. The algebraic theorem assumes unrestricted input values. Saturation can make an algebraically controlled invariant subspace nonviable for large states. Diagnostic: if amplitude or state bounds matter, use a constrained invariant-set formulation.
State availability vs. implementability. The friend is a full-state feedback map. Diagnostic: if only $y=Cx$ is measured, solve the output-feedback or observer problem rather than treating the friend as directly implementable.
Largest admissible region vs. design flexibility. The supremal $V^*(K)$ maximizes contained directions, but a smaller controlled invariant subspace may permit better pole placement or robustness. Diagnostic: separate feasibility under containment from the later controller-performance objective.
Ordinary vs. controlled invariance. Inputs can repair $A$-leakage, but that existential repair is lost if $B$ changes. Diagnostic: always state the pair $(A,B)$, not $V$ alone.

Structural–Framed Character¶

Controlled Invariant Subspace is structural, with aggregate framed score $0.10$. Membership follows from a basis-independent linear inclusion and three equivalent mathematical conditions. It does not depend on institutional designation, evaluative judgment, or a community's preference. Once $A$, $B$, and $V$ are fixed, the answer is determinate.

Its slight framed component records modeling choices, not vagueness in the definition: the analyst chooses the state coordinates, input channels, candidate constraint subspace, time model, admissible-input class, and whether disturbance or saturation constraints are in scope. After those are declared, the recognition test is formal.

Structural Core vs. Domain Accent¶

Skeletal core. A process would leave a designated region under its native dynamics; an authorized intervention can cancel every outward component; one uniform rule can realize the correction; and a local closure test certifies indefinite containment. This skeleton is recognizable as intervention-enabled invariance.

Domain-bound identity. The exact node requires a vector state space, linear maps $A$ and $B$, a linear subspace $V$, image and subspace sums, state trajectories, and a feedback matrix. Its finite algorithm depends on dimension and linear preimages. Remove those entities and the equivalence among trajectory containment, $AV\subseteq V+\operatorname{im}B$, and an invariant closed-loop map disappears.

Why not a prime. The portable residue is already represented by prime:invariance, prime:feedback, and prime:controllability. Controlled Invariant Subspace does not recur literally in unrelated domains without importing linear-system formalism. Its autonomous value lies in its geometric-control theorem package, computation, and synthesis role, so domain-specific classification is required.

Controlled Invariant Subspace strictly instantiates prime:invariance in a control-mediated form. The selected subspace is invariant under at least one closed-loop transformation $A+BF$, and equivalently the open-loop drift is invariant modulo available input directions. The parent captures preservation under transformation; the child adds linear state/input structure, an existential feedback selection, and constructive synthesis. A single strict subsumption edge to Invariance is therefore the minimal proposal.

It is related to prime:controllability, but not its subtype. Controllability concerns steering among states; controlled invariance concerns staying within one subspace, and neither implies the other for an arbitrary candidate $V$.

It is related to prime:feedback because a friend realizes containment as static state feedback, but the subspace is a feasibility certificate rather than a feedback loop itself. It is related to prime:observability through conditioned-invariant duality and output-feedback design, yet observability is not part of the defining test.

Relationships to Other Abstractions¶

Current abstraction Controlled Invariant Subspace Domain-specific

Parents (1) — more general patterns this builds on

Controlled Invariant Subspace is a kind of Invariance Prime

Controlled Invariant Subspace strictly instantiates prime:invariance in a control-mediated form.

Hierarchy path (1) — routes to 1 parentless root

Controlled Invariant Subspace → Invariance

Neighborhood in Abstraction Space¶

Controlled Invariant Subspace sits in a sparse region of the domain-specific corpus (81^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

$A$-invariant subspace: $AV\subseteq V$ without corrective input. Tell: set $B=0$ or ask whether zero input suffices.
Strongly invariant subspace: containment holds for every input, not for some selected input. In the standard linear setting this requires both $AV\subseteq V$ and $\operatorname{im}B\subseteq V$. Tell: inspect the input quantifier.
Controllability subspace: states can be steered to the origin in finite time while staying inside. Every controllability subspace is controlled invariant, not conversely. Tell: test internal reachability, not only containment.
Stabilizability subspace: there is a contained trajectory that is also stable in a specified stability domain. Tell: controlled invariance alone imposes no spectral condition.
Output-nulling controlled invariant subspace: the subspace is controlled invariant and lies inside the kernel of the relevant output map, with direct-feedthrough variants requiring modified conditions. Tell: look for an explicit output-zero constraint.
Conditioned invariant subspace: a dual observer-side object, commonly characterized by $A(S\cap\ker C)\subseteq S$ or existence of output injection making $S$ invariant. Tell: its corrective map uses measured output injection, not control input $B$.
Control-invariant set / viability kernel: a potentially nonlinear or nonconvex set preserved under constrained control. Tell: a general set need not be a subspace or satisfy the linear inclusion.
Robust controlled invariant set or subspace: containment must survive all disturbances in a declared class. Tell: locate the universal disturbance quantifier and disturbance directions.
Eigenspace: a subspace associated with eigenvectors of a fixed operator. Tell: a controlled invariant subspace can fail to be invariant, much less an eigenspace, for the open-loop $A$.
Feedback controller: one friend is a witness for the property. Tell: the controlled invariant subspace is the geometric object; the controller is one realization, and many friends may exist.
Safe set: a modeling constraint region. Tell: $K$ becomes an invariant certificate only after a controlled-invariance or viability test; merely naming it safe does not ensure it can be maintained.

References¶

[1] Harry L. Trentelman, Anton A. Stoorvogel, and Malo L. J. Hautus, Control Theory for Linear Systems (Springer, 2001), Chapter 4, doi:10.1007/978-1-4471-0339-4; author-hosted chapter text. registry ↩a ↩b ↩c ↩d ↩e ↩f

[2] Giuseppe Basile and Giovanni Marro, “Controlled and Conditioned Invariant Subspaces in Linear System Theory,” Journal of Optimization Theory and Applications 3, no. 5 (1969): 306–315, doi:10.1007/BF00931370. registry ↩

[3] W. Murray Wonham, Linear Multivariable Control: A Geometric Approach, 3^rd ed. (Springer, 1985), doi:10.1007/978-1-4612-1082-5. registry ↩a ↩b