Control-Lyapunov function¶

In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8700
Domain group: Applied Sciences & Engineering
Origin domain: Robotics & Automation
Subdomains: Control Theory, Nonlinear Control → Robotics & Automation

Core Idea¶

Control-Lyapunov function is treated here as the recurring cross-domain formal modeling identity summarized by this source-grounded definition: In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs.

In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state x \ne 0 in some domain D, then the state will remain in D for all time.

For asymptotic stability, the state is also required to converge to x = 0 . A control-Lyapunov function is used to test whether a system is asymptotically stabilizable, that is whether for any state x there exists a control u(x,t) such that the system can be brought to the zero state asymptotically by applying the control u. The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D.

For Control-Lyapunov function, the abstraction is narrower than the article's general subject matter: a positive case must preserve In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in cross-domain formal modeling, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

Always a Way Downhill

Imagine a marble on a bumpy table, and you want it to end up in one special spot. A control-Lyapunov function is like a 'how far from home' score for every place the marble could be. If, from every place, you can always give a push that makes the score go down, you know you can steer the marble home.

The Steer-Home Score

Engineers who build robots and machines want them to settle into a goal position and stay there. A Lyapunov function is like an 'energy' or 'height' score that is lowest at the goal; if the score always goes down, the system heads to the goal. A control-Lyapunov function adds steering: it's a score where, from any position, there is some control you can choose, like pushing a motor, that makes the score go down. If such a score exists, you know the system can be brought to the goal. It tests whether a system is steerable to rest, not just whether it gets there on its own.

Stabilizability Certificate Function

In control theory, a Lyapunov function V(x) is a scalar function used to test whether a dynamical system is stable: roughly, V behaves like an energy that is smallest at the equilibrium and does not increase along the system's motion. Lyapunov stability means a state starting in some region D stays in D for all time; asymptotic stability also requires the state to converge to zero. A control-Lyapunov function (CLF) extends this to systems with control inputs. Instead of asking whether the system settles on its own, it asks whether the system is asymptotically stabilizable: whether, for every state x, there exists a control u that can bring the system asymptotically to zero. A CLF is a function V for which, at each state, some choice of control makes V decrease, which certifies that stabilizing controls exist.

A control-Lyapunov function (CLF) extends the Lyapunov function V(x) to systems with control inputs, of the form dx/dt = f(x, u). An ordinary Lyapunov function certifies that an uncontrolled system is Lyapunov stable (trajectories starting in a domain D remain in D) or, more restrictively, asymptotically stable (trajectories also converge to x = 0). A CLF is instead used to test asymptotic stabilizability: whether for any state x there exists a control u(x, t) that drives the system to the zero state asymptotically. The requirement is that V be positive definite and that, at each nonzero state, some admissible input makes the derivative of V along the dynamics negative. The existential quantifier over controls is what distinguishes it from a Lyapunov function for a fixed closed loop.

Structural Signature¶

Sig role-phrases:

Defining carrier — Without loss of generality, suppose the equilibrium is at x_=0 (for an equilibrium x_\neq 0 , it can be translated to the origin by a change of variables).
Constitutive relation — Subbotin that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback.
Operating condition — For the general nonlinear system (), the input u can be found by solving a static non-linear programming problem.
Recognition evidence — This is made rigorous by Artstein's theorem.
Admissible variation — It was later shown by Francis H.
Characteristic consequence — The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D.
Failure boundary — A control-Lyapunov function is used to test whether a system is asymptotically stabilizable, that is whether for any state x there exists a control u(x,t) such that the system can be brought to the zero state asymptotically by applying the control u.

What It Is Not¶

Not the whole field of cross-domain formal modeling. The node requires the specific identity stated by In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs.
Not an over-broad reading. A control-Lyapunov function (CLF) is a function V : D \to \mathbb{R} that is continuously differentiable, positive-definite (that is, V(x) is positive for all x\in D except at x=0 where it is zero), and such that for all x \in \mathbb{R}^n (x \neq 0), there exists u\in \mathbb{R}^m such that.
Not an over-broad reading. Artstein proved that the dynamical system () has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback u(x).
Not an over-broad reading. which is a linear first order differential equation which has solution.
Not automatically Stability Theory. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Control-Lyapunov function applies literally inside cross-domain formal modeling wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable.
Documented setting. The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D.
Documented setting. A control-Lyapunov function is used to test whether a system is asymptotically stabilizable, that is whether for any state x there exists a control u(x,t) such that the system can be brought to the zero state asymptotically by applying the control u.
Theorems. Artstein proved that the dynamical system () has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback u(x).
Constructing the Stabilizing Input. It is often difficult to find a control-Lyapunov function for a given system, but if one is found, then the feedback stabilization problem simplifies considerably.
Example. Here is a characteristic example of applying a Lyapunov candidate function to a control problem.

Outside cross-domain formal modeling, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Control-Lyapunov function names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The strongest recognition evidence in the frozen account is: This is made rigorous by Artstein's theorem. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification A control-Lyapunov function (CLF) is a function V : D \to \mathbb{R} that is continuously differentiable, positive-definite (that is, V(x) is positive for all x\in D except at x=0 where it is zero), and such that for all x \in \mathbb{R}^n (x \neq 0), there exists u\in \mathbb{R}^m such that. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Control-Lyapunov function compresses multiple cross-domain formal modeling details into a stable diagnostic relation. The source shows both the central mechanism—subbotin that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback.—and the practical consequence—the theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the cross-domain formal modeling entities to which the claim applies.
State the relation. Use the source-grounded identity: In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs.
Check operation and conditions. For the general nonlinear system (), the input u can be found by solving a static non-linear programming problem.
Demand recognition evidence. This is made rigorous by Artstein's theorem.
Test variation. Change an implementation or setting while preserving it was later shown by Francis H.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Control-Lyapunov function transfers literally when a new case preserves the same carrier type, relation, and recognition test. The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D.

Beyond the home domain. No canonical parent is asserted for Control-Lyapunov function. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In the special case of a single input system (m=1) , Sontag's formula is written as. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs; recognition evidence → This is made rigorous by Artstein's theorem

Applied / In Practice¶

k(x) = \begin{cases} \displaystyle -\frac{L_{f} V(x)+\sqrt{\left[L_{f} V(x)\right]^{2}+\left[L_{g} V(x)\right]^{4}}}{L_{g} V(x)} & \text { if } L_{g} V(x) \neq 0 \. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Constructing the Stabilizing Input; invariant → In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs; boundary → the case exits the class when a control-Lyapunov function (CLF) is a function V : D \to \mathbb{R} that is continuously differentiable, positive-definite (that is, V(x) is positive for all x\in D except at x=0 where it is zero), and such that for all x \in \mathbb{R}^n (x \neq 0), there exists u\in \mathbb{R}^m such that

Structural Tensions¶

T1 — Stable identity versus admissible variation. A control-Lyapunov function (CLF) is a function V : D \to \mathbb{R} that is continuously differentiable, positive-definite (that is, V(x) is positive for all x\in D except at x=0 where it is zero), and such that for all x \in \mathbb{R}^n (x \neq 0), there exists u\in \mathbb{R}^m such that. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Artstein proved that the dynamical system () has a differentiable control-Lyapunov function if and only if there exists a regular stabilizing feedback u(x). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. which is a linear first order differential equation which has solution. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. which can then be solved using any linear differential equation methods. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Without loss of generality, suppose the equilibrium is at x_=0 (for an equilibrium x_\neq 0 , it can be translated to the origin by a change of variables). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Control-Lyapunov function literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. Subbotin that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Control-Lyapunov function distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Control-Lyapunov function is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. Its framed side is the cross-domain formal modeling vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: For the general nonlinear system (), the input u can be found by solving a static non-linear programming problem. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Without loss of generality, suppose the equilibrium is at x=0 (for an equilibrium x\neq 0 , it can be translated to the origin by a change of variables). Subbotin that every asymptotically controllable system can be stabilized by a (generally discontinuous) feedback. It further constrains recognition and variation through: For the general nonlinear system (), the input u can be found by solving a static non-linear programming problem. This is made rigorous by Artstein's theorem.

What is domain-bound. cross-domain formal modeling supplies the operative entities, technical vocabulary, warrants, and exceptions that make Control-Lyapunov function literal. Its documented scope includes the condition that The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Another bounded application condition is that The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—It was later shown by Francis H.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry presupposes Feedback.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Control-Lyapunov function. The reviewed identity is: In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Control-Lyapunov function Domain-specific

Parents (1) — more general patterns this builds on

Control-Lyapunov function presupposes Feedback Prime

Control-Lyapunov function presupposes Feedback: the parent's defining role is necessary to the child's frozen mechanism or criterion.

Hierarchy path (1) — routes to 1 parentless root

Control-Lyapunov function → Feedback

Neighborhood in Abstraction Space¶

Control-Lyapunov function sits in a crowded region of the domain-specific corpus (37^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Autonomous Control & Learning Systems (11 abstractions)

Nearest neighbors

Filling radius — 0.89
Linear-quadratic regulator rapidly exploring random tree — 0.89
Behavioral modeling — 0.88
Linear elasticity — 0.87
Homoclinic bifurcation — 0.87

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs?
Stability Theory. The mathematical framework for determining whether trajectories or solutions remain close to, converge toward, or depart from a reference behavior after small perturbations. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Backstepping. A recursive nonlinear-control design method for strict-feedback systems that constructs a stabilizing controller and Lyapunov function stage by stage. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Lyapunov redesign. A nonlinear-control method that augments a nominal stabilizing feedback law using a known Lyapunov function to preserve stability under matched uncertainty. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Control-Lyapunov function remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside cross-domain formal modeling lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Control-Lyapunov_function (revision 1365691314).
Preserved source candidate: https://link.springer.com/chapter/10.1007/978-0-8176-4759-9_3
Preserved source candidate: https://books.google.com/books?id=_eTb4Yl0SOEC
Preserved source candidate: http://www.sontaglab.org/FTPDIR/sontag_mathematical_control_theory_springer98.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.