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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.

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.

Scope of Application

  • 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.

  • 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.

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.

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.

Abstract Reasoning

  1. Type the carrier. Identify the cross-domain formal modeling entities to which the claim applies.
  2. 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.
  3. Check operation and conditions. For the general nonlinear system (), the input u can be found by solving a static non-linear programming problem.
  4. Demand recognition evidence. This is made rigorous by Artstein's theorem.
  5. Test variation.

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.

Relationships to Other Abstractions

Local relationship map for Control-Lyapunov functionParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Control-LyapunovfunctionDOMAINPrime abstraction: Feedback — presupposesFeedbackPRIME

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

Neighborhood in Abstraction Space

Control-Lyapunov function sits in a crowded region of the domain-specific corpus (37th 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

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