Lyapunov equation
E181625
The Lyapunov equation is a fundamental matrix equation in control theory and dynamical systems used to analyze the stability of equilibrium points and design stable controllers.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lyapunov equation canonical | 4 |
| continuous-time Lyapunov equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1597812 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Lyapunov equation Context triple: [Aleksandr Lyapunov, notableWork, Lyapunov equation]
-
A.
Nyquist stability criterion
The Nyquist stability criterion is a graphical frequency-domain method in control theory used to determine the stability of feedback systems by analyzing how their open-loop transfer function encircles a critical point in the complex plane.
-
B.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
-
C.
Linear Systems
"Linear Systems" is a highly influential graduate-level textbook by Thomas Kailath that rigorously develops the theory and applications of linear system analysis and control.
-
D.
Kailath factorization in linear systems
Kailath factorization in linear systems is a matrix factorization technique used in control and signal processing to efficiently analyze and solve linear dynamical systems.
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E.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Lyapunov equation Target entity description: The Lyapunov equation is a fundamental matrix equation in control theory and dynamical systems used to analyze the stability of equilibrium points and design stable controllers.
-
A.
Nyquist stability criterion
The Nyquist stability criterion is a graphical frequency-domain method in control theory used to determine the stability of feedback systems by analyzing how their open-loop transfer function encircles a critical point in the complex plane.
-
B.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
-
C.
Linear Systems
"Linear Systems" is a highly influential graduate-level textbook by Thomas Kailath that rigorously develops the theory and applications of linear system analysis and control.
-
D.
Kailath factorization in linear systems
Kailath factorization in linear systems is a matrix factorization technique used in control and signal processing to efficiently analyze and solve linear dynamical systems.
-
E.
Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
concept in control theory
ⓘ
concept in dynamical systems ⓘ matrix equation ⓘ |
| appliesTo |
linear time-invariant systems
ⓘ
state-space models ⓘ |
| assumes | Q is positive definite or positive semidefinite ⓘ |
| characterizes | asymptotic stability of linear time-invariant systems ⓘ |
| conditionFor | existence of quadratic Lyapunov functions ⓘ |
| hasForm |
A P + P A^T = -Q
ⓘ
A P A^T - P = -Q ⓘ A^T P + P A = -Q ⓘ A^T P A - P = -Q ⓘ |
| hasType |
Lyapunov equation
self-linksurface differs
ⓘ
surface form:
continuous-time Lyapunov equation
discrete-time Lyapunov equation ⓘ generalized Lyapunov equation ⓘ |
| hasVariable |
symmetric matrix P
ⓘ
symmetric matrix Q ⓘ system matrix A ⓘ |
| holdsIf |
system matrix A has eigenvalues inside the unit circle in discrete time
ⓘ
system matrix A has eigenvalues with negative real parts in continuous time ⓘ |
| implies | P is positive definite if A is stable and Q is positive definite ⓘ |
| mathematicalDomain |
linear algebra
ⓘ
matrix analysis ⓘ |
| namedAfter | Aleksandr Lyapunov ⓘ |
| relatedTo |
Lyapunov function
ⓘ
Lyapunov stability theory ⓘ Riccati equation ⓘ algebraic Riccati equation ⓘ |
| solvedBy |
Bartels–Stewart algorithm
ⓘ
Kronecker product methods ⓘ Schur decomposition methods ⓘ iterative numerical algorithms ⓘ vectorization methods ⓘ |
| usedFor |
analyzing stability of equilibrium points
ⓘ
checking internal stability of closed-loop systems ⓘ computing Lyapunov functions in quadratic form ⓘ computing controllability Gramians ⓘ computing observability Gramians ⓘ designing stable controllers ⓘ model reduction of dynamical systems ⓘ robust control analysis ⓘ |
| usedIn |
control theory
ⓘ
controller design ⓘ dynamical systems theory ⓘ mechanical system stability analysis ⓘ optimal control ⓘ signal processing ⓘ stability analysis ⓘ systems and control engineering ⓘ systems biology modeling ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lyapunov equation Description of subject: The Lyapunov equation is a fundamental matrix equation in control theory and dynamical systems used to analyze the stability of equilibrium points and design stable controllers.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
continuous-time Lyapunov equation