Brockett’s condition for smooth feedback stabilization
E615222
Brockett’s condition for smooth feedback stabilization is a fundamental result in nonlinear control theory that provides a necessary topological criterion for when a nonlinear system can be stabilized by a smooth state-feedback law.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Brockett’s condition for smooth feedback stabilization canonical | 1 |
| Brockett’s necessary condition for smooth feedback stabilization | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6735780 — 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: Brockett’s condition for smooth feedback stabilization Context triple: [Roger W. Brockett, notableWork, Brockett’s condition for smooth feedback stabilization]
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A.
Nonlinear Control Systems
Nonlinear Control Systems is a foundational textbook by Alberto Isidori that systematically develops the theory and design methods for controlling nonlinear dynamical systems.
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B.
Lyapunov stability theory
Lyapunov stability theory is a fundamental framework in dynamical systems and control theory that uses energy-like functions to assess the stability of equilibrium points without explicitly solving differential equations.
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C.
Stability of Linear Systems
"Stability of Linear Systems" is a foundational book by Eliahu I. Jury that systematically develops the theory and criteria for determining the stability of linear dynamical and control systems.
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D.
Inners and Stability of Dynamic Systems
"Inners and Stability of Dynamic Systems" is a seminal work in control theory by Eliahu I. Jury that analyzes the role of inner functions in determining the stability properties of dynamic systems.
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E.
Zames–Falb multipliers
Zames–Falb multipliers are a class of frequency-domain operators used in control theory to analyze and guarantee the stability of nonlinear and time-varying feedback systems.
- 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: Brockett’s condition for smooth feedback stabilization Target entity description: Brockett’s condition for smooth feedback stabilization is a fundamental result in nonlinear control theory that provides a necessary topological criterion for when a nonlinear system can be stabilized by a smooth state-feedback law.
-
A.
Nonlinear Control Systems
Nonlinear Control Systems is a foundational textbook by Alberto Isidori that systematically develops the theory and design methods for controlling nonlinear dynamical systems.
-
B.
Lyapunov stability theory
Lyapunov stability theory is a fundamental framework in dynamical systems and control theory that uses energy-like functions to assess the stability of equilibrium points without explicitly solving differential equations.
-
C.
Stability of Linear Systems
"Stability of Linear Systems" is a foundational book by Eliahu I. Jury that systematically develops the theory and criteria for determining the stability of linear dynamical and control systems.
-
D.
Inners and Stability of Dynamic Systems
"Inners and Stability of Dynamic Systems" is a seminal work in control theory by Eliahu I. Jury that analyzes the role of inner functions in determining the stability properties of dynamic systems.
-
E.
Zames–Falb multipliers
Zames–Falb multipliers are a class of frequency-domain operators used in control theory to analyze and guarantee the stability of nonlinear and time-varying feedback systems.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in nonlinear control theory ⓘ |
| alsoKnownAs |
Brockett’s necessary condition for smooth feedback stabilization
ⓘ
Brockett’s topological obstruction ⓘ |
| appliesTo |
control-affine systems
ⓘ
finite-dimensional dynamical systems ⓘ nonlinear control systems ⓘ |
| assumes |
continuous-time dynamical system
ⓘ
smoothness of the feedback map ⓘ |
| basedOn |
differential topology
ⓘ
properties of continuous maps ⓘ topology ⓘ |
| clarifies |
difference between linear and nonlinear stabilizability
ⓘ
limitations of smooth static state feedback ⓘ |
| concerns |
asymptotic stabilization of equilibria
ⓘ
smooth state-feedback stabilization ⓘ topological obstructions to stabilization ⓘ |
| contrastsWith |
results on discontinuous feedback stabilization
ⓘ
results on time-varying feedback stabilization ⓘ |
| doesNotGive | sufficient condition for smooth stabilizing feedback ⓘ |
| field |
applied mathematics
ⓘ
control theory ⓘ nonlinear control theory ⓘ |
| gives | necessary condition for existence of smooth stabilizing feedback ⓘ |
| hasImpactOn |
feedback design methodologies
ⓘ
theory of stabilization of nonholonomic systems ⓘ understanding of structural properties of control systems ⓘ |
| historicalPeriod | late 20th century ⓘ |
| implies | existence of topological obstructions to smooth stabilization ⓘ |
| involves |
continuous feedback law
ⓘ
continuous surjections ⓘ equilibrium point of the closed-loop system ⓘ image of vector field under feedback map ⓘ local asymptotic stability ⓘ smooth feedback law ⓘ |
| motivated |
research on discontinuous feedback stabilization
ⓘ
research on hybrid control strategies ⓘ research on time-varying feedback laws ⓘ |
| namedAfter | Roger W. Brockett ⓘ |
| relatedTo |
Brockett integrator example
ⓘ
nonholonomic integrator ⓘ |
| statesThat | for certain nonlinear systems no smooth static state feedback can render the origin asymptotically stable ⓘ |
| typeOf | necessary condition ⓘ |
| usedIn |
analysis of nonholonomic systems
ⓘ
design of control laws for mobile robots ⓘ geometric control theory ⓘ study of driftless control systems ⓘ |
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: Brockett’s condition for smooth feedback stabilization Description of subject: Brockett’s condition for smooth feedback stabilization is a fundamental result in nonlinear control theory that provides a necessary topological criterion for when a nonlinear system can be stabilized by a smooth state-feedback law.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Brockett’s condition for smooth feedback stabilization
→
alsoKnownAs
→
Brockett’s necessary condition for smooth feedback stabilization
ⓘ
linked to: Brockett’s condition for smooth feedback stabilization