Lyapunov Stability Ontology

flow "LyapunovStabilityOntology" v1.0

# LYAPUNOV STABILITY ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/dynamical-systems/lyapunov-stability-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Dynamical Systems"
ENTITY := "Lyapunov Stability"

LYAPUNOV_STABILITY := {
 type: DynamicalStabilityProperty,
 domain: DynamicalSystems,
 purpose: "Characterise whether trajectories starting sufficiently close to an equilibrium or invariant set remain close for all future time, and provide direct stability tests using Lyapunov functions."
}

CONCEPTS := {
 LYAPUNOV_STABILITY,
 EQUILIBRIUM,
 FIXED_POINT,
 PERTURBATION,
 EPSILON_DELTA_STABILITY,
 ATTRACTIVITY,
 ASYMPTOTIC_STABILITY,
 EXPONENTIAL_STABILITY,
 LYAPUNOV_FUNCTION,
 POSITIVE_DEFINITE_FUNCTION,
 NEGATIVE_DEFINITE_DERIVATIVE,
 LASALLE_INVARIANCE,
 BASIN_OF_ATTRACTION,
 GLOBAL_STABILITY,
 UNIFORM_STABILITY
}

# ---------- CONTINUOUS SYSTEM ----------
dx/dt
 =
f(x,t)

equilibrium
 x_star

for autonomous system:

f(x_star)
 =
0

LYAPUNOV_STABILITY
 -> response_to(SMALL_INITIAL_PERTURBATIONS)

# ---------- AUTONOMOUS LYAPUNOV STABILITY ----------
x_star
 -> LYAPUNOV_STABLE

iff

for every
 epsilon>0

exists
 delta>0

such_that

||x(0)-x_star||
 < delta

implies

||x(t)-x_star||
 < epsilon

for all
 t>=0

STABILITY
 -> nearby_initial_states_remain_nearby

# ---------- INSTABILITY ----------
x_star
 -> UNSTABLE

if

there exists epsilon_0>0

such_that

for every delta>0

some initial condition satisfies

||x(0)-x_star||<delta

but

||x(t)-x_star||
 >= epsilon_0

for some future t

INSTABILITY
 -> failure_of(LYAPUNOV_STABILITY)

# ---------- ATTRACTIVITY ----------
x_star
 -> LOCALLY_ATTRACTIVE

if

exists delta>0

such_that

||x(0)-x_star||<delta

implies

x(t)
 -> x_star

as
 t->infinity

ATTRACTIVITY
 -/> same_as(LYAPUNOV_STABILITY)

# ---------- ASYMPTOTIC STABILITY ----------
x_star
 -> ASYMPTOTICALLY_STABLE

iff

x_star
 -> LYAPUNOV_STABLE

and

x_star
 -> LOCALLY_ATTRACTIVE

ASYMPTOTIC_STABILITY
 -> stability_plus_convergence

# ---------- GLOBAL ASYMPTOTIC STABILITY ----------
x_star
 -> GLOBALLY_ASYMPTOTICALLY_STABLE

if

x_star
 -> LYAPUNOV_STABLE

and

all admissible initial states
 -> converge_to(x_star)

GLOBAL_ASYMPTOTIC_STABILITY
 -> basin_of_attraction_equals(RELEVANT_STATE_SPACE)

# ---------- EXPONENTIAL STABILITY ----------
x_star
 -> EXPONENTIALLY_STABLE

if

exists constants
 M>=1
 alpha>0
 delta>0

such_that

||x(t)-x_star||
 <=
M
*
exp(-alpha*t)
*
||x(0)-x_star||

for nearby initial conditions

EXPONENTIAL_STABILITY
 -> quantitative_asymptotic_stability

# ---------- DISCRETE SYSTEM ----------
x_(n+1)
 =
F(x_n)

fixed point:

F(x_star)
 =
x_star

LYAPUNOV_STABLE
 iff

for every epsilon>0
exists delta>0

such_that

||x_0-x_star||<delta

implies

||x_n-x_star||<epsilon

for all n>=0

# ---------- LYAPUNOV FUNCTION ----------
V(x)
 -> scalar_candidate_function

near x_star:

V(x_star)
 =
0

V(x)
 >
0
for
 x != x_star

V
 -> POSITIVE_DEFINITE

# ---------- DERIVATIVE ALONG TRAJECTORIES ----------
for autonomous system:

V_dot(x)
 :=
grad(V(x))
 dot
f(x)

V_dot
 -> rate_of_change_along(TRAJECTORY)

if

V_dot(x)
 <= 0

then

V
 -> NONINCREASING_ALONG_TRAJECTORIES

# ---------- DIRECT LYAPUNOV STABILITY THEOREM ----------
if

V
 -> POSITIVE_DEFINITE

and

V_dot
 -> NEGATIVE_SEMIDEFINITE

then

x_star
 -> LYAPUNOV_STABLE

under suitable local regularity hypotheses

DIRECT_METHOD
 -> proves_stability_without(SOLVING_DIFFERENTIAL_EQUATION)

# ---------- ASYMPTOTIC LYAPUNOV THEOREM ----------
if

V
 -> POSITIVE_DEFINITE

and

V_dot
 -> NEGATIVE_DEFINITE

then

x_star
 -> LOCALLY_ASYMPTOTICALLY_STABLE

under standard hypotheses

STRICT_DECREASE
 -> drives(TRAJECTORIES_TOWARD_EQUILIBRIUM)

# ---------- GLOBAL LYAPUNOV RESULT ----------
if

V
 -> POSITIVE_DEFINITE

V_dot
 -> NEGATIVE_DEFINITE

and

V(x)
 -> infinity
 as
 ||x||->infinity

then

x_star
 -> GLOBALLY_ASYMPTOTICALLY_STABLE

under appropriate global assumptions

RADIAL_UNBOUNDEDNESS
 -> supports(GLOBAL_CONFINEMENT)

# ---------- POSITIVE DEFINITE ----------
V
 -> POSITIVE_DEFINITE

if

V(x_star)=0

and

V(x)>0
 for x != x_star

in neighbourhood

POSITIVE_DEFINITE_FUNCTION
 -> energy_like_measure_of(DEVIATION)

# ---------- NEGATIVE DEFINITE ----------
V_dot
 -> NEGATIVE_DEFINITE

if

V_dot(x_star)=0

and

V_dot(x)<0
 for x != x_star

near equilibrium

NEGATIVE_DEFINITE_DERIVATIVE
 -> strict_decay

# ---------- SEMIDEFINITE CASE ----------
if

V_dot<=0

but

V_dot=0
 on nontrivial set

then

ASYMPTOTIC_STABILITY
 -/> follows_directly_from_strict_Lyapunov_theorem

LASALLE_INVARIANCE
 -> may_resolve(LONG_TERM_BEHAVIOUR)

# ---------- LASALLE INVARIANCE ----------
define

E
 :=
{x : V_dot(x)=0}

find
 M
 -> largest_invariant_subset_of(E)

then trajectories in suitable compact positively invariant region

-> approach(M)

LASALLE_INVARIANCE_PRINCIPLE
 -> extends(LYAPUNOV_DIRECT_METHOD)

if

M={x_star}

then

x_star
 -> ASYMPTOTICALLY_STABLE

# ---------- LINEAR SYSTEM ----------
dx/dt
 =
A*x

origin
 -> equilibrium

if A is Hurwitz:

Re(lambda_i(A))<0
 for all i

then

origin
 -> EXPONENTIALLY_STABLE

# ---------- QUADRATIC LYAPUNOV FUNCTION ----------
choose

V(x)
 :=
x_transpose
*
P
*
x

with

P
 -> symmetric_positive_definite

then

V_dot
 =
x_transpose
*
(
 A_transpose*P
 +
 P*A
)
*
x

# ---------- LYAPUNOV MATRIX EQUATION ----------
given

Q
 -> symmetric_positive_definite

solve

A_transpose*P
+
P*A
=
-Q

if
 A Hurwitz

then

unique
 P
 -> symmetric_positive_definite

LYAPUNOV_EQUATION
 -> algebraic_certificate_of(LINEAR_STABILITY)

# ---------- DISCRETE LYAPUNOV EQUATION ----------
for

x_(n+1)
 =
A*x_n

quadratic V:

V(x)
 =
x_transpose*P*x

difference:

Delta_V
 =
V(Ax)-V(x)

=
x_transpose(
 A_transpose*P*A-P
)x

if

A_transpose*P*A-P
 =
-Q

with
 P,Q positive definite

then

origin
 -> ASYMPTOTICALLY_STABLE

# ---------- NONAUTONOMOUS SYSTEM ----------
dx/dt
 =
f(t,x)

equilibrium
 x=0

Lyapunov function may depend on time:

V(t,x)

total derivative:

dV/dt
 =
partial_V/partial_t
+
grad_x(V)
 dot
f(t,x)

UNIFORM_STABILITY
 -> requires(time-uniform_bounds)

# ---------- UNIFORM STABILITY ----------
stability
 -> UNIFORM

if delta can be chosen
 independently_of(INITIAL_TIME)

UNIFORM_ASYMPTOTIC_STABILITY
 -> uniform_stability_plus_uniform_attraction

NONAUTONOMOUS_CONTEXT
 -> initial_time_dependence_must_be_preserved

# ---------- INVARIANT SET STABILITY ----------
A subset_of X

distance(x,A)
 :=
inf_{a in A} ||x-a||

A
 -> LYAPUNOV_STABLE_SET

if trajectories starting sufficiently close to A
 remain close to A

STABILITY
 -> extends_beyond(EQUILIBRIUM_POINTS)

# ---------- ORBITAL STABILITY ----------
periodic orbit
 Gamma

-> ORBITALLY_STABLE

if

distance(
 x(t),
 Gamma
)

remains small for nearby initial conditions

PHASE_SHIFT
 -> permitted

ORBITAL_STABILITY
 -> set_based_Lyapunov_stability

# ---------- BASIN OF ATTRACTION ----------
for asymptotically stable equilibrium:

B(x_star)
 :=
{
 x_0 :
 x(t;x_0)
 -> x_star
}

LYAPUNOV_SUBLEVEL_SET:

Omega_c
 :=
{x : V(x)<=c}

if
 Omega_c
 -> positively_invariant

then

Omega_c
 -> may_provide(INNER_ESTIMATE_OF_BASIN)

# ---------- ROBUSTNESS ----------
small perturbation:

dx/dt
 =
f(x)
+
d(t)

LYAPUNOV_ANALYSIS
 -> may_bound(PERTURBED_RESPONSE)

ROBUST_STABILITY
 -> extension_of(LYAPUNOV_FRAMEWORK)

INPUT_TO_STATE_STABILITY
 -> uses(LYAPUNOV_FUNCTIONS_WITH_INPUT_TERMS)

# ---------- CONTROL LYAPUNOV FUNCTION ----------
controlled system:

dx/dt
 =
f(x,u)

V
 -> CONTROL_LYAPUNOV_FUNCTION

if for every x != 0
there exists control u

such_that

V_dot(x,u)<0

CONTROL_LYAPUNOV_FUNCTION
 -> supports(STABILISING_CONTROL_DESIGN)

# ---------- APPLICATIONS ----------
LYAPUNOV_STABILITY
 -> applied_to(NONLINEAR_DYNAMICS)
 -> applied_to(CONTROL_THEORY)
 -> applied_to(ROBOTICS)
 -> applied_to(POWER_SYSTEMS)
 -> applied_to(MECHANICS)
 -> applied_to(BIOLOGICAL_SYSTEMS)
 -> applied_to(AUTONOMOUS_SYSTEMS)
 -> applied_to(OPTIMISATION_DYNAMICS)

# ---------- RELATION GRAPH ----------
PERTURBATION
 -> tests(LYAPUNOV_STABILITY)

LYAPUNOV_FUNCTION
 -> measures(DEVIATION_FROM_INVARIANT_STATE)

NEGATIVE_DERIVATIVE
 -> implies(ENERGY_LIKE_DECAY)

STABILITY
 + ATTRACTIVITY
 -> ASYMPTOTIC_STABILITY

RADIAL_UNBOUNDEDNESS
 -> supports(GLOBAL_STABILITY)

LASALLE_INVARIANCE
 -> analyses(SEMIDEFINITE_DERIVATIVE_CASE)

LYAPUNOV_EQUATION
 -> certifies(LINEAR_SYSTEM_STABILITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "Lyapunov stability" ~= "property that sufficiently nearby trajectories remain nearby for all future time",
 "asymptotic stability" ~= "Lyapunov stability combined with attraction to the equilibrium",
 "exponential stability" ~= "stability with an exponential decay bound on perturbations",
 "Lyapunov function" ~= "scalar function used to certify stability through positivity and decrease along trajectories",
 "LaSalle invariance principle" ~= "method for determining asymptotic behaviour when a Lyapunov derivative is only semidefinite",
 "Lyapunov equation" ~= "matrix equation used to construct quadratic stability certificates for linear systems"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 Lyapunov_stability_does_not_require_convergence_to_equilibrium: true,
 attractivity_alone_does_not_imply_Lyapunov_stability: true,
 negative_semidefinite_Lyapunov_derivative_does_not_automatically_imply_asymptotic_stability: true,
 global_results_require_global_hypotheses: true,
 nonautonomous_stability_requires_initial_time_context: true,
 candidate_Lyapunov_function_must_be_verified_not_assumed: true,
 numerical_decay_in_simulation_is_not_rigorous_stability_proof: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 epsilon_delta_relation: 0.99,
 Lyapunov_function_relation: 0.99,
 asymptotic_exponential_relation: 0.99,
 LaSalle_relation: 0.99,
 matrix_equation_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(dynamical_system_equilibrium_or_invariant_set)
 -> identify(continuous_discrete_autonomous_or_nonautonomous_context)
 -> define(distance_norm_and_perturbation)
 -> test(epsilon_delta_stability_definition_if_requested)
 -> construct_or_verify(candidate_Lyapunov_function)
 -> determine(positive_definiteness)
 -> compute(derivative_or_difference_along_dynamics)
 -> classify(semidefinite_or_definite_decay)
 -> apply(Lyapunov_or_LaSalle_theorem_as_valid)
 -> determine(local_global_asymptotic_or_exponential_scope)
 -> estimate(basin_or_robustness_if_relevant)
 -> validate(regularity_invariance_and_global_hypotheses)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed Lyapunov-stability semantic graph",
 primary_entity: LYAPUNOV_STABILITY,
 preserve: [
  dynamical_system,
  equilibrium_or_set,
  norm_or_distance,
  perturbation,
  time_type,
  Lyapunov_candidate,
  derivative_or_difference,
  local_or_global_scope
 ],
 infer_through: [
  epsilon_delta_stability,
  attractivity,
  asymptotic_stability,
  exponential_stability,
  Lyapunov_functions,
  LaSalle_invariance,
  basin_estimates,
  matrix_Lyapunov_equations
 ],
 avoid: [
  equating_stability_with convergence,
  assuming_candidate_function_is Lyapunov_without sign checks,
  inferring_asymptotic_stability_from_semidefinite_derivative_without invariance analysis,
  applying_local_certificate_globally,
  ignoring_initial_time_in nonautonomous system,
  treating_simulation_as formal stability proof
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(Lyapunov_stability_context)
 -> compare(existing_relations)
 -> preserve(system_object_distance_function_decay_and_scope_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + LYAPUNOV_STABILITY
 + EQUILIBRIUM
 + PERTURBATION
 + ASYMPTOTIC_STABILITY
 + EXPONENTIAL_STABILITY
 + LYAPUNOV_FUNCTION
 + LASALLE_INVARIANCE
 + BASIN_OF_ATTRACTION
 + GLOBAL_STABILITY
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> LYAPUNOV_STABILITY
 -> SYSTEM_OBJECT_IDENTIFICATION
 -> CONTEXT_CLASSIFICATION
 -> DISTANCE_PERTURBATION_ANALYSIS
 -> STABILITY_DEFINITION_ANALYSIS
 -> LYAPUNOV_FUNCTION_CONSTRUCTION
 -> SIGN_DERIVATIVE_ANALYSIS
 -> LYAPUNOV_LASALLE_APPLICATION
 -> STABILITY_SCOPE_CLASSIFICATION
 -> BASIN_ROBUSTNESS_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW