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Reasoning about Stability Mahesh Viswanathan University of Illinois, Urbana-Champaign MVD, October 2014 Viswanathan Reasoning about Stability Reasoning about Stability Mahesh Viswanathan P. Prabhakar G.E. Dullerud N. Roohi University of

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Reasoning about Stability

Mahesh Viswanathan University of Illinois, Urbana-Champaign

MVD, October 2014

Viswanathan Reasoning about Stability

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Reasoning about Stability

Mahesh Viswanathan

  • P. Prabhakar

G.E. Dullerud

  • N. Roohi

University of Illinois, Urbana-Champaign

MVD, October 2014

Viswanathan Reasoning about Stability

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Cyber Physical Systems

Automotive National Power

Grid Robotics Medical Devices Cyber Physical Systems

Industrial Automation

Avionics

Computing devices controlling physical processes arise everywhere

Viswanathan Reasoning about Stability

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Hybrid Automata

1

  • -1
  • -2

2

Viswanathan Reasoning about Stability

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Hybrid Automata

1

  • -1
  • -2

2

Go Ahead Turn Right

Out of the Road!

−1 ≤ x ≤ 1

−2 ≤ x ≤ −1

x ≤ −2

Safe!

x = x x = x x = x

Viswanathan Reasoning about Stability

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Hybrid Automata

1

  • -1
  • -2

2 Guard

Go Ahead Turn Right

Out of the Road!

−1 ≤ x ≤ 1

−2 ≤ x ≤ −1

x ≤ −2

Safe!

x = x x = x x = x

Reset Flow Invariant

Viswanathan Reasoning about Stability

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Equilibria in a Pendulum

Viswanathan Reasoning about Stability

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Equilibria in a Pendulum

Viswanathan Reasoning about Stability

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Equilibria in a Pendulum

Viswanathan Reasoning about Stability

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Equilibria in a Pendulum

Viswanathan Reasoning about Stability

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Equilibria in a Pendulum

Viswanathan Reasoning about Stability

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Equilibria in a Pendulum

Viswanathan Reasoning about Stability

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Lyapunov Stability [Aleksandr Lyapunov 1892]

An equilibrium point is a state from which no executions leave

Viswanathan Reasoning about Stability

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Lyapunov Stability [Aleksandr Lyapunov 1892]

An equilibrium point is a state from which no executions leave Lyapunov Stability is defined as

Viswanathan Reasoning about Stability

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Lyapunov Stability [Aleksandr Lyapunov 1892]

An equilibrium point is a state from which no executions leave Lyapunov Stability is defined as ✏ ∀✏ > 0

Viswanathan Reasoning about Stability

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Lyapunov Stability [Aleksandr Lyapunov 1892]

An equilibrium point is a state from which no executions leave Lyapunov Stability is defined as

  • ✏

∀✏ > 0∃ > 0.

Viswanathan Reasoning about Stability

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Lyapunov Stability [Aleksandr Lyapunov 1892]

An equilibrium point is a state from which no executions leave Lyapunov Stability is defined as

  • ✏

∀✏ > 0∃ > 0. ∀. ((0) ∈ B(0)) → (∀t. (t) ∈ B✏(0))

Viswanathan Reasoning about Stability

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Asymptotic Stability

Lyapunov Stability + Convergence

Viswanathan Reasoning about Stability

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Asymptotic Stability

Lyapunov Stability + Convergence

  • ∃ > 0

Viswanathan Reasoning about Stability

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Asymptotic Stability

Lyapunov Stability + Convergence

  • ∃ > 0∀. ( ∈ B(0)) → Conv()

Viswanathan Reasoning about Stability

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Asymptotic Stability

Lyapunov Stability + Convergence

  • ∃ > 0∀. ( ∈ B(0)) → Conv()

Conv() ≡ ∀✏ > 0.∃T.∀t ≥ T. (t) ∈ B✏(0)

Viswanathan Reasoning about Stability

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Talk Outline

Why is the problem of checking stability different?

Viswanathan Reasoning about Stability

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Talk Outline

Why is the problem of checking stability different? How difficult is checking stability computationally?

Viswanathan Reasoning about Stability

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Talk Outline

Why is the problem of checking stability different? How difficult is checking stability computationally? What are proof principles to check stability?

Viswanathan Reasoning about Stability

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Part I Why is stability different?

Prabhakar-Dullerud-Viswanathan

Viswanathan Reasoning about Stability

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Process Preorders and Equivalences

System Design Simplified System

Viswanathan Reasoning about Stability

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Process Preorders and Equivalences

System Design Simplified System Easy to Reason

Viswanathan Reasoning about Stability

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Process Preorders and Equivalences

System Design Simplified System Property Preserved Easy to Reason

Viswanathan Reasoning about Stability

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Process Preorders and Equivalences

System Design Simplified System Property Preserved Easy to Reason

Process preorders and equivalences characterize the relationship between complex and simple systems that preserve properties of interest

Viswanathan Reasoning about Stability

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Bisimulation

For Labeled Transition Systems

Transition Systems T = (S, Σ, − →), where S is a set of states, Σ is a set of actions, and − →⊆ S × Σ × S is the transition relation.

Viswanathan Reasoning about Stability

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Bisimulation

For Labeled Transition Systems

Transition Systems T = (S, Σ, − →), where S is a set of states, Σ is a set of actions, and − →⊆ S × Σ × S is the transition relation. Bisimulation R ⊆ S1 × S2 is a bisimulation between T1 = (S1, Σ, − →1) and T2 = (S2, Σ, − →2) iff for every (p, q) ∈ R

Viswanathan Reasoning about Stability

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Bisimulation

For Labeled Transition Systems

Transition Systems T = (S, Σ, − →), where S is a set of states, Σ is a set of actions, and − →⊆ S × Σ × S is the transition relation. Bisimulation R ⊆ S1 × S2 is a bisimulation between T1 = (S1, Σ, − →1) and T2 = (S2, Σ, − →2) iff for every (p, q) ∈ R If p

a

− →1 p0 then there is q0 ∈ S2 s.t. q

a

− →2 q0 and (p0, q0) ∈ R, and

Viswanathan Reasoning about Stability

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Bisimulation

For Labeled Transition Systems

Transition Systems T = (S, Σ, − →), where S is a set of states, Σ is a set of actions, and − →⊆ S × Σ × S is the transition relation. Bisimulation R ⊆ S1 × S2 is a bisimulation between T1 = (S1, Σ, − →1) and T2 = (S2, Σ, − →2) iff for every (p, q) ∈ R If p

a

− →1 p0 then there is q0 ∈ S2 s.t. q

a

− →2 q0 and (p0, q0) ∈ R, and If q

a

− →2 q0 then there is p0 ∈ S1 s.t. p

a

− →1 p0 and (p0, q0) ∈ R.

Viswanathan Reasoning about Stability

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Bisimulation

For Hybrid Transition Systems

Hybrid Transition Systems T = (S, Σ, − →, ∆), where S is a set of states, Σ is a set of actions, − →⊆ S × Σ × S is the transition relation, and ∆ ⊆ { | : [0, T] → S} is a set of trajectories.

Viswanathan Reasoning about Stability

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Bisimulation

For Hybrid Transition Systems

Hybrid Transition Systems T = (S, Σ, − →, ∆), where S is a set of states, Σ is a set of actions, − →⊆ S × Σ × S is the transition relation, and ∆ ⊆ { | : [0, T] → S} is a set of trajectories. Bisimulation R ⊆ S1 × S2 is a bisimulation between T1 = (S1, Σ, − →1, ∆1) and T2 = (S2, Σ, − →2, ∆2) iff R is a bisimulation w.r.t. the discrete transitions

Viswanathan Reasoning about Stability

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Bisimulation

For Hybrid Transition Systems

Hybrid Transition Systems T = (S, Σ, − →, ∆), where S is a set of states, Σ is a set of actions, − →⊆ S × Σ × S is the transition relation, and ∆ ⊆ { | : [0, T] → S} is a set of trajectories. Bisimulation R ⊆ S1 × S2 is a bisimulation between T1 = (S1, Σ, − →1, ∆1) and T2 = (S2, Σ, − →2, ∆2) iff R is a bisimulation w.r.t. the discrete transitions and for every (p, q) ∈ R If ⌧1 ∈ ∆(p) then there is ⌧2 ∈ ∆2(q) s.t. ∀t, (⌧1(t), ⌧2(t)) ∈ R, and “Conversely”

Viswanathan Reasoning about Stability

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Features of Bisimulation

The cannonical notion of equivalence between systems Preserves all categories of properties like safety, liveness, branching time, etc. Basis for minimization and decidability results

Viswanathan Reasoning about Stability

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Is stability preserved by bisimulation?

Viswanathan Reasoning about Stability

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Is stability preserved by bisimulation? No!

Viswanathan Reasoning about Stability

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Stability not Bisimulation Invariant

1

Dynamics: (x0, t) = x

1 2t Viswanathan Reasoning about Stability

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Stability not Bisimulation Invariant

1

Dynamics: (x0, t) = x

1 2t

Bisimulation: R = [0, 1] × [0, 1] is a bisimulation

Viswanathan Reasoning about Stability

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Stability not Bisimulation Invariant

1

Dynamics: (x0, t) = x

1 2t

Bisimulation: R = [0, 1] × [0, 1] is a bisimulation 1 is Lyapunov/Asymptotically stable but 0 is not!

Viswanathan Reasoning about Stability

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Need continuity!

Viswanathan Reasoning about Stability

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Need continuity!

Definition A uniformly continuous bisimulation between T1 and T2 is a binary relation R such that R is a bisimulation between T1 and T2 and R is a uniformly continuous relation

Viswanathan Reasoning about Stability

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Need continuity!

Definition A uniformly continuous bisimulation between T1 and T2 is a binary relation R such that R is a bisimulation between T1 and T2 and R is a uniformly continuous relation, i.e., ∀✏ > 0∃ > 0∀x ∈ dom(R). R(B(x)) ⊆ B✏(R(x))

Viswanathan Reasoning about Stability

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Invariance under Uniformly Continuous Bisimulations

Theorem Let T1 and T2 be hybrid transition systems with 0 as an equilibrium

  • point. Suppose R is a uniformly continuous bisimulation such that

(0, 0) ∈ R then T1 is Lyapunov stable w.r.t. 0 iff T2 is Lyapunov stable w.r.t. 0

Viswanathan Reasoning about Stability

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Invariance under Uniformly Continuous Bisimulations

Theorem Let T1 and T2 be hybrid transition systems with 0 as an equilibrium

  • point. Suppose R is a uniformly continuous bisimulation such that

(0, 0) ∈ R then T1 is Lyapunov (Asymptotically) stable w.r.t. 0 iff T2 is Lyapunov (Asymptotically) stable w.r.t. 0

Viswanathan Reasoning about Stability

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Invariance under Uniformly Continuous Bisimulations

Theorem Let T1 and T2 be hybrid transition systems with 0 as an equilibrium

  • point. Suppose R is a uniformly continuous bisimulation such that

(0, 0) ∈ R then T1 is Lyapunov (Asymptotically) stable w.r.t. 0 iff T2 is Lyapunov (Asymptotically) stable w.r.t. 0 Above observation generalizes to stronger notions of stability

Viswanathan Reasoning about Stability

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Invariance under Uniformly Continuous Bisimulations

Theorem Let T1 and T2 be hybrid transition systems with 0 as an equilibrium

  • point. Suppose R is a uniformly continuous bisimulation such that

(0, 0) ∈ R then T1 is Lyapunov (Asymptotically) stable w.r.t. 0 iff T2 is Lyapunov (Asymptotically) stable w.r.t. 0 Above observation generalizes to stronger notions of stability Uniformly continuous simulations reflect stability notions

Viswanathan Reasoning about Stability

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Lyapunov’s Second Method

System ˙ x = F(x) with solution '(x, t) Equilibrium F(0) = 0

Viswanathan Reasoning about Stability

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Lyapunov’s Second Method

System ˙ x = F(x) with solution '(x, t) Equilibrium F(0) = 0 If there exists a “Lyapunov function” for the system then it is Lyapunov stable.

Viswanathan Reasoning about Stability

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Lyapunov Function

An Illustration

Viswanathan Reasoning about Stability

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Lyapunov Function

An Illustration

Viswanathan Reasoning about Stability

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Lyapunov Function

An Illustration

Viswanathan Reasoning about Stability

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Lyapunov Function

An Illustration

Viswanathan Reasoning about Stability

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Lyapunov’s Method as an Abstraction

˙ x = F(x) ϕ

Viswanathan Reasoning about Stability

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Lyapunov’s Method as an Abstraction

˙ x = F(x) ϕ

Exists V : Rn → R+ s.t. V is positive definite C 1 ˙ V ≤ 0

Viswanathan Reasoning about Stability

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Lyapunov’s Method as an Abstraction

˙ x = F(x) ϕ V (ϕ)

Exists V : Rn → R+ s.t. V is positive definite C 1 ˙ V ≤ 0 v1

t

− → v2 iff exist x1, x2 s.t. x1

t

− → x2, V (x1) = v1 and V (x2) = v2.

Viswanathan Reasoning about Stability

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Lyapunov’s Method as an Abstraction

˙ x = F(x) ϕ V (ϕ) Easily Stable

Exists V : Rn → R+ s.t. V is positive definite C 1 ˙ V ≤ 0 v1

t

− → v2 iff exist x1, x2 s.t. x1

t

− → x2, V (x1) = v1 and V (x2) = v2.

Viswanathan Reasoning about Stability

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Lyapunov’s Method as an Abstraction

˙ x = F(x) ϕ V (ϕ)

  • Unif. Cont.

Simulation Easily Stable

Exists V : Rn → R+ s.t. V is positive definite C 1 ˙ V ≤ 0 v1

t

− → v2 iff exist x1, x2 s.t. x1

t

− → x2, V (x1) = v1 and V (x2) = v2.

Viswanathan Reasoning about Stability

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Other Characterizations

Other extensions of Lyapunov’s method to switched systems can also be understood in the abstraction setting Hartman-Grobman Theorem contructs a uniformly continuous bisimilar linearization

Viswanathan Reasoning about Stability

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Expressing Stability

Stabiity cannot be expressed in the classical modal/temporal logics like Hennessy-Milner, LTL, CTL, µ-calculus, etc.

Logic equivalence for these logics coincides with bisimulation

Viswanathan Reasoning about Stability

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Expressing Stability

Stabiity cannot be expressed in the classical modal/temporal logics like Hennessy-Milner, LTL, CTL, µ-calculus, etc.

Logic equivalence for these logics coincides with bisimulation

First order logic over appropriate topologial structures is too strong

Viswanathan Reasoning about Stability

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Expressing Stability

Stabiity cannot be expressed in the classical modal/temporal logics like Hennessy-Milner, LTL, CTL, µ-calculus, etc.

Logic equivalence for these logics coincides with bisimulation

First order logic over appropriate topologial structures is too strong Is there a (modal) logic that can express stability for which logic equivalence coincides with “continuous” bisimulations?

Viswanathan Reasoning about Stability

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S4: A Modal Logic for Space

Orlov[1928], Lewis[1932], G¨

  • del[1933], Stone[1937], Tarski[1937]

' ::= p | ¬' | '1 ∧ '2 | '1 ∨ '2 | I' | C'

Viswanathan Reasoning about Stability

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S4: A Modal Logic for Space

Orlov[1928], Lewis[1932], G¨

  • del[1933], Stone[1937], Tarski[1937]

' ::= p | ¬' | '1 ∧ '2 | '1 ∨ '2 | I' | C' Formulas interpreted as sets of points in a topological space

Viswanathan Reasoning about Stability

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S4: A Modal Logic for Space

Orlov[1928], Lewis[1932], G¨

  • del[1933], Stone[1937], Tarski[1937]

' ::= p | ¬' | '1 ∧ '2 | '1 ∨ '2 | I' | C' Formulas interpreted as sets of points in a topological space [ [I'] ]: Interior of set defined by '

Viswanathan Reasoning about Stability

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S4: A Modal Logic for Space

Orlov[1928], Lewis[1932], G¨

  • del[1933], Stone[1937], Tarski[1937]

' ::= p | ¬' | '1 ∧ '2 | '1 ∨ '2 | I' | C' Formulas interpreted as sets of points in a topological space [ [I'] ]: Interior of set defined by ' [ [C'] ]: Closure of set defined by '

Viswanathan Reasoning about Stability

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Bimodal Spatio-Temporal Logic

Add to S4 the usual temporal modalities of ⇤ and ⌃

Viswanathan Reasoning about Stability

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Bimodal Spatio-Temporal Logic

Add to S4 the usual temporal modalities of ⇤ and ⌃ Can stability be expressed in the resulting logic?

Viswanathan Reasoning about Stability

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Bimodal Spatio-Temporal Logic

Add to S4 the usual temporal modalities of ⇤ and ⌃ Can stability be expressed in the resulting logic? No!

Viswanathan Reasoning about Stability

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Bimodal Spatio-Temporal Logic

Add to S4 the usual temporal modalities of ⇤ and ⌃ Can stability be expressed in the resulting logic? No!

[Aiello-van Bentham, Davoren] Logic equivalence in this bi-modal logic coincides with bisimilarity under relations with “weak continuity” properties

Viswanathan Reasoning about Stability

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Bimodal Spatio-Temporal Logic

Add to S4 the usual temporal modalities of ⇤ and ⌃ Can stability be expressed in the resulting logic? No!

[Aiello-van Bentham, Davoren] Logic equivalence in this bi-modal logic coincides with bisimilarity under relations with “weak continuity” properties

Open Question: What is the right logic?

Viswanathan Reasoning about Stability

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Part II How difficult is checking stability computationally?

Prabhakar-Viswanathan

Viswanathan Reasoning about Stability

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What is known about checking Stability?

Viswanathan Reasoning about Stability

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What is known about checking Stability?

Very little!

Viswanathan Reasoning about Stability

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What is known about checking Stability?

Very little! Traditional control theoretic methods focus on identifying sufficient conditions that guarantee stability

Viswanathan Reasoning about Stability

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What is known about checking Stability?

Very little! Traditional control theoretic methods focus on identifying sufficient conditions that guarantee stability

Some complexity results on how difficult it is to find these sufficient conditions

Viswanathan Reasoning about Stability

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What is known about checking Stability?

Very little! Traditional control theoretic methods focus on identifying sufficient conditions that guarantee stability

Some complexity results on how difficult it is to find these sufficient conditions

[Blondel-Tsitsiklis et. al.] Prove computational lower bounds (undecidability/NP-hardness) on checking stability of special discrete time linear switched systems

Viswanathan Reasoning about Stability

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Deciding Hybrid Models

Traditional techniques to establish decidablity for various properties (like safety, liveness, etc.) of special hybrid models fail for stability Establishing stability needs new ideas

Viswanathan Reasoning about Stability

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Deciding Hybrid Models

Traditional techniques to establish decidablity for various properties (like safety, liveness, etc.) of special hybrid models fail for stability Decidability results rely on establishing the existence of an effectively constructable finite “bisimulation” quotient Establishing stability needs new ideas

Viswanathan Reasoning about Stability

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Piecewise Constant Derivative (PCD) Systems

Viswanathan Reasoning about Stability

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Piecewise Constant Derivative (PCD) Systems

Viswanathan Reasoning about Stability

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Piecewise Constant Derivative (PCD) Systems

Viswanathan Reasoning about Stability

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Piecewise Constant Derivative (PCD) Systems

Viswanathan Reasoning about Stability

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Piecewise Constant Derivative (PCD) Systems

Viswanathan Reasoning about Stability

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Piecewise Constant Derivative (PCD) Systems

Viswanathan Reasoning about Stability

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Piecewise Constant Derivative (PCD) Systems

Viswanathan Reasoning about Stability

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Stability in PCD

∀✏ > 0 ∃ > 0 ∀. ((0) ∈ B(0)) → (∀t. (t) ∈ B✏(0))

Viswanathan Reasoning about Stability

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SLIDE 90

Stability in PCD

∃ > 0. ∀✏ ∈ (0, ] ∀✏ > 0 ∃ > 0 ∀. ((0) ∈ B(0)) → (∀t. (t) ∈ B✏(0))

Viswanathan Reasoning about Stability

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Special Structure Near the Origin

The planar partition looks like “wedges”.

Viswanathan Reasoning about Stability

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Executions near 0

Viswanathan Reasoning about Stability

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Executions near 0

Viswanathan Reasoning about Stability

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Capturing distance from origin

Viswanathan Reasoning about Stability

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Capturing distance from origin

Viswanathan Reasoning about Stability

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Capturing distance from origin

Viswanathan Reasoning about Stability

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Capturing distance from origin

w(e) = |d1| |d2|

Viswanathan Reasoning about Stability

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SLIDE 98

Stability Analysis

Weighted Graph

Viswanathan Reasoning about Stability

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SLIDE 99

Stability Analysis

Weighted Graph

= (p1, p2)(p2, p3) · · · (p4, p5)

Viswanathan Reasoning about Stability

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SLIDE 100

Stability Analysis

Weighted Graph

= (p1, p2)(p2, p3) · · · (p4, p5) w() = |d((T))|

|d((0))|

Viswanathan Reasoning about Stability

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SLIDE 101

Stability Analysis

Weighted Graph

= (p1, p2)(p2, p3) · · · (p4, p5) w() = |d((T))|

|d((0))| = w(p1, p2)w(p2, p3) · · · w(p4, p5)

Viswanathan Reasoning about Stability

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Example 1

Viswanathan Reasoning about Stability

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Example 1

Viswanathan Reasoning about Stability

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Example 1

Lyapunov stable but not asymptotically stable.

Viswanathan Reasoning about Stability

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SLIDE 105

Example 2

Viswanathan Reasoning about Stability

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SLIDE 106

Example 2

Viswanathan Reasoning about Stability

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Example 2

Lyapunov and asymptotically stable.

Viswanathan Reasoning about Stability

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SLIDE 108

Example 3

Viswanathan Reasoning about Stability

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SLIDE 109

Example 3

Viswanathan Reasoning about Stability

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Example 3

Neither Lyapunov nor asymptotically stable.

Viswanathan Reasoning about Stability

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SLIDE 111

Stability of PCD

Theorem A piecewise constant derivative system is Lyapunov stable iff the weighted graph does not have any cycles of weight > 1 .

Viswanathan Reasoning about Stability

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SLIDE 112

Stability of PCD

Theorem A piecewise constant derivative system is Lyapunov (asymptotically) stable iff the weighted graph does not have any cycles of weight > 1 (≥ 1).

Viswanathan Reasoning about Stability

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SLIDE 113

Decidability Results

Observations can be extend to show that the stability problem is decidable for planar rectangular switched systems with polyhedral invariants and guards

Viswanathan Reasoning about Stability

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Decidability Results

Observations can be extend to show that the stability problem is decidable for planar rectangular switched systems with polyhedral invariants and guards Stability problem is undecidable for PCD in 5 dimensions.

Viswanathan Reasoning about Stability

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Stability, Safety, and Abstraction

Is stability computationally harder than checking reachability?

Viswanathan Reasoning about Stability

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Stability, Safety, and Abstraction

Is stability computationally harder than checking reachability? Undecidability proof obtained by reducing reachability to stability

Viswanathan Reasoning about Stability

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Stability, Safety, and Abstraction

Is stability computationally harder than checking reachability? Undecidability proof obtained by reducing reachability to stability However, reachability is not known to be decidable for planar rectangular switched systems with polyhedral guards and resets

Viswanathan Reasoning about Stability

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Stability, Safety, and Abstraction

Is stability computationally harder than checking reachability? Undecidability proof obtained by reducing reachability to stability However, reachability is not known to be decidable for planar rectangular switched systems with polyhedral guards and resets; stability algorithm exploits special structure near origin

Viswanathan Reasoning about Stability

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Stability, Safety, and Abstraction

Is stability computationally harder than checking reachability? Undecidability proof obtained by reducing reachability to stability However, reachability is not known to be decidable for planar rectangular switched systems with polyhedral guards and resets; stability algorithm exploits special structure near origin Stability algorithm can be thought of as constructing a “quantitative predicate abstraction”.

Viswanathan Reasoning about Stability

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Stability, Safety, and Abstraction

Is stability computationally harder than checking reachability? Undecidability proof obtained by reducing reachability to stability However, reachability is not known to be decidable for planar rectangular switched systems with polyhedral guards and resets; stability algorithm exploits special structure near origin Stability algorithm can be thought of as constructing a “quantitative predicate abstraction”.Can be exploited to do abstraction-based checking of stabiity of more general hybrid systems.

Viswanathan Reasoning about Stability

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Proof Rules for reasoning about Stability

[Roohi-Dullerud-Viswanathan]

Traditional proof principles for stability are too conservative to reason about systems in the presence of “fair” controllers

Viswanathan Reasoning about Stability

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SLIDE 122

Proof Rules for reasoning about Stability

[Roohi-Dullerud-Viswanathan]

Traditional proof principles for stability are too conservative to reason about systems in the presence of “fair” controllers Compositional reasoning principles

Viswanathan Reasoning about Stability

slide-123
SLIDE 123

Conclusions

Stability requires new analysis techniques and a number of questions remain open

Viswanathan Reasoning about Stability

slide-124
SLIDE 124

Conclusions

Stability requires new analysis techniques and a number of questions remain open Concrete abstraction based methods to verify stability Logic to specify stability Computational complexity of checking stability Relationship between Safety and Stability Compositional reasoning principles

Viswanathan Reasoning about Stability