The Flow of ODEs Fabian Immler & Christoph Traut ITP 2016 e l - PowerPoint PPT Presentation

The Flow of ODEs Fabian Immler & Christoph Traut ITP 2016 e l l e b a ∀ s I = α λ β →

Introduction Motivation ◮ Lorenz attractor, chaos 1 / 14

Introduction Motivation ◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof 1 / 14

Introduction Motivation ◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) 1 / 14

Introduction Motivation ◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions 1 / 14

Introduction Motivation ◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program 1 / 14

Introduction Motivation ◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program Contribution 1 / 14

Introduction Motivation ◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program Contribution ◮ formalization of flow : general theory for dependence on initial conditions 1 / 14

Introduction Motivation ◮ Lorenz attractor, chaos ◮ Tucker’s computer-aided proof ◮ goal: formal verification of program (and proof) ◮ ODE’s sensitive dependence on initial conditions ◮ numerical bounds from computer program Contribution ◮ formalization of flow : general theory for dependence on initial conditions ◮ use existing verified ODE-solver [Immler, TACAS 2015]: bounds on variational equation 1 / 14

Structure Flow Dependence on Initial Condition Numerics 2 / 14

Structure Flow Dependence on Initial Condition Numerics 3 / 14

The Flow of ODEs x ( t ) = f ( x ( t )) ˙ ◮ ordinary differential equation (ODE) x ∈ R n t ∈ R 3 / 14

The Flow of ODEs x ( t ) = f ( x ( t )) ˙ ◮ ordinary differential equation (ODE) ◮ [Immler, H¨ olzl @ ITP 2012]: initial value problems x ∈ R n t ∈ R 3 / 14

The Flow of ODEs x ( t ) = f ( x ( t )) ˙ ◮ ordinary differential equation (ODE) ◮ [Immler, H¨ olzl @ ITP 2012]: initial value problems x ∈ R n ◮ formalize flow ϕ ( x 0 , t ): solution w.r.t. initial condition t ∈ R 3 / 14

The Flow of ODEs x ( t ) = f ( x ( t )) ˙ ◮ ordinary differential equation (ODE) ◮ [Immler, H¨ olzl @ ITP 2012]: initial value problems x ∈ R n ◮ formalize flow ϕ ( x 0 , t ): solution w.r.t. initial condition ◮ formalize dependence on initial condition t ∈ R 3 / 14

The Flow of ODEs x ( t ) = f ( x ( t )) ˙ ◮ ordinary differential equation (ODE) ◮ [Immler, H¨ olzl @ ITP 2012]: initial value problems x ∈ R n ◮ formalize flow ϕ ( x 0 , t ): solution w.r.t. initial condition ◮ formalize dependence on initial condition ◮ qualitative: continuous t ∈ R 3 / 14

The Flow of ODEs x ( t ) = f ( x ( t )) ˙ ◮ ordinary differential equation (ODE) ◮ [Immler, H¨ olzl @ ITP 2012]: initial value problems x ∈ R n ◮ formalize flow ϕ ( x 0 , t ): solution w.r.t. initial condition ◮ formalize dependence on initial condition ◮ qualitative: continuous ◮ quantitative: differentiable t ∈ R 3 / 14

The Flow of ODEs x ( t ) = f ( x ( t )) ˙ ◮ ordinary differential equation (ODE) ◮ [Immler, H¨ olzl @ ITP 2012]: initial value problems x ∈ R n ◮ formalize flow ϕ ( x 0 , t ): solution w.r.t. initial condition ◮ formalize dependence on initial condition ◮ qualitative: continuous ◮ quantitative: differentiable t ∈ R 3 / 14

Formalization ◮ continuity and differentiability are “natural” properties (chapter 7): 4 / 14

Formalization ◮ continuity and differentiability are “natural” properties (chapter 7): ◮ continuous ϕ 4 / 14

Formalization ◮ continuity and differentiability are “natural” properties (chapter 7): ◮ continuous ϕ ◮ differentiable ϕ 4 / 14

Formalization ◮ continuity and differentiability are “natural” properties (chapter 7): ◮ continuous ϕ ◮ differentiable ϕ ◮ technicalities demand “a firm and extensive background in the principles of real analysis.” 4 / 14

Formalization ◮ continuity and differentiability are “natural” properties (chapter 7): ◮ continuous ϕ ◮ differentiable ϕ ◮ technicalities demand “a firm and extensive background in the principles of real analysis.” ◮ proofs in chapter 17 4 / 14

Formalization ◮ continuity and differentiability are “natural” properties (chapter 7): ◮ continuous ϕ ◮ differentiable ϕ ◮ technicalities demand “a firm and extensive background in the principles of real analysis.” ◮ proofs in chapter 17 ◮ interface to the rest of the theory that hides technical constructions 4 / 14

The Interface: ex-ivl and ϕ ◮ locally Lipschitz continuous f : R n → R n (on open set X ) x ( t ) = f ( x ( t )) ˙ ϕ ( x 0 , t ) x 0 0 t 5 / 14

The Interface: ex-ivl and ϕ ◮ locally Lipschitz continuous f : R n → R n (on open set X ) x ( t ) = f ( x ( t )) ˙ ◮ ϕ ( x 0 , t ) := “unique solution of IVP x ( t ) = f ( x ( t )) ∧ x (0) = x 0 ” ˙ ϕ ( x 0 , t ) x 0 0 t 5 / 14

The Interface: ex-ivl and ϕ ◮ locally Lipschitz continuous f : R n → R n (on open set X ) x ( t ) = f ( x ( t )) ˙ ◮ ϕ ( x 0 , t ) := “unique solution of IVP x ( t ) = f ( x ( t )) ∧ x (0) = x 0 ” ˙ ϕ ( x 0 , t ) ◮ maximal existence interval ex-ivl x 0 0 t 5 / 14

The Interface: ex-ivl and ϕ ◮ locally Lipschitz continuous f : R n → R n (on open set X ) x ( t ) = f ( x ( t )) ˙ ◮ ϕ ( x 0 , t ) := “unique solution of IVP x ( t ) = f ( x ( t )) ∧ x (0) = x 0 ” ˙ ◮ maximal existence interval ex-ivl ◮ t ∗ ∈ ex-ivl ( x 1 ) ◮ t ∗ �∈ ex-ivl ( x 2 ) x 2 x 1 t ∗ 5 / 14

The Interface: ex-ivl and ϕ ◮ locally Lipschitz continuous f : R n → R n (on open set X ) x ( t ) = f ( x ( t )) ˙ ◮ ϕ ( x 0 , t ) := “unique solution of IVP x ( t ) = f ( x ( t )) ∧ x (0) = x 0 ” ˙ ϕ ( x 0 , t ) ◮ maximal existence interval ex-ivl x 0 ◮ t ∗ ∈ ex-ivl ( x 1 ) ◮ t ∗ �∈ ex-ivl ( x 2 ) Theorem (flow solves IVP) 0 t For t ∈ ex-ivl ( x 0 ) : 5 / 14

The Interface: ex-ivl and ϕ ◮ locally Lipschitz continuous f : R n → R n (on open set X ) x ( t ) = f ( x ( t )) ˙ ◮ ϕ ( x 0 , t ) := “unique solution of IVP x ( t ) = f ( x ( t )) ∧ x (0) = x 0 ” ˙ ϕ ( x 0 , t ) ◮ maximal existence interval ex-ivl x 0 ◮ t ∗ ∈ ex-ivl ( x 1 ) ◮ t ∗ �∈ ex-ivl ( x 2 ) Theorem (flow solves IVP) 0 t For t ∈ ex-ivl ( x 0 ) : ◮ ˙ ϕ ( x 0 , t ) = f ( ϕ ( x 0 , t )) 5 / 14

The Interface: ex-ivl and ϕ ◮ locally Lipschitz continuous f : R n → R n (on open set X ) x ( t ) = f ( x ( t )) ˙ ◮ ϕ ( x 0 , t ) := “unique solution of IVP x ( t ) = f ( x ( t )) ∧ x (0) = x 0 ” ˙ ϕ ( x 0 , t ) ◮ maximal existence interval ex-ivl x 0 ◮ t ∗ ∈ ex-ivl ( x 1 ) ◮ t ∗ �∈ ex-ivl ( x 2 ) Theorem (flow solves IVP) 0 t For t ∈ ex-ivl ( x 0 ) : ◮ ˙ ϕ ( x 0 , t ) = f ( ϕ ( x 0 , t )) ◮ ϕ ( x 0 , 0) = x 0 5 / 14

Flow property x t t + s 0 Theorem (Flow property) ( t ∈ ex-ivl ( x ) ∧ s ∈ ex-ivl ( ϕ ( x , t ))) = ⇒ ϕ ( x , t + s ) = ϕ ( ϕ ( x , t ) , s ) 6 / 14

Structure Flow Dependence on Initial Condition Numerics 7 / 14

Structure Flow Dependence on Initial Condition Numerics 8 / 14

Technical Lemmas ◮ Gr¨ onwall lemma continuous-on [0; a ] g = ⇒ � t ∀ t . 0 ≤ g ( t ) ≤ C + K · g ( s ) d s = ⇒ 0 ∀ t ∈ [0; a ] . g ( t ) ≤ C · e K · t 8 / 14

Technical Lemmas ◮ Gr¨ onwall lemma ◮ exponential sensitivity O ( e t ) x 2 x 1 ⇒ | ϕ ( x 1 , t ) − ϕ ( x 2 , t ) | ∈ O (e t ) t ∈ ex-ivl ( x 1 ) ∩ ex-ivl ( x 2 ) = 8 / 14

Technical Lemmas ◮ Gr¨ onwall lemma ◮ exponential sensitivity ◮ same existence interval in neighborhood x 2 x 1 t ∗ 8 / 14

Technical Lemmas ◮ Gr¨ onwall lemma ◮ exponential sensitivity ◮ same existence interval in neighborhood x 1 t ∗ 8 / 14

Technical Lemmas ◮ Gr¨ onwall lemma ◮ exponential sensitivity ◮ same existence interval in neighborhood ◮ continuous ϕ at ( x 1 , t ∗ ) x 1 t ∗ ∀ ε > 0 . ∃ δ. ϕ ( U δ ( x 1 , t ∗ )) ⊆ U ε ( ϕ ( x 1 , t ∗ )) 8 / 14

Technical Lemmas ◮ Gr¨ onwall lemma ◮ exponential sensitivity ◮ same existence interval in neighborhood ◮ continuous ϕ at ( x 1 , t ∗ ) ◮ continuity w.r.t. right-hand side of ODE x ( t ) = f ( x ( t )); ˙ x ( t ) = g ( x ( t )) ˙ ⇒ | ϕ f ( x 1 , t ) − ϕ g ( x 1 , t ) | ∈ O (e t ) | f − g | < ε = 8 / 14

Differentiability x ( t ) = f ( x ( t )) with f ′ ( x ) derivative of f : R → R ODE ˙ Variational Equation ( R ) � u ( t ) = f ′ ( ϕ ( x 0 , t )) · u ( t ) ˙ u (0) = 1 9 / 14