Differential algebraic equations and distributional solutions Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Oberseminseminar Analysis Technische Universit¨ at Dresden, 07.01.2010
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Content Introduction 1 System class Simple example Distributions as solutions 2 Review: classical distribution theory Restriction of distributions Piecewise smooth distributions Solution theory for switched DAEs 3 Impulse and jump freeness of solutions 4 Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Switched DAEs DAE = Differential algebraic equation Homogeneous switched linear DAE E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) ( swDAE ) or short E σ ˙ x = A σ x with Switching signal σ : R → { 1 , 2 , . . . , N } piecewise constant, right continuous locally finitely many jumps matrix pairs ( E 1 , A 1 ) , . . . , ( E N , A N ) E p , A p ∈ R n × n , p = 1 , . . . , N ( E p , A p ) regular, i.e. det( E p s − A p ) �≡ 0 or more general: E p , A p ∈ ( C ∞ ) n × n Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Motivation and questions Why switched DAEs E σ ˙ x = A σ x ? Modelling electrical circuits 1 DAEs E ˙ x = Ax + Bu with switched feedback 2 u ( t ) = F σ ( t ) x ( t ) or u ( t ) = F σ ( t ) x ( t ) + G σ ( t ) ˙ x ( t ) Approximation of time-varying DAEs E ( t )˙ x = A ( t ) x by 3 piecewise-constant DAEs Questions 1) Solution theory 2) Impulse free solutions 3) Stability Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Example: Electrical circuit with coil i i u L u u u L L L 1 0 0 0 0 0 1 0 0 0 0 0 A 2 = A 1 = 0 L 0 0 0 1 E 2 = E 1 = 0 L 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Solution of example L d u = 0, ˙ d t i = u L , 0 = u + u L or 0 = i L Assume: u (0) = u 0 , i (0) = 0 � 1 , t < t s switch at t s > 0: σ ( t ) = t ≥ t s 2 , u L ( t ) i ( t ) − u 0 t t t s t s δ t s Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Distribution theorie - basic ideas Distributions - overview Generalized functions Arbitrarily often differentiable Dirac-Impulse δ 0 is “derivative” of jump function ✶ [0 , ∞ ) Two different formal approaches Functional analytical: Dual space of the space of test functions 1 (L. Schwartz 1950) Axiomatic: Space of all “derivatives” of continuous functions 2 (J.S. Silva 1954) Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Distributions - formal Definition (Test functions) C ∞ := { ϕ : R → R | ϕ is smooth with compact support } 0 Definition (Distributions) D := { D : C ∞ → R | D is linear and continuous } 0 Definition (Regular distributions) f ∈ L 1 , loc ( R → R ): f D : C ∞ → R , ϕ �→ � R f ( t ) ϕ ( t )d t ∈ D 0 Definition (Derivative) Dirac Impulse at t 0 ∈ R δ t 0 : C ∞ D ′ ( ϕ ) := − D ( ϕ ′ ) → R , ϕ �→ ϕ ( t 0 ) 0 Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Multiplication with functionen Definition (Multiplication with smooth functions) α ∈ C ∞ : ( α D )( ϕ ) := D ( αϕ ) (swDAE) E σ ˙ x = A σ x Coefficients not smooth ∈ C ∞ Problem: E σ , A σ / Observation: E σ ˙ x = A σ x ⇔ ∀ i ∈ Z : ( E p i ˙ x ) [ t i , t i +1 ) = ( A p i x ) [ t i , t i +1 ) i ∈ Z : σ [ t i , t i +1 ) ≡ p i New question: Restriction of distributions Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Desired properties of distributional restriction Distributional restriction: { M ⊆ R | M interval } × D → D , ( M , D ) �→ D M and for each interval M ⊆ R D �→ D M is a projection (linear and idempotent) 1 ∀ f ∈ L 1 , loc : ( f D ) M = ( f M ) D 2 � � supp ϕ ⊆ M ⇒ D M ( ϕ ) = D ( ϕ ) ∀ ϕ ∈ C ∞ : 3 0 supp ϕ ∩ M = ∅ ⇒ D M ( ϕ ) = 0 ( M i ) i ∈ N pairwise disjoint, M = � i ∈ N M i : 4 � D M 1 ∪ M 2 = D M 1 + D M 2 , D M = D M i , ( D M 1 ) M 2 = 0 i ∈ N Theorem Such a distributional restriction does not exist. Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Proof of non-existence of restriction Consider the following distribution(!): d i := ( − 1) i � D := d i δ d i , i + 1 i ∈ N 1 1 2 4 1 1 0 3 Properties 2 and 3 give � D (0 , ∞ ) = d 2 k δ d 2 k k ∈ N Choose ϕ ∈ C ∞ such that ϕ [0 , 1] ≡ 1: 0 1 � � 2 k + 1 = ∞ D (0 , ∞ ) ( ϕ ) = d 2 k = k ∈ N k ∈ N Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Dilemma Distributions Switched DAEs Distributional restriction not Examples: distributional possible solutions Multiplication with non-smooth Multiplication with non-smooth coefficients not possible coefficients Initial value problems cannot be Or: Restriction on intervals formulated Underlying problem Space of distributions too big. Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Piecewise smooth distributions Define a suitable smaller space: Definition (Piecewise smooth distributions D pw C ∞ ) f ∈ C ∞ � pw , � � � T ⊆ R locally finite , D pw C ∞ := f D + D t � � ∀ t ∈ T : D t = � n t i δ ( i ) i =0 a t t ∈ T � t f D D t i − 1 D t i +1 D t i t i − 1 t i t i +1 Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Properties of D pw C ∞ C ∞ pw “ ⊆ ” D pw C ∞ D ∈ D pw C ∞ ⇒ D ′ ∈ D pw C ∞ Restriction D pw C ∞ → D pw C ∞ , D �→ D M for all intervals M ⊆ R well defined Multiplication with C ∞ pw -functions well defined Left and right sided evaluation at t ∈ R : D ( t − ) , D ( t +) Impulse at t ∈ R : D [ t ] (swDAE) E σ ˙ x = A σ x Application to (swDAE) x ∈ ( D pw C ∞ ) n and (swDAE) holds in D pw C ∞ x solves (swDAE) : ⇔ Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Relevant questions Consider E σ ˙ x = A σ x with regular matrix pairs E p , A p . Existence of solutions? Uniqueness of solutions? Inconsistent initial value problems? Jumps and impulses in solutions? Conditions for jump and impulse free solutions? Theorem (Existence and uniqueness) ∀ x 0 ∈ ( D pw C ∞ ) n ∀ t 0 ∈ R ∃ ! x ∈ ( D pw C ∞ ) n : x ( −∞ , t 0 ) = x 0 ( −∞ , t 0 ) ( E σ ˙ x ) [ t 0 , ∞ ) = ( A σ x ) [ t 0 , ∞ ) : ⇔ Remark: x is called consistent solution E σ ˙ x = A σ x on whole R . Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
Introduction Distributions as solutions Solution theory Impulse- and jump freeness Content Introduction 1 System class Simple example Distributions as solutions 2 Review: classical distribution theory Restriction of distributions Piecewise smooth distributions Solution theory for switched DAEs 3 Impulse and jump freeness of solutions 4 Stephan Trenn Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Differential algebraic equations and distributional solutions
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