Linear differential-algebraic equations with piecewise smooth - PowerPoint PPT Presentation

Linear differential-algebraic equations with piecewise smooth coefficients Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Perugia, 20 th June 2007

Content Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

A simple example t = 0 R + u − i c u c C Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

A simple example t = 1 R + u − i c u c C Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

A simple example t = 1 R + u − i c u c C Capacitor equation: C d d t u c ( t ) = i c ( t ), t ∈ R � u ( t ) − Ri c ( t ) , t ∈ [0 , 1) Kirchhoff’s law: u c ( t ) = 0 , otherwise Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Linear time-varying DAE Definition (Linear time-varying DAE) E ( · )˙ x = A ( · ) x + f Example: x 1 = u c , x 2 = i c � �  0 1  , t ∈ [0 , 1)   1 R    � C � 0  E ( t ) = , A ( t ) = 0 0 � �  0 1    , otherwise   1 0  � u ( t ) , t ∈ [0 , 1) f ( t ) = 0 , otherwise Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Solution of example u c ( t ) t = 0 t = 1 Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Solution of example u c ( t ) t = 0 t = 1 i c ( t ) t = 0 t = 1 Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Solution of example u c ( t ) t = 0 t = 1 i c ( t ) t = 0 t = 1 Conclusion Solution theory of DAEs needs distributional solutions. Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Content Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Basic properties of distributions Distributions - informal Generalized functions Arbitrarily often differentiable Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Basic properties of distributions Distributions - informal Generalized functions Arbitrarily often differentiable Definition (Test functions) Φ := { ϕ : R → R | ϕ is smooth with bounded support } Definition (Distributions) D := { D : Φ → R | D is linear und continuous } = Φ ′ Definition (Support of distribution) supp D := ( � { M ⊆ R | ∀ ϕ ∈ Φ : supp ϕ ⊆ M ⇒ D ( ϕ ) = 0 } ) C Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Basic properties of distributions Definition (Regular distributions) � f ∈ L 1 , loc ( R → R ): f D : Φ → R , ϕ �→ R ϕ ( t ) f ( t )d t Dirac impulse at t ∈ R δ t : Φ → R , ϕ �→ ϕ ( t ) Definition (Derivative of distributions) D ′ ( ϕ ) := − D ( ϕ ′ ) Definition (Multiplication with smooth function a : R → R ) ( aD )( ϕ ) := D ( a ϕ ) Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Distributional DAEs Definition (Distributional DAE) E ( · ) X ′ = A ( · ) X + f D , X ∈ D n Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Distributional DAEs Definition (Distributional DAE) E ( · ) X ′ = A ( · ) X + f D , X ∈ D n Problem Only well defined if E and A are constant or smooth! ⇒ Multiplication aD for non-smooth a : R → R must be studied. Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Content Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Multiplication with non-smooth functions Question Is it possible to define aD for non-smooth a and arbitrary D ∈ D ? Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Multiplication with non-smooth functions Question Is it possible to define aD for non-smooth a and arbitrary D ∈ D ? Answer: NO (already for piecewise constant functions a ) Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Multiplication with non-smooth functions Question Is it possible to define aD for non-smooth a and arbitrary D ∈ D ? Answer: NO (already for piecewise constant functions a ) Therefore, consider a subset of D : Definition (Piecewise W n distributions) D ∈ D pw W n : ⇔ D = f D + � i D i Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Multiplication with non-smooth functions Question Is it possible to define aD for non-smooth a and arbitrary D ∈ D ? Answer: NO (already for piecewise constant functions a ) Therefore, consider a subset of D : Definition (Piecewise W n distributions) D ∈ D pw W n : ⇔ D = f D + � i D i , where f ∈ W n pw ( R → R ) ⊆ L 1 , loc ( R → R ), i.e. piecewise n -times weakly differentiable Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Multiplication with non-smooth functions Question Is it possible to define aD for non-smooth a and arbitrary D ∈ D ? Answer: NO (already for piecewise constant functions a ) Therefore, consider a subset of D : Definition (Piecewise W n distributions) D ∈ D pw W n : ⇔ D = f D + � i D i , where f ∈ W n pw ( R → R ) ⊆ L 1 , loc ( R → R ), i.e. piecewise n -times weakly differentiable D i ∈ D , i ∈ Z , are distributions with point support { t i } Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Multiplication with non-smooth functions Question Is it possible to define aD for non-smooth a and arbitrary D ∈ D ? Answer: NO (already for piecewise constant functions a ) Therefore, consider a subset of D : Definition (Piecewise W n distributions) D ∈ D pw W n : ⇔ D = f D + � i D i , where f ∈ W n pw ( R → R ) ⊆ L 1 , loc ( R → R ), i.e. piecewise n -times weakly differentiable D i ∈ D , i ∈ Z , are distributions with point support { t i } the support of all D i has no accumulation points Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

✶ Properties of piecewise W n distributions Piecewise regular distributions W n pw ( R → R ) ⊆ W 0pw ( R → R ) = L 1 , loc ( R → R ) D pw := D pw W 0 - piecewise regular distributions Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

✶ Properties of piecewise W n distributions Piecewise regular distributions W n pw ( R → R ) ⊆ W 0pw ( R → R ) = L 1 , loc ( R → R ) D pw := D pw W 0 - piecewise regular distributions Lemma D ′ ∈ D pw W n . D ∈ D pw W n +1 ⇒ Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Properties of piecewise W n distributions Piecewise regular distributions W n pw ( R → R ) ⊆ W 0pw ( R → R ) = L 1 , loc ( R → R ) D pw := D pw W 0 - piecewise regular distributions Lemma D ′ ∈ D pw W n . D ∈ D pw W n +1 ⇒ Definition (Restriction of piecewise regular distributions) D = f D + � i D i ∈ D pw , M ⊆ R � D M := ( f M ) D + ✶ M ( t i ) D i i Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients

Multiplication with non-smooth functions Definition (Piecewise smooth functions) a ∈ C ∞ pw ( R → R ) : ⇔ a = � j ✶ I j a j , where a j ∈ C ∞ ( R → R ) and I j = [ t j , t j +1 ) for j ∈ Z . Note: Representation is not unique! Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Linear differential-algebraic equations with piecewise smooth coefficients