Distributional solution theory of linear DAEs Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau GAMM 2008, Bremen, 01.04.2008, 11:40 - 12:00
Motivation Piecewise smooth distributions Solution theory: First results Contents Motivation 1 Piecewise smooth distributions 2 Solution theory: First results 3 Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Motivation E ( · ) ˙ x = A ( · ) x + B ( · ) u E singular (1) y = C ( · ) x Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Motivation E ( · ) ˙ x = A ( · ) x + B ( · ) u E singular (1) y = C ( · ) x Equivalence: = ( SAT − SET ′ ) z + SBu SET ˙ z x = Tz (1) ⇐ ⇒ y = CTz for invertible matrices S , T Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Motivation E ( · ) ˙ x = A ( · ) x + B ( · ) u E singular (1) y = C ( · ) x Equivalence: = ( SAT − SET ′ ) z + SBu SET ˙ z x = Tz (1) ⇐ ⇒ y = CTz for invertible matrices S , T Assumption “Type” of transformation matrices S , T equal to “type” of coefficient matrices E , A , B , C . Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Assumptions z = ( SAT − SET ′ ) z + SBu SET ˙ y = CTz “Negative” assumptions Coefficients time-varying and not necessarily continuous Inhomogenity not necessarily continuous Initial values not necessarily consistent Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Assumptions z = ( SAT − SET ′ ) z + SBu SET ˙ y = CTz “Negative” assumptions Coefficients time-varying and not necessarily continuous Inhomogenity not necessarily continuous Initial values not necessarily consistent Goal Solution theory under this assumptions. Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Assumptions z = ( SAT − SET ′ ) z + SBu SET ˙ y = CTz “Negative” assumptions Coefficients time-varying and not necessarily continuous Inhomogenity not necessarily continuous Initial values not necessarily consistent Goal Solution theory under this assumptions. Consequences: Distributional solutions Distributional coefficients Multiplication of distributions! Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Contents Motivation 1 Piecewise smooth distributions 2 Solution theory: First results 3 Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Distributions revisited Definition Test functions: C ∞ := { ϕ ∈ C ∞ ( R → R ) | supp f is compact } 0 Distributions: D := { D : C ∞ → R | D is linear and continuous } 0 Distributions with given support M ⊆ R : D M := { D ∈ D | supp D ⊆ M } Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Distributions revisited Definition Test functions: C ∞ := { ϕ ∈ C ∞ ( R → R ) | supp f is compact } 0 Distributions: D := { D : C ∞ → R | D is linear and continuous } 0 Distributions with given support M ⊆ R : D M := { D ∈ D | supp D ⊆ M } Theorem (Distributions with point support) i =0 α i δ ( i ) ∃ α 0 , . . . , α n ∈ R : D = � n D ∈ D { t } , t ∈ R ⇒ t Dirac-impulse and its derivatives: δ ( i ) t ( ϕ ) = ( − 1) i ϕ ( i ) ( t ) Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Piecewise smooth distributions Definition (Piecewise smooth distributions D pw C ∞ ) D ∈ D pw C ∞ ⊂ D is a piecewise smooth distribution : ⇐ ⇒ ∃ f ∈ C ∞ � � ∃ feasible T ⊆ R ∃ D t ∈ D { t } | t ∈ T : pw � D = f D + D 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 Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Properties of piecewise smooth distributions Theorem (Properties of D pw C ∞ ) Let F = f D + � t ∈ T F t ∈ D pw C ∞ . Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Properties of piecewise smooth distributions Theorem (Properties of D pw C ∞ ) Let F = f D + � t ∈ T F t ∈ D pw C ∞ . Closed under differentiation and integration: F ′ ∈ D pw C ∞ and � t 0 F ∈ D pw C ∞ Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Properties of piecewise smooth distributions Theorem (Properties of D pw C ∞ ) Let F = f D + � t ∈ T F t ∈ D pw C ∞ . Closed under differentiation and integration: F ′ ∈ D pw C ∞ and � t 0 F ∈ D pw C ∞ Pointwise evaluation : t 0 ∈ R : F ( t 0 − ) , F ( t 0 +) ∈ R und F [ t 0 ] ∈ D { t 0 } Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Properties of piecewise smooth distributions Theorem (Properties of D pw C ∞ ) Let F = f D + � t ∈ T F t ∈ D pw C ∞ . Closed under differentiation and integration: F ′ ∈ D pw C ∞ and � t 0 F ∈ D pw C ∞ Pointwise evaluation : t 0 ∈ R : F ( t 0 − ) , F ( t 0 +) ∈ R und F [ t 0 ] ∈ D { t 0 } Restriction to intervals: M ⊆ R interval : F M ∈ D pw C ∞ ∩ D M Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Properties of piecewise smooth distributions Theorem (Properties of D pw C ∞ ) Let F = f D + � t ∈ T F t ∈ D pw C ∞ . Closed under differentiation and integration: F ′ ∈ D pw C ∞ and � t 0 F ∈ D pw C ∞ Pointwise evaluation : t 0 ∈ R : F ( t 0 − ) , F ( t 0 +) ∈ R und F [ t 0 ] ∈ D { t 0 } Restriction to intervals: M ⊆ R interval : F M ∈ D pw C ∞ ∩ D M Associative multiplication (Fuchssteiner multiplication): G ∈ D pw C ∞ : FG ∈ D pw C ∞ with ( FG ) ′ = F ′ G + FG ′ , ∀ f , g ∈ C ∞ ( fg ) D = f D g D pw , δ t F = F ( t − ) and F δ t = F ( t +) Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results The Fuchssteiner multiplication Recall: F ∈ D pw C ∞ ⇔ F = f D + F [ · ], where f ∈ C ∞ pw and F [ · ] = � t ∈ T F [ t ] i δ ( i ) F [ t ] = � n t i =0 α t t Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results The Fuchssteiner multiplication Recall: F ∈ D pw C ∞ ⇔ F = f D + F [ · ], where f ∈ C ∞ pw and F [ · ] = � t ∈ T F [ t ] i δ ( i ) F [ t ] = � n t i =0 α t t Definition (Multiplication by Dirac impulses) For F ∈ D pw C ∞ and t ∈ R let δ t F := F ( t − ) δ t and F δ t := F ( t +) δ t and for n ∈ N � ′ � ′ � � δ ( n +1) δ ( n ) − δ ( n ) F δ ( n +1) F δ ( n ) − F ′ δ ( n ) t F ′ , F := := t F t . t t t Hence for F , G ∈ D pw C ∞ : FG = ( fg ) D + f D F [ · ] + G [ · ] g D Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
Motivation Piecewise smooth distributions Solution theory: First results Distributional DAEs E ˙ x = A x + Bu y = Cx E , A , B , C matrices with D pw C ∞ -entries x , y , u vectors with D pw C ∞ -entries Stephan Trenn Institut f¨ ur Mathematik, Technische Universit¨ at Ilmenau Distributional solution theory of linear DAEs
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