Fractional Delay Equations in the Young Sense Jorge A. Len - PowerPoint PPT Presentation

Fractional Delay Equations in the Young Sense Jorge A. León Departamento de Control Automático Cinvestav del IPN Spring School “Stochastic Control in Finance”, Roscoff 2010 Jointly with Samy Tindel Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 1 / 64

Contents Introduction 1 Preliminaries 2 Young Integral 3 Delay Equations in the Young sense 4 Young and Fractional Integrals 5 Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 2 / 64

Contents Introduction 1 Preliminaries 2 Young Integral 3 Delay Equations in the Young sense 4 Young and Fractional Integrals 5 Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 3 / 64

Equation We consider the equation � t 0 f ( Z y y t = ξ 0 + s ) dx s , t ∈ [ 0 , T ] , Z 0 = ξ. Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 4 / 64

Equation We consider the equation � t 0 f ( Z y y t = ξ 0 + s ) dx s , t ∈ [ 0 , T ] , Z y = ξ. 0 Here x ∈ C ν ([ 0 , T ]) , f : C ν ([ − h , 0 ]) → R , ξ ∈ C ν ([ − h , 0 ]) . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 5 / 64

Equation We consider the equation � t 0 f ( Z y y t = ξ 0 + s ) dx s , t ∈ [ 0 , T ] , Z y = ξ. 0 Here x ∈ C ν ([ 0 , T ]) , f : C ν ([ − h , 0 ]) → R and ξ ∈ C ν ([ − h , 0 ]) , with Z y ν > 1 / 2 and s ( θ ) = y ( s + θ ) , θ ∈ [ − h , 0 ] . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 6 / 64

Equation We consider the equation � t 0 f ( Z y y t = ξ 0 + s ) dx s , t ∈ [ 0 , T ] , Z y = ξ. 0 Here x ∈ C ν ([ 0 , T ]) , f : C ν ([ − h , 0 ]) → R and ξ ∈ C ν ([ − h , 0 ]) , with Z y ν > 1 / 2 s ( θ ) = y ( s + θ ) , θ ∈ [ − h , 0 ] . and The integral is a Young one Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 7 / 64

Contents Introduction 1 Preliminaries 2 Young Integral 3 Delay Equations in the Young sense 4 Young and Fractional Integrals 5 Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 8 / 64

Increments We consider g : [ 0 , T ] k → R : � � g t 1 ,..., t k = 0 if t i = t i + 1 C k ( R ) = for some i ∈ { 1 , . . . , k − 1 } Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 9 / 64

Increments We consider g : [ 0 , T ] k → R : � � g t 1 ,..., t k = 0 if t i = t i + 1 C k ( R ) = for some i ∈ { 1 , . . . , k − 1 } and δ : C k ( R ) → C k + 1 ( R ) Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 10 / 64

Increments We consider g : [ 0 , T ] k → R : � � g t 1 ,..., t k = 0 if t i = t i + 1 C k ( R ) = for some i ∈ { 1 , . . . , k − 1 } and k + 1 � ( − 1 ) k − i g t i ,..., ˆ ( δ g ) t 1 ,..., t k + 1 = t i ,..., t k + 1 . i = 1 Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 11 / 64

Properties of δ We consider g : [ 0 , T ] k → R : � � g t 1 ,..., t k = 0 if t i = t i + 1 C k ( R ) = for some i ∈ { 1 , . . . , k − 1 } and k + 1 � ( − 1 ) k − i g t i ,..., ˆ ( δ g ) t 1 ,..., t k + 1 = t i ,..., t k + 1 . i = 1 δδ = 0 . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 12 / 64

Properties of δ We consider g : [ 0 , T ] k → R : � � g t 1 ,..., t k = 0 if t i = t i + 1 C k ( R ) = for some i ∈ { 1 , . . . , k − 1 } and k + 1 � ( − 1 ) k − i g t i ,..., ˆ ( δ g ) t 1 ,..., t k + 1 = t i ,..., t k + 1 . i = 1 δδ = 0 . Let Z C k ( R ) = C k ( R ) ∩ ker δ and B C k ( R ) = C k ( R ) ∩ Im δ . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 13 / 64

Properties of δ We consider g : [ 0 , T ] k → R : � � g t 1 ,..., t k = 0 if t i = t i + 1 C k ( R ) = for some i ∈ { 1 , . . . , k − 1 } and k + 1 � ( − 1 ) k − i g t i ,..., ˆ ( δ g ) t 1 ,..., t k + 1 = t i ,..., t k + 1 . i = 1 δδ = 0 . Let Z C k ( R ) = C k ( R ) ∩ ker δ and B C k ( R ) = C k ( R ) ∩ Im δ . Then, Z C k ( R ) = B C k ( R ) . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 14 / 64

Properties of δ We consider g : [ 0 , T ] k → R : � � g t 1 ,..., t k = 0 if t i = t i + 1 C k ( R ) = for some i ∈ { 1 , . . . , k − 1 } and k + 1 ( − 1 ) k − i g t i ,..., ˆ � ( δ g ) t 1 ,..., t k + 1 = t i ,..., t k + 1 . i = 1 δδ = 0 . Let Z C k ( R ) = C k ( R ) ∩ ker δ and B C k ( R ) = C k ( R ) ∩ Im δ . Then, Z C k ( R ) = B C k ( R ) . Let k ≥ 1 and h ∈ Z C k + 1 ( R ) . Then there exists a (nonunique) f ∈ C k ( R ) such that h = δ f . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 15 / 64

Properties of δ We consider k + 1 � ( − 1 ) k − i g t i ,..., ˆ ( δ g ) t 1 ,..., t k + 1 = t i ,..., t k + 1 . i = 1 δδ = 0 . Let Z C k ( R ) = C k ( R ) ∩ ker δ and B C k ( R ) = C k ( R ) ∩ Im δ . Then, Z C k ( R ) = B C k ( R ) . Let k ≥ 1 and h ∈ Z C k + 1 ( R ) . Then there exists a (nonunique) f ∈ C k ( R ) such that h = δ f . For g ∈ C 1 ( R ) and h ∈ C 2 ( R ) , ( δ g ) st = g t − g s and ( δ h ) sut = h st − h su − h ut . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 16 / 64

Notation For v ∈ R , C µ v , a 1 , a 2 ( R ) = { g : [ a 1 , a 2 ] → R : g a 1 = v , || g || µ, [ a 1 , a 2 ] < ∞} . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 17 / 64

Notation For v ∈ R , C µ v , a 1 , a 2 ( R ) = { g : [ a 1 , a 2 ] → R : g a 1 = v , || g || µ, [ a 1 , a 2 ] < ∞} , and, for ρ ∈ C µ 1 ([ a 1 − h , a 1 ]) , ρ, a 1 , a 2 ( R ) = { ξ ∈ C µ C µ 1 ([ a 1 − h , a 2 ]) : ξ = ρ on [ a 1 − h , a 1 ] } . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 18 / 64

Notation For v ∈ R , C µ v , a 1 , a 2 ( R ) = { g : [ a 1 , a 2 ] → R : g a 1 = v , || g || µ, [ a 1 , a 2 ] < ∞} , and, for ρ ∈ C µ 1 ([ a 1 − h , a 1 ]) , C µ ρ, a 1 , a 2 ( R ) = { ξ ∈ C µ 1 ([ a 1 − h , a 2 ]) : ξ = ρ on [ a 1 − h , a 1 ] } . These are complete metric spaces with respec to d µ ( f , g ) = || f − g || µ . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 19 / 64

Notation For f ∈ C 2 ([ a 1 , a 2 ]; R ) , we define | f r , t | || f || µ, [ a 1 , a 2 ] = sup | t − r | µ . r , t ∈ [ a 1 , a 2 ] Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 20 / 64

Notation For f ∈ C 2 ([ a 1 , a 2 ]; R ) , we define | f r , t | || f || µ, [ a 1 , a 2 ] = sup | t − r | µ . r , t ∈ [ a 1 , a 2 ] and C µ � � 2 ([ a 1 , a 2 ]) = f ∈ C 2 ( R ) : || f || µ, [ a 1 , a 2 ] < ∞ Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 21 / 64

Notation For f ∈ C 2 ([ a 1 , a 2 ]; R ) , we define | f r , t | || f || µ, [ a 1 , a 2 ] = sup | t − r | µ . r , t ∈ [ a 1 , a 2 ] and C µ � � 2 ([ a 1 , a 2 ]) = f ∈ C 2 ( R ) : || f || µ, [ a 1 , a 2 ] < ∞ Similarly, for h ∈ C 3 ([ a 1 , a 2 ]) , we define | h sut | || h || ν,ρ, [ a 1 , a 2 ] = sup | u − s | ν | t − u | ρ . s , u , t ∈ [ a 1 , a 2 ] Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 22 / 64

Notation For h ∈ C 3 ([ a 1 , a 2 ]) , we define | h sut | || h || ν,ρ, [ a 1 , a 2 ] = sup | u − s | ν | t − u | ρ , s , u , t ∈ [ a 1 , a 2 ] the norm � � || h || µ, [ a 1 , a 2 ] = inf { || h i || ρ i ,µ − ρ 1 ; h = h i , 0 < ρ i < µ } . i i Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 23 / 64

Notation For h ∈ C 3 ([ a 1 , a 2 ]) , we define | h sut | || h || ν,ρ, [ a 1 , a 2 ] = sup | u − s | ν | t − u | ρ , s , u , t ∈ [ a 1 , a 2 ] the norm � � || h || µ, [ a 1 , a 2 ] = inf { || h i || ρ i ,µ − ρ 1 ; h = h i , 0 < ρ i < µ } . i i and C µ 3 ([ a 1 , a 2 ]) = { h ∈ C 3 ([ a 1 , a 2 ]) : || h || µ, [ a 1 , a 2 ] < ∞} . Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 24 / 64

Notation For h ∈ C 3 ([ a 1 , a 2 ]) , we define | h sut | || h || ν,ρ, [ a 1 , a 2 ] = sup | u − s | ν | t − u | ρ , s , u , t ∈ [ a 1 , a 2 ] the norm � � || h || µ, [ a 1 , a 2 ] = inf { || h i || ρ i ,µ − ρ 1 ; h = h i , 0 < ρ i < µ } . i i and C µ 3 ([ a 1 , a 2 ]) = { h ∈ C 3 ([ a 1 , a 2 ]) : || h || µ, [ a 1 , a 2 ] < ∞} . We use the notation C µ C 1 + � = k ([ a 1 , a 2 ]) , k = 2 , 3 . k µ> 1 Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 25 / 64

Inverse of δ We use the notation C 1 + C µ � = k ([ a 1 , a 2 ]) , k = 2 , 3 . k µ> 1 Proposition (Gubinelli) Let 0 ≤ a 1 < a 2 ≤ T. Then, there exists a unique linear map Λ : Z C 1 + 3 ([ a 1 , a 2 ]) → C 1 + 2 ([ a 1 , a 2 ]) such that δ Λ = Id Z C 1 + 3 ([ a 1 , a 2 ]) and || Λ h || µ, [ a 1 , a 2 ] ≤ || h || µ, [ a 1 , a 2 ] . 2 µ − 2 Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 26 / 64

Inverse of δ Proposition (Gubinelli) Let 0 ≤ a 1 < a 2 ≤ T. Then, there exists a unique linear map Λ : Z C 1 + 3 ([ a 1 , a 2 ]) → C 1 + 2 ([ a 1 , a 2 ]) such that δ Λ = Id Z C 1 + 3 ([ a 1 , a 2 ]) and || Λ h || µ, [ a 1 , a 2 ] ≤ || h || µ ;[ a 1 , a 2 ] . 2 µ − 2 Remark For any h ∈ C 1 + 3 ([ a 1 , a 2 ]) such that δ h = 0, there exists a unique g = Λ( h ) ∈ C 1 + such that δ g = h . 2 Jorge A. León (Cinvestav–IPN) Delay Equations Roscoff 2010 27 / 64