Draft EE 8235: Lectures 10 & 11 1 Lectures 10 & 11: - PowerPoint PPT Presentation

Draft EE 8235: Lectures 10 & 11 1 Lectures 10 & 11: Semigroup Theory • Want to generalize matrix exponential to infinite dimensional setting • Strongly continuous ( C 0 ) semigroup ⋆ Extension of matrix exponential • Infinitesimal generator of a C 0 -semigroup • Examples and conditions

Draft EE 8235: Lectures 10 & 11 2 Solution to abstract evolution equation • Abstract evolution equation on a Hilbert space H d ψ ( t ) = A ψ ( t ) , ψ (0) ∈ H d t Dilemma: how to define ” e A t ”? Finite dimensional case: ∞ ( M t ) k e M t = � M ∈ C n × n ⇒ k ! k = 1

Draft EE 8235: Lectures 10 & 11 3 d ψ ( t ) = A ψ ( t ) , ψ (0) ∈ H d t • Assume: ⋆ For each ψ (0) ∈ H , there is a unique solution ψ ( t ) ⋆ There is a well defined mapping T ( t ) : H − → H ψ ( t ) = T ( t ) ψ (0) T ( t ) - time-parameterized family of linear operators on H ⋆ Solution varies continuously with initial state T ( t ) : a bounded operator (on H ) � T ( t ) f � � T ( t ) � = < ∞ sup � f � f ∈ H

Draft EE 8235: Lectures 10 & 11 4 Strongly continuous semigroups • Properties of T ( t ) : ψ ( t ) = T ( t ) ψ (0) • Initial condition: T (0) = I • Semigroup property: for all t 1 , t 2 ≥ 0 T ( t 1 + t 2 ) = T ( t 2 ) T ( t 1 ) = T ( t 1 ) T ( t 2 ) , T( ) t 1 T( ) t 2 t 1 t 1 + t 2 T( ) t 1 + t 2 • Strong continuity: t → 0 + � T ( t ) ψ (0) − ψ (0) � = 0 , for all ψ (0) ∈ H lim a weaker condition than: � ( T ( t ) − I ) f � t → 0 + � T ( t ) − I � = lim t → 0 + sup lim = 0 � f � f ∈ H

Draft EE 8235: Lectures 10 & 11 5 Examples • Linear transport equation  d ψ ( t ) ± c d � = d x ψ ( t ) φ t ( x, t ) = ± c φ x ( x, t )   d t ⇒ f ( x ) , x ∈ R φ ( x, 0) =  ψ (0) = f ∈ L 2 ( −∞ , ∞ )  • Consider: φ ( x, t ) = [ T ( t ) f ] ( x ) = f ( x ± ct ) In class: T ( t ) defines a C 0 -semigroup on L 2 ( −∞ , ∞ ) • The infinitesimal generator of a C 0 -semigroup T ( t ) on H T ( t ) f − f A f = lim t t → 0 + � � T ( t ) f − f D ( A ) = f ∈ H ; lim exists t t → 0 +

Draft EE 8235: Lectures 10 & 11 6 • A couple of additional notes ⋆ Change of coordinates: � � φ t ( x, t ) = ± c φ x ( x, t ) φ t ( z, t ) = 0 z = x ± ct − − − − − − → φ ( x, 0) = f ( x ) , x ∈ R φ ( z, 0) = f ( z ) , z ∈ R ⋆ Reaction-convection equation: � φ t ( x, t ) = ± c φ x ( x, t ) + a φ ( x, t ) φ ( x, 0) = f ( x ) , x ∈ R C 0 -semigroup: φ ( x, t ) = [ T ( t ) f ] ( x ) = e a t f ( x ± ct ) a > 0 exponentially growing traveling wave a < 0 exponentially decaying traveling wave

Draft EE 8235: Lectures 10 & 11 7 Infinite number of decoupled scalar states • Abstract evolution equation on ℓ 2 ( N )       ψ 1 ( t ) ψ 1 ( t ) a 1 d d ψ ( t ) ψ 2 ( t ) ψ 2 ( t ) a 2  = ⇔ = A ψ ( t )       d t d t . .      ... . . . . Solution e a 1 t       ψ 1 ( t ) ψ 1 (0) e a 2 t ψ 2 ( t ) ψ 2 (0) ψ ( t ) =  =  = T ( t ) ψ (0)       . .     ... . . . . • In class: conditions for well-posedness on ℓ 2 ( N )

Draft EE 8235: Lectures 10 & 11 8 • Half-plane condition: Re ( a n ) < M < ∞ sup n Im Im x x x x x x x x x x x x x x x Re Re x x x x M x x x x x x x x x x x x x x x x x (a) (b) Same condition for: ∞ e a n t v n � v n , f � � T ( t ) f = n = 1

Draft EE 8235: Lectures 10 & 11 9 Continuum of decoupled scalar states ˙ ψ ( κ, t ) = a ( κ ) ψ ( κ, t ) , κ ∈ R Solution ψ ( κ, t ) = [ T ( t ) ψ ( · , 0)] ( κ ) = e a ( κ ) t ψ ( κ, 0) • Homework: conditions for well-posedness on L 2 ( −∞ , ∞ ) Half-plane condition: sup Re ( a ( κ )) < M < ∞ κ ∈ R

Draft EE 8235: Lectures 10 & 11 10 Hille-Yosida Theorem closed, densely defined operator A on H : A - infinitesimal generator of a C 0 -semigroup with � T ( t ) � ≤ M e ω t � M every real λ > ω is in ρ ( A ) and � ( λI − A ) − n � ≤ ( λ − ω ) n for all n ≥ 1 • Difficult to check • Important consequence: a method for computing T ( t ) � − N � t T ( t ) = lim I − N A N → ∞ Implicit Euler: d ψ ( t ) ψ ( t + ∆ t ) − ψ ( t ) = A ψ ( t ) ⇒ = A ψ ( t + ∆ t ) d t ∆ t

Draft EE 8235: Lectures 10 & 11 11 Lumer-Phillips Theorem closed, densely defined operator A on H : Re ( � ψ, A ψ � ) ≤ ω � ψ � 2 for all ψ ∈ D ( A ) ψ, A † ψ ≤ ω � ψ � 2 �� �� for all ψ ∈ D ( A † ) Re ⇓ A - infinitesimal generator of a C 0 -semigroup with � T ( t ) � ≤ e ω t • Examples:  � d f � [ A f ] ( x ) = ( x )    d x  � � f ∈ L 2 [ − 1 , 1] , d f  D ( A ) d x ∈ L 2 [ − 1 , 1] , f (1) = 0 =     � d 2 f � [ A f ] ( x ) = ( x )   d x 2   f ∈ L 2 [ − 1 , 1] , d 2 f � �  D ( A ) d x 2 ∈ L 2 [ − 1 , 1] , f ( ± 1) = 0 =   