A semigroup approach to boundary feedback systems. Alessandro - PowerPoint PPT Presentation

Introduction ACP Well-posedness Spectral theory Positivity Example revisited A semigroup approach to boundary feedback systems. Alessandro Arrigoni, Klaus Engel University of L’Aquila Hamburg, September 2th 2016.

Introduction ACP Well-posedness Spectral theory Positivity Example revisited A family ( T ( t )) t ≥ 0 of bounded linear operators on some Banach space X is called a C 0 − semigroup, if it satisfies the functional equation � T ( t + s ) = T ( t ) T ( s ) , t ≥ 0 , (FE) T (0) = I and the maps t �→ T ( t ) x (1) are continuous from R + into X for all x ∈ X . For a bounded operator A ∈ L ( X ), ( T ( t )) t ≥ 0 is defined by an operator-valued exponential function: t k A k T ( t ) := e tA = � k ! , t ≥ 0 . (2) k

Introduction ACP Well-posedness Spectral theory Positivity Example revisited An introduction to boundary problems. We are interested in the study of a class of boundary systems with unbounded and delayed feedback using the theory of strongly continuous semigroups. More precisely we consider systems of the form  u ( t ) = A m u ( t ) , ˙ t ≥ 0 ,    x ( t ) = Bx ( t ) + Cu ( t ) + Φ u t , ˙ t ≥ 0 ,  (ABFSD) Lu ( t ) = x ( t ) , t ≥ 0 ,    u 0 ( • ) = v 0 , u (0) = f 0 , Lf 0 = x 0  where u ( t ) ∈ F (e.g. L 2 ( I )) and x ( t ) ∈ ∂ F (e.g. C ), both Banach spaces.

Introduction ACP Well-posedness Spectral theory Positivity Example revisited As a possible application let us consider the following 1-D diffusion equation; we fix F := L 2 [0 , π ] and ∂ F := C 2 , A m := d 2 ds 2 and � δ 0 � L := : D ( A m ) → ∂ F . (3) δ π  u t ( t , s ) = u ss ( t , s ) ,   u t ( t , 0) = � n  k =1 α k u ( t − t k , s k ) ,  (ME) u t ( t , π ) = � n k =1 β k u ( t − t k , s k ) ,     u ( t , s ) = v 0 ( t , s ) , ( t , s ) ∈ [ − 1 , 0] × [0 , π ] with s k ∈ [0 , π ], t k ∈ [0 , 1], α k , β k ∈ C , defined on the product state-space X .

Introduction ACP Well-posedness Spectral theory Positivity Example revisited The Abstract Cauchy Problem. In order to tackle this problem by semigroup methods we have to resolve the following Problem Rewrite (ABFSD) as an (ACP) � ˙ u ( t ) = Gu ( t ) , t ≥ 0 (ACP) u (0) = u 0 for a proper operator G on a suitable Banach space X . The definition of G is in general not unique, i.e. the entries of the matrix operator may not be unique.

Introduction ACP Well-posedness Spectral theory Positivity Example revisited We investigate the boundary problem (ABFSD) via semigroups theory. (I) Equivalence of (ACP) and (ABFSD), (II) Well-posedness of (ABFSD) ⇐ ⇒ G is the generator of C 0 − semigroup, (III) Spectral theory for G , hence possibly asymptotic behaviour, (IV) Qualitative properties of solutions (positivity). (V) Stability of solution, (VI) Apply abstract results to (ME).

Introduction ACP Well-posedness Spectral theory Positivity Example revisited (ABFSD) ⇐ ⇒ (ACP) By equivalence of the systems we mean that (ABFSD) and (ACP) share the same solution. In particular we have the following Theorem (ABFSD) is well-posed iff ´ ( e t G ) t ≥ 0 ‘ is a C 0 − semigroup, where  d  0 0 ds  , G = (4) 0 A m 0  Φ C B �� v � � � ∈ W 1 , p ( I , Z ) × D ( A m ) × D ( B ) D ( G ) = � v (0) = f , Lf = x , f � x D ( G ) ⊂ X := L p ([ − 1 , 0] , F ) × F × ∂ F. Moreover, we have u ( t ) := π 2 ( u ( t ))is solution of (ABFSD).

Introduction ACP Well-posedness Spectral theory Positivity Example revisited Via a decomposition we impose that G is a generator. The idea is to write G as a ´structured perturbation’ G = D + B · C (5) where D is a diagonal operator with nice properties and the product B · C is a perturbation. We point out that the decomposition may not be unique. To impose the generator property of G we use a variation of the Staffans-Weiss theorem

Introduction ACP Well-posedness Spectral theory Positivity Example revisited To study the spectral theory of G we make use of the factorization. Given the resolvent set � � � ρ ( G ) := λ ∈ C � λ − G : D ( G ) → X is bijective � the spectrum is its complement: σ ( G ) ∪ ρ ( G ) = C . The location in the complex plane, of the spectrum of a generator, may determine the qualitative behaviour of the solution of the (ACP). The idea for studying it, is to factorize λ − G .

Introduction ACP Well-posedness Spectral theory Positivity Example revisited More precisely, we have Lemma For λ ∈ ρ ( A ) one has   λ − D 0 0 • R λ . λ − G = L λ • 0 λ − A 0 (6)   0 0 λ − ( B + Φ( ε λ ⊗ L λ )) where both L λ and R λ are bijective. i.e. λ ∈ σ ( G ) ⇐ ⇒ λ ∈ σ ( B + Φ( ε λ ⊗ L λ )) (7)

Introduction ACP Well-posedness Spectral theory Positivity Example revisited Description of the fine structure of the spectrum. Theorem It is possible to get a finer description of the spectrum; Let λ in ρ ( A ) . λ ∈ σ ⋆ ( G ) ⇐ ⇒ λ ∈ σ ⋆ ( B + Φ( ε λ ⊗ L λ )) (8) for ⋆ ∈ { p , a , c , r , ess } , i.e. they share injectivity, surjectivity, dense range, closed range.

Introduction ACP Well-posedness Spectral theory Positivity Example revisited Moreover, if λ ∈ ρ ( A ) ∩ ρ ( G ) then we get an expression for the resolvent     Id E ǫ ⊗ Id Z ε λ ⊗ L λ R ( λ, D ) 0 0 R ( λ, G ) = 0 Id Z L λ 0 R ( λ, A ) 0  •    0 0 Id ∂ F ∆( λ )Φ ∆( λ )( ǫ ⊗ R ( λ, A )) ∆( λ ) Therefore we may impose positivity: e t G ≥ 0 ⇐ ⇒ 0 ≤ R ( λ, G ) ∀ λ large . (9)

Introduction ACP Well-posedness Spectral theory Positivity Example revisited The Motivating Example revisited. Letting G be the operator associated to (ME) we obtain: (i) G is a generator = ⇒ (ME) is well-posed; (ii) ∂ F ( i . e . C 2 ) is finite dimension = ⇒ ∃ Q : ρ ( A ) → M 2 ( C ) s.t. λ ∈ ρ ( A ) ∩ ρ ( G ) ⇐ ⇒ det ( Q ( λ )) = 0; (10) (iii) 0 ≤ e t G ⇐ ⇒ α k , β k ≥ 0; (iv) s ( G ) ≤ 0 ⇐ ⇒ f ( α k , β k ) ≤ 0.

Introduction ACP Well-posedness Spectral theory Positivity Example revisited Thank you! A. Arrigoni and K.-J. Engel. A semigroup approach to boundary feedback systems with delay . [AE], to be submitted.