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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 C0−semigroup, if it satisfies the functional equation

• T(t + s) = T(t)T(s),

t ≥ 0, T(0) = I (FE) 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(t) := etA =

• k

tkAk k! , t ≥ 0. (2)

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) = Amu(t), t ≥ 0, ˙ x(t) = Bx(t) + Cu(t) + Φut, t ≥ 0, Lu(t) = x(t), t ≥ 0, u0(•) = v0, u(0) = f0, Lf0 = x0 (ABFSD) where u(t) ∈ F (e.g. L2(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 := L2[0, π] and ∂F := C2, Am := d2

ds2 and

L := δ0 δπ

• : D(Am) → ∂F.

(3)          ut(t, s) = uss(t, s), ut(t, 0) = n

k=1 αku(t − tk, sk),

ut(t, π) = n

k=1 βku(t − tk, sk),

u(t, s) = v0(t, s), (t, s) ∈ [−1, 0] × [0, π] (ME) with sk ∈ [0, π], tk ∈ [0, 1], αk, βk ∈ 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 u(0) = u0 (ACP) 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

• perator 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 C0−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 ´(etG)t≥0‘ is a C0−semigroup, where G =  

d ds

Am Φ C B   , (4) D(G) = v

f x

• ∈ W 1,p(I, Z) × D(Am) × D(B)
• v(0) = f , Lf = x
• ,

D(G) ⊂ X := Lp([−1, 0], F) × F × ∂F. Moreover, we have u(t) := π2(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 − G = Lλ •   λ − D λ − A λ − (B + Φ(ελ ⊗ Lλ))  

• Rλ.

(6) where both Lλ and Rλ are bijective. i.e. λ ∈ σ(G) ⇐ ⇒ λ ∈ σ (B + Φ(ελ ⊗ Lλ)) (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(ελ ⊗ Lλ)) (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 R(λ, G) =   IdE ǫ ⊗ IdZ ελ ⊗ Lλ IdZ Lλ Id∂F   •   R(λ, D) R(λ, A) ∆(λ)Φ ∆(λ)(ǫ ⊗ R(λ, A)) ∆(λ)   Therefore we may impose positivity: etG ≥ 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. C2) is finite dimension = ⇒ ∃ Q : ρ(A) → M2(C) s.t. λ ∈ ρ(A) ∩ ρ(G) ⇐ ⇒ det(Q(λ)) = 0; (10) (iii) 0 ≤ etG ⇐ ⇒ αk, βk ≥ 0; (iv) s(G) ≤ 0 ⇐ ⇒ f (αk, βk) ≤ 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.