MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROLPROBLEMS

KLAUS-JOCHEN ENGEL, MARJETA KRAMAR FIJAVZ,BERND KLOSS, RAINER NAGEL AND ESZTER SIKOLYA

Abstract. We develop a semigroup approach to abstract boundary control problemswhich allows to characterize the space of all approximately reachable states. We thenintroduce the ”maximal reachability space” giving an upper bound for this space. Theabstract results are applied to the flow in a network controlled in a single vertex.

1. Introduction

In this paper we develop an abstract approach towards a wide range of boundary controlproblems. The classical treatment of J.L. Lions for differential operators on domains hasbeen described in great detail in the monographs by I. Lasiecka–R. Triggiani [LT00]. Amore abstract approach to boundary control systems has been initiated by H. Fattoriniand systematized by D. Salamon and D.L. Russel (see [Sal87], [Rus73], [Rus78]). Otherapproaches are due to G. Weiss [Wei89], who uses his concept of admissibility (see the sur-vey article by Jacob–Partington [JP04] and the monograph by O.J. Staffans [Sta05]), whileH.J.A.M. Heijmans [Hei87] worked in sun-reflexive spaces. W. Desch et al. [DLS85], [DS89],[DMS01] applied multiplicative perturbations. L. Maniar and his coauthors S. Boulite etal. ([BIM05], [BIM06]), as well as O.J. Staffans, G. Weiss et al. ([MSW06], [STW01]) andD. Salamon [Sal87] worked in abstract extrapolation spaces.

To us, the following idea of D. Salamon and G. Greiner [Gre87] seems to be the most naturaland efficient to treat a multitude of problems as “abstract boundary control problems”.

Besides the state space X on which the dynamics acts, choose a “boundaryspace” ∂X on which the control takes place.

Under very natural assumptions G. Greiner ([Gre87], [Gre89], [GK91]) and others (e.g.,[CENN03], [CENP05]) have shown the value of this approach to study well-posedness,spectral properties and regularity. J. Malinen and O.J. Staffans in [MaSt06] and [MaSt07]handled boundary control problems in Hilbert spaces in this way but they were mainlyconcerned with well-posedness. S. Krause [Kra90] used it to characterize approximate con-trollability of abstract boundary control problems.We complete results of [Kra90] and give a very precise characterization of the approximatereachability space in Theorem 2.12. In particular, we obtain a new and useful upper boundfor the approximate reachability space in terms of the eigenvectors of the corresponding“maximal operator” (Corollary 2.13). We call this space the maximal reachability space and,due to its simplicity, it plays a crucial role. Indeed, given boundary conditions constrain thestate space X to the maximal reachability space, and only this space, and not the wholestate space X, is to be considered while aiming for controllability of a boundary problem.

1991 Mathematics Subject Classification. 93B05, 47N70, 34B45.Key words and phrases. boundary control, operator semigroups, reachability spaces, Kalman condition,

flows in networks.1

2 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

The paper is organized as follows. We first recall in Subsection 2.a the setting and themain results on approximate controllability of standard linear control systems (see [CZ95]or [EN00, Sect. VI.8]). In Subsection 2.b we explain what we mean by an abstract boundarycontrol system and make the appropriate assumptions on the operators involved. Using thecorresponding “Dirichlet operators” (see Lemma 2.4) we can associate a standard controlsystem to our boundary problem (see Proposition 2.6). This leads in particular to thevariation of parameters formulas (2.9), (2.11) and the characterization of the approximatereachability space (Theorem 2.12) in Subsection 2.c. Our definitions and results oftencorrespond to interpretations of abstract boundary control systems as “system nodes”or as “well-posed systems”, see [MSW06] or [Sal87, Chapt. 4.7]. Finally, the maximalreachability space and a maximally controllable system are introduced, see Definition 2.14.Their importance becomes manifest in concrete boundary control problems such as controlproblems for flows in networks (see Section 3). A second application is made in [EKNS08].Both lead to a Kalman-type criterion for (maximal) controllability of the given problem.The paper concludes with a brief appendix on Favard spaces needed in this work.

2. Abstract Boundary Control Systems

2.a. Classical Situation. We start from a control system Σ(A,B), associated to thecontrolled abstract Cauchy problem on the Banach space X with control space U , definedas {

x(t) = Ax(t) +Bu(t), t ≥ 0,

x(0) = x0.(2.1)

Here, the operator (A,D(A)) is the generator of the strongly continuous semigroup (T (t))t≥0

on X, and B ∈ L(U,X). We recall from [EN00, Sect. VI.8.a] or [CZ95, Def. 4.1.3] thatfor t ≥ 0 the controllability maps Bt ∈ L(L1([0, t], U), X) of Σ(A,B) are well-defined andgiven by

(2.2) Btu :=

∫ t

0

T (t− s)Bu(s) ds, u ∈ L1([0, t], U).

This formula gives the so-called mild solution of the problem for x0 = 0. If u ∈W1,1loc(R+, U),

then the function t 7→ Btu =: x(t) is even a classical solution of (2.1), i.e., it is continuouslydifferentiable with respect to t, x(t) ∈ D(A), and it satisfies (2.1). We are mainly concernedwith the approximate reachability space (see [CZ95, Def. 4.1.17]) defined as

(2.3) R :=⋃t≥0

ran(Bt).

If R = X, the control system Σ(A,B) is called approximately controllable.For the following alternative description of R we choose some constant ω ≥ ω0(A) whereω0(A) denotes the growth bound of A.

Proposition 2.1. The approximate reachability space R coincides with

(i) the smallest closed, (T (t))t≥0-invariant subspace of X containing ran(B), or(ii) the smallest closed, R(µ,A)-invariant, µ > ω, subspace of X containing ran(B).

Proof. (i) is proved in [CZ95, Lem. 4.1.19]. Using the integral representation of the resolvent

R(µ,A) =

∫ ∞0

e−µtT (t) dt for µ > ω0(A)

MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROL PROBLEMS 3

(see [EN00, Thm. II.1.10]) it follows from (i) that R is R(µ,A)-invariant for each µ > ω0(A).Similarly, the Post–Widder inversion formula

T (t) = limn→∞

[ntR(nt, A)]n

for t > 0

(see [EN00, Cor. III.5.5]) implies that R is also given as the smallest closed subspace of Xcontaining ran(B) and being R(µ,A)-invariant for µ > ω. �

2.b. Boundary Control. Our aim is to treat linear control systems with control operatorsacting on the boundary only. In the spirit of G. Greiner [Gre87] we introduce our

Abstract Framework 2.2. We consider

(i) three Banach spaces X, ∂X and U , called the state, boundary and control space, resp.;(ii) a closed, densely defined system operator Am : D(Am) ⊆ X → X;(iii) a boundary operator Q ∈ L([D(Am)], ∂X);(iv) a control operator B ∈ L(U, ∂X).

For these operators and spaces and a control function u ∈ L1loc(R+, U) we then consider

the abstract Cauchy problem with boundary control

(2.4)

x(t) = Amx(t), t ≥ 0,Qx(t) = Bu(t), t ≥ 0,x(0) = x0.

A function x(·) = x(·, x0, u) ∈ C1(R+, X) with x(t) ∈ D(Am) for all t ≥ 0 satisfying (2.4)is called a classical solution. Moreover, we denote the abstract boundary control systemassociated to (2.4) by ΣBC(Am, B,Q).

In order to investigate (2.4) we make the following assumption ensuring in particular thatthe uncontrolled abstract Cauchy problem, i.e., (2.1) or (2.4) with B = 0, is well-posed.

Main Assumptions 2.3.

(i) The restricted operator A ⊂ Am with domain D(A) := kerQ generates a stronglycontinuous semigroup (T (t))t≥0 on X;

(ii) the boundary operator Q : D(Am)→ ∂X is surjective.

Under these assumptions the following properties have been shown by Greiner [Gre87,Lem. 1.2, Lem. 1.3] and will be the key tools for our studies.

Lemma 2.4. Let Assumptions 2.3 be satisfied. Then the following assertions are true foreach λ ∈ ρ(A).

(i) D(Am) = D(A)⊕ ker(λ− Am);(ii) Q|ker(λ−Am) is invertible and the operator Qλ := (Q|ker(λ−Am))

−1 : ∂X → ker(λ−Am) ⊆X is bounded;

(iii) Pλ := QλQ ∈ L([D(Am)]) is a projection onto ker(λ− Am) along D(A) = kerQ;(iv) R(µ,A)Qλ = 1

µ−λ(Qµ −Qλ) = R(λ,A)Qµ for all λ, µ ∈ ρ(A), λ 6= µ.

The following operators are essential to obtain an explicit representation of the solutionsof (2.4).

Definition 2.5. For λ ∈ ρ(A) we call the operator Qλ, introduced in Lemma 2.4(ii), theDirichlet operator and define

Bλ := QλB ∈ L(U, ker(λ− Am)

).

4 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

The following result associates the boundary control system (2.4) to the system{x(t) = A

(x(t)−Bλu(t)

)+ λBλu(t), t ≥ 0,

x(0) = x0

(2.5)

for some λ ∈ ρ(A). Here x(·) = x(·, x0, u, λ) ∈ C1(R+, X) is called a classical solution of(2.5) if x(t)−Bλu(t) ∈ D(A) for all t ≥ 0 and x(·) satisfies (2.5).

Proposition 2.6. Let Assumptions 2.3 be satisfied and let x0 ∈ X, λ ∈ ρ(A) and u ∈L1

loc(R+, U). Then x(·) = x(·, x0, u) is a classical solution of (2.4) if and only if it is aclassical solution of the system (2.5).

Proof. Let x(·) be a classical solution of (2.4). Using the projection Pλ = QλQ ∈ L([D(Am)])defined in Lemma 2.4 satisfying ran(Pλ) = ker(λ−Am) and kerPλ = ran(I − Pλ) = D(A)we can decompose

x(t) = (I − Pλ)x(t) + Pλx(t).

Then by (2.4) we obtain

(2.6)

x(t) = Amx(t) = Am((I − Pλ)x(t) + Pλx(t)

)= A

(x(t)−QλQx(t)

)+ λQλQx(t)

= A(x(t)−Bλu(t)

)+ λBλu(t),

which is the first line of (2.5).Let now x(·) be a classical solution of (2.5). Then

x(t)−Bλu(t) ∈ D(A), t ≥ 0.

By QQλ = I∂X we have 0 = Q(x(t)− Bλu(t)) = Qx(t)−QQλBu(t) = Qx(t)− Bu(t) andso Qx(t) = Bu(t). Therefore the boundary condition in (2.4) is satisfied. Finally, reversingthe computations in (2.6) we conclude

x(t) = A(x(t)−Bλu(t)

)+ λBλu(t) = Amx(t),

so x(·) is a classical solution of (2.4). �

Our aim is now to find explicit representations for the solutions of the boundary controlsystem (2.4). To this end we first transform (2.5) into a standard control system in a biggerstate space. Using (2.5) we obtain

(2.7)x(t) = A

(x(t)−Bλu(t)

)+ λBλu(t)

= A−1x(t) + (λ− A−1)Bλu(t),

where A−1 denotes the generator of the extrapolated semigroup (T−1(t))t≥0 with domainD(A−1) = X on X−1, cf. [EN00, Sect. II.5.a]. Next we solve this system as indicated inSubsection 2.a.

Proposition 2.7 (Variation of parameters formula, extrapolated version). Let x0 ∈ X,u ∈ L1

loc(R+, U) and λ ∈ ρ(A). If x(·) = x(·, x0, u) is a classical solution (2.4), then it isgiven by the variation of parameters formula

(2.8) x(t) = T (t)x0 + (λ− A−1)

∫ t

0

T (t− s)Bλu(s) ds, t ≥ 0.

MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROL PROBLEMS 5

Proof. By the usual variation of parameters formula, see [EN00, Sect. VI.7.a], applied tothe inhomogeneous Cauchy problem (2.7) in X−1 we obtain

x(t) = T (t)x0 +

∫ t

0

T−1(t− s)(λ− A−1)Bλu(s) ds,

= T (t)x0 + (λ− A−1)

∫ t

0

T (t− s)Bλu(s) ds

where the first integral is understood in X−1 while the second one converges in X. �

In order to obtain from (2.8) solutions having values in X we have to impose additionalassumptions concerning either

• time regularity, i.e., on the smoothness of the control function u(·), or• space regularity, i.e., on the smoothness of the elements in ker(λ− Am).

We start by assuming time regularity generalizing the results in [Kra90, Thm. 1.8]. Notethat in order to obtain classical solutions of (2.4) we have to impose the compatibilitycondition Qx0 = Bu(0) which follows from the boundary condition Qx(t) = Bu(t) fort = 0.

Proposition 2.8 (Variation of parameters formula, time regularity). Let x0 ∈ X, u ∈W1,1

loc(R+, U) and λ ∈ ρ(A). If x(·) = x(·, x0, u) is a classical solution of (2.4), then

(2.9) x(t) =

∫ t

0

T (t− s)Bλ

(λu(s)− u′(s)

)ds+ T (t)

(x0 −Bλu(0)

)+Bλu(t), t ≥ 0.

On the other hand, if x0 ∈ D(Am) and u ∈ W2,1loc(R+, U) satisfy Qx0 = Bu(0), then x(·)

defined by (2.8) or (2.9) is a classical solution of (2.4).

Proof. If u ∈W1,1loc(R+, U), (2.9) follows from (2.8) by integration by parts.

Now assume that x0 ∈ D(Am), u ∈ W2,1loc(R+, U), Qx0 = Bu(0) and let x(·) be defined by

(2.9). Then λu(·)− u′(·) ∈W1,1loc(R+, U) and from [EN00, Cor. VI.7.6] it follows that

[0,+∞) 3 t 7→ z(t) :=

∫ t

0

T (t− s)Bλ

(λu(s)− u′(s)

)ds ∈ D(A) ⊂ X

is continuously differentiable and satisfies

(2.10) z(t) = Az(t) +Bλ

(λu(t)− u′(t)

), t ≥ 0.

Since Q(x0−Bλu(0)

)= Qx0−Bu(0) = 0 we conclude x0−Bλu(0) ∈ ker(Q) = D(A) and

hence x(t)−Bλu(t) = z(t) + T (t)(x0 −Bλu(0)

)∈ D(A) for all t ≥ 0. Moreover,

[0,+∞) 3 t 7→ x(t) = z(t) + T (t)(x0 −Bλu(0)

)+Bλu(t) ∈ X

is continuously differentiable with derivative

x(t) = z(t) + AT (t)(x0 −Bλu(0)

)+Bλu

′(t)

= Az(t) +Bλ

(λu(t)− u′(t)

)+ AT (t)

(x0 −Bλu(0)

)+Bλu

′(t)

= A(x(t)−Bλu(t)

)+ λBλu(t), t ≥ 0.

Since x(0) = x0 we conclude that x(·) is a classical solution of (2.5) and the assertionfollows from Proposition 2.6. �

6 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

Next we replace the time regularity assumption on the control function u(·) by a spaceregularity assumption on the elements in D(Am) = D(A)⊕ker(λ−Am). For the definitionof the Favard class we refer to Appendix A.

Proposition 2.9 (Variation of parameters formula, space regularity). Let ker(λ−Am) ⊆FA

1 for some λ ∈ ρ(A), x0 ∈ X and u ∈ L1loc(R+, U). If x(·) = x(·, x0, u) is a classical

solution of (2.4), then

(2.11) x(t) = T (t)x0 + (λ− A)

∫ t

0

T (t− s)Bλu(s) ds, t ≥ 0.

On the other hand, if x0 ∈ D(Am) and u ∈ W1,1loc(R+, U) satisfy Qx0 = Bu(0), then

x(·) = x(·, x0, u) defined by (2.11) is a classical solution of (2.4).

Proof. Since D(Am) = D(A)⊕ ker(λ− Am) ⊆ FA1 the closed graph theorem implies Bλ ∈

L(U, FA1 ) and hence Bλu ∈ L1

loc(R+, FA1 ). Formula (2.11) then follows from Proposition 2.7

and Lemma A.3 for v = Bλu.

To prove the second part we first observe that A−1 ∈ L(FA1 , F

A0 ). Together with Bλu ∈

W1,1loc(R+, F

A1 ) this implies f := (λ−A−1)Bλu ∈W1,1

loc(R+, FA0 ). Moreover, the compatibility

condition Qx0 = Bu(0) yields

A−1x0 + f(0) = A(I − Pλ)x0 + λPλx0 ∈ X.

Therefore we can apply [EN00, Cor. VI.7.13] to conclude that x(·) ∈ C1(R+, X)∩C(R+, FA1 )

is a classical solution of the inhomogeneous Cauchy problem

(2.12) x(t) = A−1x(t) + (λ− A−1)Bλu(t), t ≥ 0.

Moreover, Qx0 = Bu(0) also implies x0 − Bλu(0) = (I − Pλ)x0 ∈ D(A). Since D(A) =ker(Q) is (T (t))t≥0-invariant from (2.9) and Lemma A.3 for v = Bλ(λu−u′) ∈ L1

loc(R+, FA1 )

it follows that

(2.13) x(t)−Bλu(t) =

∫ t

0

T (t− s)Bλv(s) ds+ T (t)(x0 −Bλu(0)

)∈ D(A), t ≥ 0.

This shows that x(·) ∈ C1(R+, X) is a classical solution of (2.5) and hence by Proposi-tion 2.6 a classical solution of (2.4) as well. �

Remark 2.10. If x1(·) and x2(·) are classical solutions of (2.4), then their difference x(·) :=x1(·)− x2(·) satisfies x(t) = Amx(t), t ≥ 0,

Qx(t) = 0, t ≥ 0,x(0) = 0,

Since the restriction of Am on kerQ is a generator, classical solutions of this problem areunique, see [EN00, Prop. II.6.2], and therefore x(·) ≡ 0. By Proposition 2.7 it followsthat classical solutions of (2.4) given by (2.8), (2.9) or (2.11) are unique as well, henceindependent of the choice of λ ∈ ρ(A).

2.c. Approximate Reachability Spaces. Having found the solutions of (2.4) for a largeclass of initial values x0 and control functions u(·), we now investigate which states in Xcan be approximately reached from x0 = 0 by solutions of (2.4). Since we are interestedonly in approximate controllability, we can restrict ourself to sufficiently regular controlfunctions u(·) and define the following.

MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROL PROBLEMS 7

For fixed λ ∈ ρ(A) and t > 0 the operators BBCt ∈ L(W2,1([0, t], U), X) given by

(2.14) BBCt u :=

∫ t

0

T (t− s)Bλ

(λu(s)− u′(s)

)ds+Bλu(t)− T (t)Bλu(0),

u ∈ W2,1([0, t], U), are called the controllability maps of the system ΣBC(Am, B,Q), cf.(2.2) and (2.9). By Remark 2.10 this definition is independent of the particular choice ofλ ∈ ρ(A) and gives the (unique) classical solution of (2.4) for given u ∈W2,1([0, t], U) withx0 = 0.

Definition 2.11. The approximate reachability space of ΣBC(Am, B,Q) is defined by

(2.15) RBC :=⋃t≥0

ran(BBCt ).

Before characterizing RBC we consider the standard control system Σ(A,Bλ) for someλ > ω0(A) associated to the controlled abstract Cauchy problem{

x(t) = Ax(t) +Bλu(t), t ≥ 0,

x(0) = x0,(2.16)

cf. (2.1). As in (2.2), the controllability maps Bλt ∈ L(L1([0, t], U), X) of Σ(A,Bλ) are given

by

(2.17) Bλt u :=

∫ t

0

T (t− s)Bλu(s) ds, u ∈ L1([0, t], U).

Moreover, the corresponding approximate reachability space of Σ(A,Bλ) is

Rλ :=⋃t≥0

ran(Bλt ),

cf. (2.3).

It turns out that RBC and Rλ coincide and can be described in various ways.

Theorem 2.12. The approximate reachability space RBC of ΣBC(Am, B,Q) coincides with

(i) Rλ for each λ > ω0(A),(ii) the smallest closed, (T (t))t≥0-invariant subspace of X containing ran(Bµ) for all µ

sufficiently large,(iii) the smallest closed, R(µ,A)-invariant, µ > ω > ω0(A), subspace of X containing

ran(Bµ) for all µ sufficiently large, and(iv) span

( ⋃λ>ω ran(Bλ)

)for some ω > ω0(A).

Proof. (i) Observe that using (2.14) and (2.17) we have

(2.18) BBCt u = λ ·Bλ

t u−Bλt u′ +Bλu(t)− T (t)Bλu(0), u ∈W2,1([0, t], U).

Hence, in order to show that RBC ⊆ Rλ, by (2.15) it suffices to verify that ran(Bλ) ⊂ Rλ

and T (t) ran(Bλ) ⊂ Rλ. These are immediate consequences of Proposition 2.1(i).

For the converse inclusion, observe that (2.18) can be rewritten as

Bλt ·(λ− d

ds

)= BBC

t −Bλ ◦ δt + T (t)Bλ ◦ δ0.

8 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

Since the operator (λ− d

ds

): W2,1([0, t], U)→ L1([0, t], U)

has dense range, we have

Rλ =⋃t≥0

ran(Bλt ) =

⋃t≥0

ran

(Bλt ·(λ− d

ds

)).

Hence to prove Rλ ⊆ RBC it is enough to show that ranBλ ⊂ RBC and T (t) ran(Bλ) ⊂ RBC

for all t > 0 and λ ≥ ω0(A).

For u0 ∈ U define ut(s) := eλ(s−t)u0. Then

BBCt ut = Bλut(t)− T (t)Bλut(0) = Bλu0 − e−λtT (t)Bλu0 ∈ RBC.

Taking the limit as t→∞ we obtain

limt→∞

BBCt ut = Bλu0 ∈ RBC

for all λ > ω0(A) and u0 ∈ U .

The Post–Widder inversion formula

T (t) = limn→∞

[ntR(nt, A)]n

for t > 0

(see [EN00, Cor. III.5.5]) implies that in order to show T (t) ran(Bλ) ⊂ RBC, it is enough toprove that R(µ,A)k ran(Bλ) ⊂ RBC for µ big enough and any k ∈ N. Using Lemma 2.4(iv)we have

R(µ,A)Bλu =1

µ− λ(Bµ −Bλ)u ∈ span

( ⋃µ>ω

ran(Bµ))

for µ, λ > ω ≥ ω0(A) and u ∈ U . Since ranBλ ⊂ RBC we obtain what we wanted for k = 1.For larger k’s the statement follows by induction.

(ii) and (iii) now follow immediately from Proposition 2.1 and part (i).

(iv) Let C := span(∪λ>ω ran(Bλ)). Then by part (iii), ran(Bλ) ⊆ RBC for all λ > ω andhence C ⊆ RBC. To show the converse inclusion we first note that by Lemma 2.4(iv) thespace C is R(µ,A)-invariant for µ > ω. Since R(µ,A) is bounded, this implies that C

is a closed, R(µ,A)-invariant, µ > ω, subspace of X containing ran(Bλ) for all λ > ω.However, again by part (iii), RBC is the smallest space having these properties and henceRBC ⊆ C. �

The following corollary is a consequence of Theorem 2.12(iv).

Corollary 2.13. For each boundary control system we always have

RBC ⊂ span⋃

λ>ω0(A)

ker(λ− Am).

This shows that there is an upper bound for the reachability space depending on theeigenvectors of Am, but independent of the control operator B.

Definition 2.14. The maximal reachability space of ΣBC(Am, B,Q) is defined by

(2.19) RBCmax := span

⋃λ>ω0(A)

ker(λ− Am).

The system ΣBC(Am, B,Q) is called maximally controllable if RBC = RBCmax.

MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROL PROBLEMS 9

Let us stress that RBCmax 6= X may happen, hence the relevant question about controllability

is indeed answered by comparing RBC to the space RBCmax and not to the whole space X, as

it is usually done in the classical situation. A boundary system is maximally controllableif the largest possible set of states RBC

max can be approximately reached by applying theboundary control B.

3. Control of Flows in Networks with Dynamic Ramification Nodes

To show how our approach can be applied to concrete situations we study the control of aflow in a network. The following model is taken from the work of E. Sikolya (see [Sik05]),where it is explained in detail and where well-posedness and asymptotical behavior arestudied. A related example can be found in [EKNS08].

3.a. The Basic Model. We consider a flow in a strongly connected network, modeledby a simple directed graph consisting of the set of edges E := {ej : j = 1, . . . ,m} andthe set of vertices V := {vi : i = 1, . . . , n}. We parameterize the edges on the interval[0, 1], contrary to their direction, i.e., ej(1) will denote the tail of the j-th edge while ej(0)will denote its head. For the definition of the necessary graph-theoretical objects, such asincidence matrices Φ−, Φ+

ω and Φ−ω , we refer to [Sik05, Sect. 2 and Def. 4.2]. Moreover, wedefine the following two objects.

Definition 3.1. (i) To simplify the notation we use the abbreviation Ψ :=(Φ−ω)>

.(ii) The weighted adjacency matrix of the graph is defined by

A := Φ+ωΨ ∈ Cn×n.

Remark 3.2. One easily verifies that Ψ is column stochastic and therefore

Φ−Ψ = Idn.

With this notation we we will examine the following system (see [Sik05, Sect. 3] with allcj = 1 and without a boundary control term).

(3.1)

(FE) zj(s, t) = z′j(s, t) , j = 1, . . . ,m, s ∈ [0, 1], t ≥ 0,(BC1) Φ−z(1, t) = Φ+

ω z(0, t), t ≥ 0,(BC2) z(1, t) ∈ ran Ψ, t ≥ 0,(IC1) zj(s, 0) = f0(s), s ∈ [0, 1],(IC2) Φ−z(1, 0) = d0.

Note that (BC1) describes a delay equation for the process in the tails of the edges. Thisinterpretation will gain more importance later in this section. We are now interested in theboundary controllability of this system, i.e., we characterize the possible mass distributionsof material in the network after injecting a control only in the i-th vertex. To this end wefirst reformulate the problem in a Banach-space setting by choosing

(i) the state space Y := L1([0, 1],Cm);(ii) the boundary space ∂Y := Cn; and

(iii) the control space U := C.

On these spaces we then consider

(i) the operator (Aω, D(Aω)) on Y defined by the diagonal matrixAω := diag( dds, . . . , d

ds)

with domain D(Aω) := {f ∈ (W1,1([0, 1],C))m : f(1) ∈ ran Ψ};(ii) the operator L ∈ L(D(Aω),Cn) given by Lf = Φ−f(1);

(iii) the delay operator M ∈ L(D(Aω),Cn) given by Mf = Φ+ω f(0); and

10 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

(iv) the control operator B ∈ L(C,Cn) given by Bz = z · bi, where bi denotes the i-thcanonical unit vector of Cn.

Finally we set y(t) := z(·, t). If we now impose a control in the i-th vertex, (3.1) becomes

(3.2)

y(t) = Aωy(t), t ≥ 0,

Ly(t) = My(t) +Bv(t), t ≥ 0,y(0) = f0,

Ly(0) = d0.

Before we proceed we define classical solutions of (3.2).

Definition 3.3. For v ∈ L1loc(R+,C) we call a function y(·) = y(·, f0, d0, v) : R+ → X a

classical solution of (3.2) if

(i) y ∈ C1(R+, X),(ii) y(t) ∈ D(Aω) for all t ≥ 0, and

(iii) y fulfills (3.2).

The main question we want to investigate is now the following:

Which mass distributions in L1([0, 1],Cm) can be approximately reached inany time from initial data zero using a control function v ∈ L1

loc(R+,C)?

More precisely, we want to determine the space of approximately reachable mass distribu-tions

Rmass :={y(t0, 0, 0, v) : v ∈ L1

([0, t0],C

), t0 > 0

},

where y(·) = y(·, 0, 0, v) denotes the state trajectory of (3.2) associated to the initial valuesf0 = 0 = d0 and the control function v ∈ L1

loc(R+,C).In order to tackle this problem, in the next subsection we associate to (3.2) an equation inthe form of an abstract boundary control system (2.4).

3.b. The Abstract Setting. Following ideas of A. Batkai, S. Piazerra and E. Sikolya(see [BP05], [Sik05]) we consider the problem (3.2) on an appropriate product space. Moreprecisely, we define

(i) the product state space X := Y × ∂Y = L1([0, 1],Cm)× Cn;(ii) the system operator (Am, D(Am)) given by

Am =

(Aω 0M 0

)with domain D(Am) := D(Aω)× Cn;

(iii) the boundary operator Q ∈ L(D(Am),Cn) given by Q(fd

):= Lf − d; and

(iv) the operator (A,D(A)) given by the restriction of Am to the kernel of Q, i.e.,

Ax = Amx for x ∈ D(A) := kerQ ⊂ D(Am).

Now we consider the abstract boundary control problem

(3.3)

x(t) = Amx(t), t ≥ 0,Qx(t) = Bu(t), t ≥ 0,x(0) = x0,

where x(·) : R+ → X and x0 :=(f0d0

). In the sequel we will denote the projections from

X = Y × ∂Y to the first and second coordinate by Π1 and Π2, respectively. In order toapply the results of the previous section we have to check the following prerequisites.

Proposition 3.4. (i) The boundary operator Q is surjective.

MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROL PROBLEMS 11

(ii) A generates a strongly continuous semigroup (T (t))t≥0 on X.(iii) For t = 1 the semigroup generated by A is given by[

T (1)(fd

)](s) =

(Ψ(Φ+ω

∫ s0f(r) dr + d

)Φ+ω

∫ 1

0f(r) dr + d

), 0 ≤ s ≤ 1.

Consequently, we have

Π2T (1)(fd

)= LΠ1T (1)

(fd

).

Proof. (i) is clear and (ii) is proved in [Sik05, Thm. 4.5]. To show (iii) we integrate theboundary condition (BC1) in (3.1) from 0 to t obtaining∫ t

0

Φ+ω z(0, r) dr =

∫ t

0

Φ−z(1, r) dr = Φ−z(1, t)− Φ−z(1, 0).

By [Sik05, Thm. 4.1] it follows that for(fd

)∈ D(A),

Π2T (t)(fd

)= Φ−z(1, t) = Φ−z(1, 0) + Φ+

ω

∫ t

0

z(0, r) dr.

Now using [Sik05, Thm. 4.1 and Lem. 6.1] we have

z(0, r) =[Π1T (r)

(fd

)](0) = f(r), 0 ≤ r ≤ 1.

Also, sinceΦ−z(1, 0) = Π2T (0)

(fd

)= Π2

(fd

)= d

it follows that

Π2T (t)(fd

)= Φ+

ω

∫ t

0

f(r) dr + d, 0 ≤ t ≤ 1,

which implies the statement for the second coordinate. The statement for the first coordi-nate also follows from [Sik05, Lem. 6.1], since[

Π1T (1)(fd

)](s) = ΨΠ2T (s)

(fd

)= Ψ

(Φ+ω

∫ s

0

f(r) dr + d

)for 0 ≤ s ≤ 1. Hence, since the statement holds for all

(fd

)in the dense subset D(A) of X,

the claim follows from the boundedness of T (1). �

Now we know that (3.3) fits into our general framework presented in Subsection 2.b and wecan move on to show the connection between the originally posed controllability question for(3.2) and equation (3.3). The proceeding result is similar to [BP05, Cor. 3.5 and Cor. 3.9].

Proposition 3.5. Let f0 ∈ D(Aω), d0 ∈ Cn. For v ∈ L1loc(R+,C) define u(t) :=

∫ t0v(s) ds,

i.e., u ∈W1,1loc(R+,C), u = v and u(0) = 0. Then for x0 =

(f0d0

)∈ X the following holds.

(i) If x(·) = x(·, x0, u) is a classical solution of system (3.3), then

y(·) = y(·, f0, d0, v) := Π1x(·, x0, u)

is a classical solution of system (3.2).(ii) If y(·) = y(·, f0, d0, v) is a classical solution of system (3.2) then

x(·) = x(·, x0, u) :=

(y(·, f0, d0, v)

Ly(·, f0, d0, v)(·)−Bu(·)

)gives a classical solution of system (3.3).

12 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

Proof. (i) By assumption y(·) ∈ C1(R+, Y ) ∩ C(R+, D(Aω)) and fulfills

y(t) = Aωy(t).

Moreover, we have

MΠ1x(t) = My(t) = Π2x(t), t ≥ 0

and

Qx(t) = LΠ1x(t)− Π2x(t) = Bu(t), t ≥ 0.

Consequently

Qx(t) = LΠ1x(t)− Π2x(t) = Ly(t)−My(t) = Bu(t) = Bv(t),

i.e., Ly(t) = My(t) + Bv(t). Finally, since Qx(0) = Lf0 − d0 = Bu(0) = 0 also the lastequation in (3.2) is satisfied.

(ii) By assumption, x(·) ∈ C1(R+, X) ∩ C(R+, D(Am)) and

x(t) =

(y(t)

Ly(t)−Bv(t)

)=

(Aωy(t)

My(t)

)= Amx(t), t ≥ 0.

Moreover

Qx(t) = Ly(t)− (Ly(t)−Bu(t)) = Bu(t), t ≥ 0.

Finally, x(0) =(

y(0)Ly(0)−Bu(0)

)=(f0d0

)= x0 which gives the last equation in (3.3). This

completes the proof. �

As we will see in the next section (see Lemma 3.8(ii) below) the boundary control prob-lem (3.3) provides space regularity as stated in Proposition 2.9. So if we apply W1,1

loc-controlsu(·), there always exists a unique classical solution to the problem (3.3). Combining this ob-servation with the previous result and using the terminology of Subsection 2.c we thereforeobtain the following characterization for Rmass.

Corollary 3.6. The space of approximately reachable mass distributions is given by

Rmass = Π1 RBC.

In order to solve the original problem for (3.2) it thus suffices to study equation (3.3).

3.c. Maximal Controllability for the Flow. We will now examine (3.3) on maximalapproximate (boundary) controllability using the representation of the reachability spaceprovided in Theorem 2.12. Moreover, we will give an explicit representation of the maximalspace of reachable mass distributions in L1([0, 1],Cm).

In the following we denote by ελ, λ ∈ C, the exponential function

ελ(s) := eλs, s ∈ [0, 1].

Moreover, for a function f : [0, 1] → C and a vector d ∈ Cn we define the vector-valuedfunction f ⊗ d : [0, 1]→ Cn by (f ⊗ d)(s) := f(s) · d.

Proposition 3.7. The maximal space of reachable mass distributions associated to (3.2)is given by

Π1 RBCmax = L1

([0, 1],C

)⊗ ran Ψ.

Proof. We first have to calculate the eigenvectors of Am. Let λ 6= 0. The element(fv

)belongs to ker(λ− Am) if and only if

MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROL PROBLEMS 13

(1) f ∈W1,1([0, 1],Cm), (3) λv = Mf ,(2) λf = Aωf , (4) f(1) ∈ ran Ψ.

(1) and (2) are fulfilled if and only if f = ελ ⊗ c for some c ∈ Cm. Then condition (4) isfulfilled if and only if f = ελ ⊗Ψd for some d ∈ Cn. Moreover, from (3) we derive

v = 1λ

Φ+ωΨd = 1

λAd, d ∈ Cn.

Altogether we thus have

(3.4) ker(λ− Am) =

{(ελ ⊗Ψd

1λAd

): d ∈ Cn

}.

Since L1([0, 1],C) ⊗ ran Ψ is a closed subspace of Y , the definition of RBCmax (see Defini-

tion 2.14) now impliesΠ1 RBC

max ⊆ L1([0, 1],C

)⊗ ran Ψ.

The converse inclusion follows by the Stone-Weierstrass theorem. �

Note that from Propositions 3.6 and 3.7 combined with Corollary 2.13 it follows thatsystem (3.2) will never be approximately controllable, i.e. Rmass = Y does not occur,unless the matrix Ψ is surjective which can happen only if the graph has equally manyedges and vertices.

In order to proceed we need an explicit representation of the Dirichlet operator.

Lemma 3.8. (i) For 0 6= λ ∈ ρ(A), the Dirichlet operator Qλ ∈ L(Cn, X) is given by

(3.5) Qλ =

(λελ ⊗ΨR(λeλ,A)

AR(λeλ,A)

).

(ii) The problem (3.3) provides space regularity, i.e., D(Am) ⊆ FA1 .

Proof. (i) First note that for λ ∈ ρ(A), λeλ ∈ ρ(A) (see [Sik05, Prop. 5.2]) and that (3.5)defines a bounded linear operator mapping from Cn into ker(λ−Am). Using (3.4) one caneasily check that this operator is indeed the inverse of Q|ker(λ−Am).

(ii) To see this we will show condition (d) in Proposition A.2, i.e., we verify that there existconstants c > 0 and ω1 > 0 such that

‖Qλ‖ ≤c

λfor all λ > ω1.

First observe that by the Neumann expansion of the resolvent of A for λ sufficiently large

(3.6) ‖R(λeλ,A)‖ ≤ 1

λeλ − ‖A‖=

1

λeλ· 1

1− ‖A‖λeλ

.

If we choose ω1 large enough, the second term is uniformly bounded by the constant 2 forall λ > ω1. Now given the representation in (i) we obtain

‖Qλ‖ ≤ max

{λ‖ελ‖L1 · ‖Ψ‖ · 2

λeλ, ‖A‖ · 2

λeλ

}≤ max

{(eλ − 1) · ‖Ψ‖ · 2

λeλ, ‖A‖ · 2

λ

}≤ max

{2‖Ψ‖, 2‖A‖

}︸ ︷︷ ︸:=c

·1λ

for all λ > ω1 and the claim follows. �

14 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

Note that Lemma 3.8(ii) combined with Proposition 2.9 implies that (3.3) has a classicalsolution for every control function u ∈ W1,1

loc(R+,C). Moreover, from Lemma 3.8(i) andTheorem 2.12(iv) we immediately obtain a description for the reachability space RBC.

Corollary 3.9. The reachability space associated to (3.3) is given by

RBC = span

{(ελ ⊗ΨR(λeλ,A)bi

1λAR(λeλ,A)bi

): λ > ω

}where bi denotes the i-th canonical unit vector and arbitrary ω > ω0(A).

Using Corollary 3.6 we now return to the original problem and obtain a more concreterepresentation of the approximately reachable mass distributions in L1([0, 1],Cm).

Theorem 3.10. The approximately reachable mass distributions associated to (3.2) aregiven by

Rmass = L1([0, 1],C

)⊗Ψ span

{bi,Abi, . . . ,An−1bi

}.

Proof. “⊆”: Since by the Cayley-Hamilton theorem Akbi ∈ span{bi,Abi, . . . ,An−1bi} for allk ∈ N, we can use the Neumann series representation for R(λeλ,A) to obtain

ελ ⊗ΨR(λeλ,A)bi ∈ L1([0, 1],C

)⊗Ψ span

{bi,Abi, . . . ,An−1bi

}for all λ > ω and the inclusion follows.

“⊇”: First note that

L1([0, 1],C

)⊗Ψ span

{bi,Abi, . . . ,An−1bi

}= span

{ελ ⊗ΨAkbi : k = 0, . . . , n− 1, λ > ω

},

hence it suffices to show that there exist vectors dk,λ ∈ Cn such that(ελ ⊗ΨAkbi

dk,λ

)∈ RBC for λ > ω and k = 0, . . . , n− 1.

Now observe that

λeλR(λeλ,A)Akbi −R(λeλ,A)Ak+1bi = Akbi.

Since RBC is a linear subspace we are thus done once we show(ελ ⊗ΨR(λeλ,A)Akbi

eλR(λeλ,A)Akbi

)∈ RBC, k = 1, 2, 3, . . .

To this end we proceed by induction and use the semigroup representation for T (1) fromProposition 3.4 (iii) and the invariance of the reachability space under the semigroup (seeTheorem 2.12 (ii)).k = 1: Using Proposition 3.4 (iii) we compute

λT (1)

(ελ ⊗ΨR(λeλ,A)bi

1λAR(λeλ,A)bi

)=

(ελ ⊗ΨR(λeλ,A)Abi

eλR(λeλ,A)Abi

)∈ RBC.

Note that we only need to do the computations for the first coordinate and then useΠ2T (1) = LΠ1T (1) for the second one.

k → (k + 1): We proceed as in the base step obtaining

λT (1)

(ελ ⊗ΨR(λeλ,A)Akbi

eλR(λeλ,A)Akbi

)=

(ελ ⊗ΨR(λeλ,A)Ak+1bi + 1⊗ΨAkbi

eλR(λeλ,A)Ak+1bi + Akbi

)∈ RBC.

MAXIMAL CONTROLLABILITY FOR BOUNDARY CONTROL PROBLEMS 15

The estimate (3.6) implies that the first term in both coordinates converges to zero asλ→∞. Using this we derive that (

1⊗ΨAkbiAkbi

)∈ RBC

and the induction can be completed. �

Using Proposition 3.7, Theorem 3.10 and the fact that Ψ is left invertible, we finally obtaina characterization of maximal controllability for system (3.2) by a Kalman-type conditionfor the graph matrix A.

Corollary 3.11. The following assertions are equivalent.

(i) The system (3.2) is maximally controllable in the vertex vi, i.e.

Rmass = Π1 RBCmax.

(ii) span{bi,Abi, . . . ,An−1bi} = Cn.

Appendix A

For the convenience of the reader we collect some facts concerning Favard classes. For amore detailed study we refer to [EN00, Sect. II.5.b] and the papers cited below.

Definition A.1. Let (T (t))t≥0 be a strongly continuous semigroup on a Banach space Xwith generator A. Then we define its Favard class as

FA1 :=

{x ∈ X : sup

t∈(0,1]

t−1 ·∥∥T (t)x− x

∥∥ <∞} ⊂ X,

which becomes a Banach space with respect to the norm

‖x‖FA1 := ‖x‖+ supt∈(0,1]

t−1 ·∥∥T (t)x− x

∥∥.We note that for reflexive Banach spaces X one always has FA

1 = D(A) (see [EN00,Cor. II.5.21]), hence Favard spaces are interesting only in nonreflexive spaces.

Similarly one can define the Favard space FA0 of the extrapolated semigroup (T−1(t))t≥0.

In this paper we deal with restrictions A of some “maximal operator” Am where the domainof A is given as the kernel of a boundary operator Q. The following result from [DS89,Thm. 9] is quite useful in order to check if D(Am) is contained in the Favard class of A.

Lemma A.2. With the notions introduced in our Abstract Framework 2.2 assume thatA ⊂ Am with domain D(A) := kerQ generates a strongly continuous semigroup on X.Then the following conditions are equivalent.

(a) D(Am) ⊂ FA1 .

(b) ker(λ− Am) ⊂ FA1 for some λ ∈ ρ(A).

(c) There exist γ > 0 and λ0 ∈ R such that ‖Qx‖ ≥ γλ·‖x‖ for all λ > λ0, x ∈ ker(λ−Am).(d) There exist c > 0 and w > 0 such that ‖Qλ‖ ≤ c · λ−1 for all λ > w.

The importance of the Favard class for our purposes stems from the regularizing effect ofthe variation of parameters formula. For its proof we refer to [NS94, Prop. 3.3].

Lemma A.3. If v ∈ L1loc(R+, F

A1 ), then∫ t

0

T (t− s)v(s) ds ∈ D(A) for all t ≥ 0.

16 K.-J. ENGEL, M. KRAMAR FIJAVZ, B. KLOSS, R. NAGEL, E. SIKOLYA

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Klaus-Jochen Engel, University of L’AquilaE-mail address: engel@mat.univaq.it

Marjeta Kramar Fijavz, University of LjubljanaE-mail address: marjeta.kramar@fgg.uni-lj.si

Bernd Kloss, University of TubingenE-mail address: bekl@fa.uni-tuebingen.de

Rainer Nagel, University of TubingenE-mail address: rana@fa.uni-tuebingen.de

Eszter Sikolya, Eotvos Lorand University BudapestE-mail address: seszter@cs.elte.hu