Jérôme Lohéac

Enrique Zuazua

Norm saturating property of time optimal controls for wave-type equations *

Keywords: Wave equations, Optimal control, Open loop control systems, Point-to-point control, Reachable states, Norm-optimal controls, Minimal control time

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Norm saturating property of time optimal controls for wave-type equations

Introduction

Time optimal control problems have been intensively studied for finite dimensional systems showing that the optimal control satisfies a Pontryagin maximum principle, it is bang-bang and unique. For a survey of these results, we refer to the books [START_REF] Lee | Foundations of optimal control theory[END_REF] and [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF] and to the original work by [START_REF] Bellman | On the "bang-bang" control problem[END_REF]. These results have been extended in [START_REF] Fattorini | Time-optimal control of solutions of operational differential equations[END_REF] to infinite dimensional systems and reported in the books by [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF] and [START_REF] Fattorini | Infinite dimensional linear control systems. The time optimal and norm optimal problems[END_REF].

Many new results have been obtained for parabolic type systems; see for instance [START_REF] Mizel | An abstract bang-bang principle and timeoptimal boundary control of the heat equation[END_REF], [START_REF] Wang | L ∞ -null controllability for the heat equation and its consequences for the time optimal control problem[END_REF], [START_REF] Phung | An observability estimate for parabolic equations from a measurable set in time and its applications[END_REF] and [START_REF] Kunisch | Time optimal control of the heat equation with pointwise control constraints[END_REF]. However only few results exist for conservative systems and they only concern distributed controls; see for instance [START_REF] Fattorini | The time optimal problem for distributed control of systems described by the wave equation. Control theory of systems governed by partial differential equations[END_REF], [START_REF] Lohéac | Maximum principle and bang-bang property of time optimal controls for Schrödinger-type systems[END_REF] and [START_REF] Kunisch | On time optimal control of the wave equation, its regularization and optimality system[END_REF][START_REF] Kunisch | On time optimal control of the wave equation and its numerical realization as parametric optimization problem[END_REF].

In all the above mentioned works, in order to ensure the existence of a time optimal control, the controls are assumed to be bounded in L ∞ . But for the wave equation, due to the finite velocity of propagation, the main difficulty arises from the fact that it is globally controllable only for a large enough control time.

In the present work, in order to analyse this delicate issue, we chose an Hilbertian approach and assume that the control is bounded in L 2 . This simplification allows us to easily consider the case of boundary control operators. In section 4, we consider the example of the string equation with Dirichlet boundary control, where some computations are explicit.

According to [START_REF] Gugat | L ∞ -norm minimal control of the wave equation: on the weakness of the bang-bang principle[END_REF] (Theorem 3.1), the string equation with Dirichlet boundary control cannot be controlled with classical bang-bang controls, i.e. controls taking their values in {-1, 1} for almost every time. In addition, for norm optimal control problems, which is a problem related to the one of finding time optimal controls as we will see later, [START_REF] Bennighof | Exact minimum-time control of a distributed system using a traveling wave formulation[END_REF] consider a string equation with Newman control at both ends and prove that for constant state targets (with constant initial data), the time optimal controls are of bang-off-bang type, i.e. controls taking values in {-1, 0, 1} for almost every time. Thus, even for constant data, time optimal controls are not, in general, of bang-bang form. A more general result on L ∞ -norm optimal controls, for the same system, can be found in [START_REF] Gugat | Analytic solutions of L ∞ optimal control problems for the wave equation[END_REF] and its generalisation to any L p -norm optimal controls in [START_REF] Gugat | Solutions of L p -norm-minimal control problems for the wave equation[END_REF].

In order to give a precise statement of our result, let us first recall some classical definitions and notations from control theory, see for instance [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF].

Throughout this paper, X and U are real Hilbert spaces identified with their duals. We denote by T = (T t ) t 0 a strongly continuous semigroup on X generated by an operator A : D(A) → X. In all this paper, we assume that A is skew-adjoint with nonempty resolvent ρ(A). The notation X 1 stands for D(A) equipped with the norm z X1 = (βId -A)z X , where β ∈ ρ(A) is fixed, while X -1 is the completion of X with respect to the norm z X-1 = (βId -A) -1 z X . Let us also denote by A and T the extensions of A to X and T to X -1 . Let us now introduce the control operator B ∈ L(U, X -1 ). Then the infinite dimensional system under consideration is:

y = Ay + Bu y(0) = y 0 , (1)
where y is the state of the system and u ∈ L 2 (R + , U ) is the control and y 0 ∈ X is the initial state position. The solution of (1) is:

y(t) = T t y 0 + Φ t u (t 0) , where Φ t ∈ L L 2 ([0, t], U ), X -1
is the input to state map defined by:

Φ t u = t 0 T t-s Bu(s) ds (t 0 , u ∈ L 2 ([0, t], U )) .
We will say that B is an admissible control operator for T if there exists t > 0 such that Ran Φ t ⊂ X and in the sequel we will assume that the pair (T, B) satisfies this condition. Finally, we will say that the pair (A, B) is exactly controllable in time T (T > 0) if Ran Φ T = X. In the sequel, we will assume that the pair (A, B) is exactly controllable in some time T > 0 and we define the universal controllability time:

T * = inf{T > 0 , Ran Φ T = X} 0 . (2)
To be more precise, the time optimal control problem we address in this work is the following: Problem 1. Given E > 0 and y 0 , y 1 ∈ X with y 0 = y 1 , find the minimal time T > 0 such that there exists u ∈ L 2 ([0, T ], U ) satisfying:

• u L 2 ([0,T ],U ) E;
• the solution y of (1) with control u and initial condition y 0 satisfies y(T ) = y 1 .

In all this note, E defines a given nonnegative constant. Our first result is as follows:

Theorem 1.1. Let y 0 , y 1 ∈ X with y 0 = y 1 .

Assume that the pair (A, B) is exactly controllable and fix T > 0. Assume that a control u ∈ L 2 ([0, T ], U ) with u L 2 ([0,T ],U ) E steering y 0 to y 1 in time T exists.

Then there exists a minimal time τ > 0 such that y 0 can be steered to y 1 in time τ = τ (y 0 , y 1 ; E) preserving this bound, i.e.

τ = min T > 0 , ∃u ∈ L 2 ([0, T ], U ) , u L 2 ([0,T ],U ) E and Φ T u = y 1 -T T y 0 } . (3) Moreover, if τ > T * (with T * 0 defined by (2)), there exists a unique control u ∈ L 2 ([0, τ ], U ) with u L 2 ([0,τ ],U ) E steering y 0 to y 1 in time τ . In addition, we have u L 2 ([0,τ ],U ) = E (4)
and there exists η ∈ X \ {0} such that:

u = Φ * τ η . (5)
Let us remind that for every η ∈ X,

(Φ * T η)(t) = B * z(t) (t ∈ [0, T ]) where z is solution of: z = -A * z , z(T ) = η .
The proof of the characterization of the optimal control Theorem 1.1 when τ > T * is similar to the one by [START_REF] Lohéac | Maximum principle and bang-bang property of time optimal controls for Schrödinger-type systems[END_REF]Theorem 1.4] and is not repeated here. We only give the key argument for the existence of τ in section 2, see Proposition 2.1.

Remark 1.1. If τ > T * the minimal time control is the minimal norm control in time τ steering y 0 to y 1 . That is to say that, if τ > T * , the time optimal control is L 2 ([0, τ ], U )-norm optimal. This fact gives the same result as the one in [START_REF] Wang | On the equivalence of minimal time and minimal norm controls for internally controlled heat equations[END_REF] for the heat equation, where we have T * = 0.

Theorem 1.1 does not give any relevant information when τ T * . In fact when τ T * the situation is less clear. In section 3, we show in Proposition 3.1, under suitable assumptions on the reachable set, that for τ < T * there exists a time optimal control u ∈ L 2 ([0, τ ], U ) with u L 2 ([0,τ ],U ) < E. That is to say that, when τ < T * , there exist time optimal controls which do not satisfy the norm saturating property (4). This situation appears at least when y 0 = 0 and it is a consequence of the following two properties of the reachable sets: They are closed and strictly increasing as a function of t < T * .

More precisely, the way we build a non saturation time optimal control is by choosing a target y 1 ∈ X so that y 1 is accessible from 0 in a time t > 0 but not for times s < t. In this case, it is clear that τ (0, y 1 ; E) t whatever E > 0 is. Choosing such a target y 1 and choosing a constant E > 0 large enough, we will obtain that τ (0, y 1 ; E) = t and the existence of a time optimal control whose norm is not E. In section 4, we will show that the assumptions made in Proposition 3.1 are fulfilled for the string equation with Dirichlet boundary control.

Well posedness

In this paragraph, we will prove that τ defined by (3) exists, i.e., the set

T > 0 , ∃u ∈ L 2 ([0, T ], U ) , u L 2 ([0,T ],U ) E and y 1 -T T y 0 = Φ T u admits a minimum.
Before going further, let us introduce some ad hoc notations and spaces. Let us define the set of points which can be reached from 0,

R 2 t = Φ t L 2 ([0, t], U ) (t > 0) , (6)
with the convention R 2 0 = {0}. Endowed with the norm:

y R 2 t = inf u L 2 ([0,T ],U ) , u ∈ L 2 ([0, T ], U ) , y = Φ t u , it constitutes a Banach space.
We also define the closed ball of radius E in R 2 t :

B 2 t (E) = y ∈ R 2 t , y R 2 t E (t > 0) .
Using these notations, we have:

τ (y 0 , y 1 ; E) = inf T > 0 , y 1 -T T y 0 ∈ B 2 T (E) . (7)
By linearity of Φ t , for every E > 0 and t > 0, B 2 t (E) is a convex set of X. Let us first give a basic property on the sets R 2 t . Lemma 2.1. Let 0 < s t, then we have

R 2 t = T s R 2 t-s + R 2 s ,
and the inclusions R 2 s ⊂ R 2 t ⊂ X are continuous. In addition, the universal time T * defined by ( 2) is given by

T * = inf T > 0 , R 2 T = X .
By a simple adaptation of [START_REF] Fattorini | Infinite dimensional linear control systems. The time optimal and norm optimal problems[END_REF] (Lemma 3.1.1 and Theorem 3.1.2), we obtain:

Proposition 2.1. Let y 0 , y 1 ∈ X with y 0 = y 1 and assume that a time T > 0 and a control u

∈ L 2 ([0, T ], U ) with u L 2 ([0,T ],U ) E steering y 0 to y 1 in time T exist. Then the set T > 0 , y 1 -T T y 0 ∈ B 2
T (E) admits a minimum τ . This result ensures the existence of a time optimal control u ∈ L 2 ([0, τ ], U ) whatever the minimal time τ > 0 is.

3 Lack of norm optimality when τ T * Proposition 3.1. Assume that T * , defined by (2), is positive. Let 0 < t T * and assume there exists y 1 ∈ R 2 t such that,

y 1 ∈ R 2 s (s ∈ (0, t)) . (8)
Then for every γ ∈ R * , the minimal time τ = τ (0, γy 1 ; E) (defined by (3)) needed to steer 0 to γy 1 is greater that t.

In addition, for |γ| > 0 small enough, we have τ (0, γy 1 ; E) = t and there exists a time optimal control u ∈

L 2 ([0, t], U ) such that u L 2 ([0, t],U ) < E. Proof. Since R 2
t is a vector space, it is clear that for every γ ∈ R, γy 1 ∈ R 2 t and, similarly, γy 1 ∈ R 2 s for every s ∈ (0, t). Consequently, τ (0, γy 1 ; E) t. Since y 1 ∈ R 2 t , for every ε > 0, there exists ūε

∈ L 2 ([0, t], U ) such that y 1 = Φt ūε and ūε L 2 ([0, t],U ) y 1 R 2 t + ε. For every γ ∈ R, it is obvious that γy 1 = Φt(γ ūε ) and γ ūε L 2 ([0, t],U ) γy 1 R 2 t + |γ|ε. Consequently, for 0 < |γ| < E y 1 R 2 t + ε , we found a control u = γ ūε ∈ L 2 ([0, t], U ) steering 0 to y 1 satisfying u L 2 ([0, t],U < E.
This ends the proof, since τ (0, γy 1 ; E) t.

Remark 3.1. With the assumptions of Proposition 3.1 and if, in addition, Ker Φt = {0}, then, for |γ| > 0 small enough, there exists an infinite number of time optimal controls steering 0 to γy 1 .

Let us also notice that with some closure property on R 2 t , there exists a target y 1 such that the assumption (8) holds.

Lemma 3.1. Assume that for every t > 0, R 2 t is a closed set in X and T * defined by (2) is nonnegative. Then, there exists y 1 ∈ X \ {0}, such that the set t > 0 , y 1 ∈ R 2 t admits a minimum.

Proof. For every y ∈ X\{0} and every t > 0, let us define f y (t) = sup

ϕ∈R 2 t y 1 , ϕ y 1 X ϕ X .
We have f y (t) ∈ [0, 1] and for every t > T * , f y(t) = 1. Thus, t y = inf {t > 0 , f y (t) = 1} exists.

By contradiction, assume that t y = 0, that is to say that for every t > 0 and every y ∈ X \ {0},

sup ϕ∈R 2 t y 1 , ϕ y 1 X ϕ X = 1
and hence, R 2 t is dense in X. Finally, using the closure property of R 2 t , we obtain R 2 t = X for every t > 0 and this is in contradiction with T * > 0. Consequently, there exists y 1 ∈ X \ {0} such that t y1 > 0. In particular, there exists ε 0 > 0 such that f y1 (ε 0 ) < 1 and, consequently, y 1 ∈ R 2 ε0 and since R 2 ε0 is a closed set, we can choose y 1 ∈ X \ {0} such that y 1 ∈ R 2 ε0 ⊥ . In addition, for every ε < ε 0 , using

R 2 ε ⊂ R 2 ε0 , we have y 1 ∈ R 2 ε ⊥ .
Due to the closure property, to end the proof, we only need to prove that f y1 (t y1 ) = 1. For every 0 < ε ε 0 , we have:

1 = f y1 (t y1 + ε) = sup ϕ∈R 2 ty 1 +ε ϕ, y 1 ϕ X y 1 X .
But, according to Lemma 2.1,

1 = f y1 (t y1 + ε) = sup ϕ1∈R 2 ty 1 ϕ0∈R 2 ε T ε ϕ 1 + ϕ 0 , y 1 T ε ϕ 1 + ϕ 0 X y 1 X = sup ϕ1∈R 2 ty 1 ϕ0∈R 2 ε T ε ϕ 1 , y 1 T ε ϕ 1 + ϕ 0 X y 1 X sup ϕ1∈R 2 ty 1 T ε ϕ 1 , y 1 T ε ϕ 1 X y 1 X sup ϕ1∈R 2 ty 1 ϕ 1 , T * ε y 1 ϕ 1 X y 1 X = f T * ε y1 (t y1 ) .
Using that lim ε→0 T * ε y 1 = y 1 , it is easy to see that lim

ε→0 f T * ε y1 (t y1 ) = f y1 (t y1
). Consequently, we have proved that f y1 (t y1 ) = 1 and this ends the proof.

In the next section, we show that the assumptions of Proposition 3.1 are fulfilled for the string equation with Dirichlet boundary control.

Time optimal control of the string equation

In this section, we focus on the string equation with Dirichlet boundary control:

ẅ(t, x) = ∂ 2 x w(t, x) (t > 0 , x ∈ (0, 1)) , (9a)
w(t, 0) = u(t) (t > 0) , (9b)
w(t, 1) = 0 (t > 0) . ( 9c
)
This system is suplemented with initial conditions:

w(0, x) = w 0 (x) and ẇ(0, x) = w 1 (x) (x ∈ (0, 1)) . (9d)
In this paragraph, we chose a Dirichlet boundary control at one end, but similar results could have been obtained with Newman or Dirichlet boundary control at one or both ends.

Using the formal representation introduced in section 1, see also § 10.3 by [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF], we have w(t) ẇ(t) = T t w 0 w 1 + Φ t u with the operator A defined by:

A : D(A) → X := L 2 (0, 1) × H -1 (0, 1) f g → 0 Id -A 0 0 f g , with D(A) = H 1 0 (0, 1)×L 2 (0, 1) ,
where A 0 is the operator with values in L 2 (0, 1) defined by:

D(A 0 ) = H 2 (0, 1) ∩ H 1 0 (0, 1) and A 0 f = - d 2 dx 2 f .
According to [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF], see § 11.6.2, A can be seen has a skew-adjoint operator on X.

The control operator B is given by its adjoint:

B * : D(A) → R ϕ ψ → d dx A -1 0 ψ x=0 .
For T > 0, let us define:

(Ψ T z)(t) = B * T t z (t ∈ [0, T ] , z ∈ D(A)) .
We have [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Proposition 4.4.1],

Ψ * T = Φ T R T , (10)
with R T ∈ L(L 2 (0, T )) defined by R T u(t) = u(T -t).

For every z = z 1 z 2 ∈ D(A) = H 1 0 (0, 1) × L 2 (0, 1), we have:

(Ψ T z)(t) = d dx ζ(t) x=0 , ( 11
)
where ζ is solution of:

ζ(t, x) = ∂ 2 x ζ(t, x) ((t, x) ∈ (0, T ) × (0, 1)) , (12a)
ζ(t, 0) = ζ(t, 1) = 0 (t ∈ [0, T ]) , (12b)
ζ(0, •) = A -1 0 z 2 ∈ H 1 0 (0, 1) and ζ(0, •) = -z 1 ∈ L 2 (0, 1) . ( 13
)
Decomposing any solution of ( 12) in Fourier series and using Ingham inequalities (see [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] and, for result related to control theory, see for instance [START_REF] Jaffard | Estimates of the constants in generalized Ingham's inequality and applications to the control of the wave equation[END_REF] and § 8.1 of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]), we obtain that the pair (A * , B * ) is exactly observable in any time T > T * = 2, i.e., there exists c(T ) > 0 such that:

c(T ) Ψ T z 2 L 2 (0,T ) z 2 X (z ∈ X) .
In addition, the observability constant c(T ) is of order 1 T when T goes to infinity. Thus for every T > 2, using a HUM control, there exists a control with L 2 (0, T )norm of order 1 √ T when T goes to infinity [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF]Remark 2.3]. We sum up the above remarks in the following lemma: Lemma 4.1. For every y 0 , y 1 ∈ X and E > 0, there exists T > 0 large enough such that y 1 -T T y 0 ∈ B 2

T (E).

The above lemma combined with Proposition 2.1 ensures that a time optimal control steering y 0 to y 1 exists.

For this particular control system, with the help of D'Alembert formula, it is possible to compute the reachable space R 2 T for all times T (including T T * = 2). To this end, let us first compute the kernel of Ψ T . Lemma 4.2.

• If T ∈ (0, 1), Ker Ψ T = z 1 z 2 ∈ L 2 (0, 1) × H -1 (0, 1) , 1 [0,T ] z 1 (x) -d dx A -1 0 z 2 = 0 ; • If T ∈ [1, 2), Ker Ψ T = d dx A -1 0 z 2 z 2 , z 2 ∈ H -1 (0, 1) , z 2 1 [2-T,1] = 0 ; • If T ∈ [2, ∞), Ker Ψ T = {0}. Proof. Ker Ψ T is the set of points (-ζ 2 , A 0 ζ 1 ) ∈ L 2 (0, 1) × H -1 (0, 1), where (ζ 1 , ζ 2 ) ∈ H 1 0 (0, 1) × L 2 (0, 1) are such that the solution ζ of (12), with initial conditions ζ(0, •) = ζ 1 and ζ(0, •) = ζ 2 , satisfies ∂ x ζ(t, •)| x=0 = 0 (t ∈ [0, T ] a.e.) .
It is natural to extend ζ 1 and ζ 2 as odd functions on [-1, 1] and then as 2-periodic functions on R. Consequently, using D'Alembert formula, it follows that ζ(t, x) is given by:

ζ(t, x) = 1 2 ζ 1 (x + t) + ζ 1 (x -t) + x+t x-t ζ 2 (s) ds .
From the above formula, we obtain:

∂ x ζ(t, •)| x=0 = ζ 1 (t) + ζ 2 (t) . (14)
Let us now compute the set

K T of points (ζ 1 , ζ 2 ) such that ∂ x ζ(t, •)| x=0 = 0 for almost every t ∈ [0, T ] for different values of T .
• If T ∈ (0, 1),

K T = (ζ 1 , ζ 2 ) ∈ H 1 0 (0, 1) × L 2 (0, 1) , ∀x ∈ [0, T ], ζ 2 (x) = -ζ 1 (x) ; • If T ∈ [1, 2), set T = 1 + θ, then (ζ 1 , ζ 2 ) ∈ K if
ζ i (1 + s) = -ζ i (1 -s) (i ∈ {1, 2}) .
This implies that:

K T = (ζ 1 , -ζ 1 ) , ζ 1 ∈ H 1 0 (0, 1) , ∀x ∈ [2 -T, 1], ζ 1 (x) = 0 ; • If T 2, K T = {0}.
From K T , we easily obtain Ker Ψ T . This ends the proof.

By duality, the reachable state R 2 T is: Proposition 4.1.

• If T ∈ (0, 1], R 2 T = f g ∈ L 2 (0, 1) × H -1 (0, 1) , 1 [T,1] f = 1 [T,1] d dx A -1 0 g = 0 , 1 [0,T ] f - d dx A -1 0 g = 0 . • If T ∈ (1, 2], R 2 T = f g ∈ L 2 (0, 1) × H -1 (0, 1) , 1 [0,2-T ] f - d dx A -1 0 g = 0 .
• If T > 2, R 2 T = X = L 2 (0, 1) × H -1 (0, 1). Proof. According to [START_REF] Gugat | Solutions of L p -norm-minimal control problems for the wave equation[END_REF], R 2 t = Ran Φ t is equal to Ran Ψ * t (since R t is a unitary operator). If Ψ T is coercive on X/Ker Ψ T (with the quotient norm) then,

Ran Ψ * T = Ker Ψ T ⊥ .
Let us then show that Ψ T is coercive on X/Ker Ψ T .

To this end, define the two semi-norms on X = L 2 (0, 1) × H -1 (0, 1):

N 1 (z 1 , z 2 ) = z 1 + d dx A -1 0 z 2 L 2 (0,1) and N 2 (z 1 , z 2 ) = z 1 -d dx A -1 0 z 2 L 2 (0,1) ((z 1 , z 2 ) ∈ X) .
Clearly, for every (z 1 , z 2 ) ∈ X,

N 1 (z 1 , z 2 ) 2 + N 2 (z 1 , z 2 ) 2 = 2 z 1 2 L 2 (0,1) + z 2 H -1 (0,1) = 2 (z 1 , z 2 ) 2 X .
For every z = z 1 z 2 ∈ X, using ( 11) and ( 14), we have:

Ψ T z 2 L 2 (0,T ) = T 0 |ζ 1 (x) + ζ 2 (x)| 2 dx , with ζ 1 (resp. ζ 2 ) defined by ζ 1 = A -1 0 z 2 (resp. ζ 2 = -z 1
) on (0, 1) and extend as an odd function on [-1, 1] and then extended by 2-periodicity on R. Thus,

• if T ∈ (0, 1], we have, Ψ T z 2 L 2 (0,T ) = d dx A -1 0 z 2 -z 1 2 L 2 (0,T ) and 2 z 2 X/Ker Φ T = 2 inf z∈Ker Φ T z + z 2 X = inf z∈Ker Φ T z 1 + z1 + d dx A -1 0 (z 2 + z2 2 L 2 (0,1) + z 1 + z1 -d dx A -1 0 (z 2 + z2 2 L 2 (0,1)
= inf z1∈L 2 (0,T )

2z 1 + z 1 + d dx A -1 0 z 2 2 L 2 (0,T ) + z 1 -d dx A -1 0 z 2 2 L 2 (0,T ) = Ψ T z 2 L 2 (0,T ) . • if T ∈ (1, 2], we have, Ψ T z 2 L 2 (0,T ) = 1 0 |ζ 1 (x) + ζ 2 (x)| 2 dx + T 1 |ζ 1 (x) + ζ 2 (x)| 2 dx = 1 0 |ζ 1 (x) + ζ 2 (x)| 2 dx + 1 2-T |ζ 1 (x) -ζ 2 (x)| 2 dx , that is to say, Ψ T z 2 L 2 (0,T ) = z 1 -d dx A -1 0 z 2 L 2 (0,1) 2 + z 1 + d dx A -1 0 z 2 2 L 2 (2-T,1)
and, as for the previous item, we obtain:

2 z 2 X/Ker Φ T = Ψ T z 2 L 2 (0,T ) . • if T > 2, we have Ker Ψ T = {0} and Ψ T z 2 L 2 (0,T ) Ψ T z 2 L 2 (0,2) = 2 z 2 X .
All in all, we have proved that, for every T > 0,

Ψ T z L 2 (0,T ) 2 z X/Ker Ψ T (z ∈ X) .
Consequently, R 2 T = (Ker Ψ T ) ⊥ and a straight forward computation leads to the result. Consequently, by considering for instance the target

y 1 = (-2x + 1)1 [0, t] (x) 2 ,
with t ∈ (0, 1), we have y 1 ∈ R 2 2-t and y 1 ∈ R 2 s for every s < 2 -t. Thus, using Proposition 3.1, there exists γ = 0, such that the minimal time τ (0, γy 1 ; E) needed to steer 0 to γy 1 is 2-t and in addition, there exists a control ū ∈ L 2 (0, 2 -t) steering 0 to γy 1 in time t together with ū L 2 (0,2-t) < E.

Concluding remarks

In this note we proved, on the example of the string equation with Dirichlet boundary control, that there exist time optimal controls with L 2 -norm strictly lower than E. More generally, if, τ > T * , time optimal controls are L 2 ([0, τ ], U )-norm optimal controls, but when τ < T * , time optimal controls and L 2 ([0, τ ], U )-norm optimal controls can differ. In particular, in the situation of Remark 3.1, there is an infinite number of time optimal controls which are not norm optimal ones. This result is due to the fact that the reachable space is not the full space in arbitrary small time. This is a great difference between Schrödinger and wave systems from the point of view of time optimal controllability. For the first one exact controllability holds in all time and, therefore, the reachable space is the full space for all time and, in particular, time independent, something that does not occur for the wave equation.

One important question which is not addressed here, and that constitutes still and interesting open problem, is the effective numerical computation of the minimal control time and of time optimal controls. In fact, this issue seems to be highly complex.

For instance for the string equation with boundary control, in order to compute time optimal controls, we should design a numerical method able to build controls even for times t < T * and, as far as we know, this has not been done so far.

Indeed, there has been extensive work done on the development of efficient numerical methods for approximating wave control problems. But this has been always done beyond the universal control time T * , i.e. for time intervals [0, T ] with T > T * . In fact, the existing methods try to mimic the behavior of the continuous wave equation and, in particular, its finite velocity of propagation property and, therefore, do not allow yielding any effective approximation method to build controls, specific to given data, when the control time is strictly smaller than T * . A survey of these methods can be found in [START_REF] Ervedoza | Numerical approximation of exact controls for waves[END_REF]. Another approach, based on Russell's principle,"stabilization implies controllability", can be found in [START_REF] Cîndea | An approximation method for exact controls of vibrating systems[END_REF]. But this method also needs the control time to be larger than the universal controllability time T * .

In addition, with the result of Proposition 4.1, we have a lower bound on the minimal time needed to steer 0 to a given final target. But we do not have a lower bound on the minimal time needed to steer a non null starting point to a given target. Further developments in this direction could be derived out of Theorem 2.1 of [START_REF] Gugat | L ∞ -norm minimal control of the wave equation: on the weakness of the bang-bang principle[END_REF].

  and only if their extensions satisfy ζ 2 (s) = -ζ 1 (s) for almost every s ∈ (0, 1 + θ). But, for almost every s ∈ [0, θ],

Acknowledgements

The authors thank M. Tucsnak for valuable discussions.

* This work was partially supported by Grants FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR and the MTM2014-52347 Grants of the MINECO (Spain). CNRS 6597 (Institut de Recherche en Communica-


Guillaume Savelli
email: guillaume.savelli@cea.fr

Sergio Silveira Stein

Guillaume Bernard-Granger

Pascal Faucherand

Laurent Montès

Stefan Dilhaire

Gilles Pernot

Mikko Ruoho

Taneli Juntunen

Ilkka Tittonen

Gabi Schierning

Ruben Chavez

Roland Schmechel

Titanium-based silicide quantum dot superlattices for thermoelectrics applications

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Thermoelectricity is one of the most promising research fields for future applications of clean energy generation. Thermoelectric systems can convert thermal energy into electrical energy for energy-harvesting applications (the Seebeck effect), or reciprocally for cooling applications (the Peltier effect). Thermoelectric materials are evaluated by their figure of merit, ZT, which is defined in equation (1), where S is the Seebeck coefficient, λ is the thermal conductivity, ρ is the electrical resistivity, and T is the temperature.

(

)

ZT S T ( ) (1)
2

ρ λ = × ×
A high ZT value directly captures the efficiency of a material used as a thermoelectric device.

Two of the most widespread thin film materials currently used are Bi 2 Te 3 and PbTe, both of which include rare or toxic elements [1,2].

An environmentally clean alternative to these materials is a doped SiGe film with low concentrations of Ge. This material has several advantages such as being abundant, relatively low in cost, well studied, and compatible with the microelectronics industry, but it exhibits poor thermoelectric performance at room temperature.

Among the different strategies used to improve the figure of merit of the materials, one of the most encouraging is the use of nanostructure engineering [3].

Nanostructuration of materials could improve ZT materials by increasing the power factor (defined as S 2 /ρ) and/or reducing the thermal conductivity.

At the short scale, SiGe-based nanostructured materials have already shown a significant decrease in thermal conductivity compared to similar SiGe thin films, Si/SiGe superlattices, and Si x Ge 1-x /Si y Ge 1-y quantum dot superlattices (QDSLs) [4 6]. This is mainly due to a reduction of the crystal's lattice thermal conductivity, which is obtained by adding phonon diffusion mechanisms as layer interfaces and nanodots.

It has recently been shown that the inclusion of metallic silicide nanodots (like TiSi 2 ) in a SiGe matrix could theoretically lead to an up to fourfold reduction of the material's thermal conductivity without altering its electrical conductivity or its Seebeck coefficient [7]. In their study, these authors proposed silicide dots with diameters and thicknesses in the 5 50 nm and 40 50 nm ranges, respectively, with a 3% quantum dot (QD) density. This result was recently experimentally demonstrated at the bulk scale, but it has never been demonstrated at the thin film scale [8] because these nanostructured thin film materials have never been grown or presented in the literature.

In this paper, we present the first instance of the growth of QDSL thin films based on titanium silicides inside doped SiGe mono-and polycrystalline matrixes, as well as a complete study of their chemical structures and thermoelectric (TE) properties.

QDSL growth mechanism

This section describes the QDSL growth mechanism, which is illustrated in figure 1. The deposition of QDSL by reducedpressure chemical vapor deposition (CVD) can be divided into three steps. The first step is the deposition of Ti nanoislands onto a SiGe thin film. In the second step, the nanoislands are embedded by the deposition of a SiGe thin film, simultaneously forming QDs by the reaction between the Ti atoms and the Si atoms. Finally, these steps are repeated to produce a QDSL.

In this paper, we differentiated the expression 'nanoisland' from the term 'QD', in the sense that the former is related to the first steps of the growth, where the deposited Ti atoms form quasiplanar nanometric structures. Once they react with the Si atoms and form silicides, spherical crystalline structures called QDs are also formed.

All QDSL and reference samples are grown in a modified Epi-Centura reduced-pressure CVD industrial cluster tool from Applied Materials. Two different substrates are used. The polycrystalline samples are grown on 100 mm, lightly doped Si wafers where a 200 nm SiO 2 insulating layer was previously grown by plasma-enhanced CVD. The monocrystalline samples are epitaxially grown on lightly doped silicon on insulator (SOI) wafers.

This technique has already been used to successfully grow nanostructured silicon-based thin films [4,[START_REF] Savelli | AIP Conf. Proc[END_REF][START_REF] Hauser | [END_REF].

The precursor chosen for the Ti-based QD growth is TiCl 4 , which is liquid at ambient temperatures. A specific, adapted apparatus allowing the controlled recovery of the precursor vapor was built and coupled to the Centura tool. This apparatus has a temperature and pressure control system, and H 2 is employed as the carrier gas to send the gaseous portion of the precursor to the reaction chamber.

The SiGe layers are grown in the same chamber using SiH 4 and GeH 4 as precursors. The samples were doped in situ using PH 3 gas. The Ge content of the QDSL SiGe layers is fixed at around 13%. As shown in figure 1, to create a QDSL, a doped SiGe layer is first grown on the substrate. After that, the SiH 4 , GeH 4 , and PH 3 gas flows are stopped and the TiCl 4rich H 2 is sent to the chamber for different periods of time to produce the Ti-based QDs. Then, another doped SiGe layer is grown to embed the nanoparticles, and so on. Deposition temperature and pressure are 800 °C and 10 Torr, respectively. The total thickness of the QDSL and the reference sample is 1 μm. The QDSLs are composed of 16 alternating QD and SiGe layers. A 200 nm SiGe buffer layer is deposited before the QDSL. This is done to reduce potential dislocations due to the lattice mismatch between the SiGe thin films and the Si substrate, to lower the interdiffusion between the Ge atoms from the QDSL and the Si atoms from the substrate, and to maintain a constant Ge content trough in all SiGe layers of the QDSL.

QDSL structural analyses

The QDSL structures are analyzed by transmission electronic microcopy (TEM). These TEM analyses are performed at 200 kV, including high-angle annular dark field (HAADF) images in the scanning mode (STEM), which allow us to acquire both high-resolution images and chemical mapping using the energy-dispersive x-ray spectroscopy (EDS) technique. The samples are prepared using a tripod polishing technique. These features allow us to acquire three types of images. The first one consists of global cross-plane views of the QDSL, which permit us to measure the QDs' diameters, each layer's thickness, and the total thickness of the samples. The second mode used involves high-resolution images, which allow us to analyze individual QDs and the distance between crystallographic planes, and to determine the phase of the particle. The third class of TEM consists of the chemical mappings, which are obtained with the EDS technique coupled with the STEM mode. This technique is based on x-ray emissions due to the relaxation of excited electrons, which assume discrete and particular energy values for each element of the periodic table. The obtained chemical mapping produces high-spatial-resolution images of the position of the different elements forming the material, estimates the Ge content of each samples, and confirms the phase of the QDs.

The results of these analyses are presented for both mono-and polycrystalline QDSLs. At the end of this section, we compare the results and the growth of each sample.

Monocrystalline sample

The overall view of the monocrystalline sample is presented in figure 2. An initial observation is that the particles of the lower layers appear to be smaller than the particles of the upper layers. This occurs because of the high mobility of the dots inside the SiGe matrix. When we embed the nanoislands with the successive SiGe monolayers (from step (a) to step (b) in figure 1), a large percentage of the particles migrates to the surface of the sample. Thus, during the stacking of the layers, the same mechanism occurs, but instead of migrating up to the surface, the QDs coalesce inside the material, resulting in particles with larger diameters.

Surface images of the QDSL show no QDs on the surface, so one can conclude that when the QDs reach a critical diameter, their mobility inside the SiGe matrix is lowered and they remain inside the material. In figure 2 Figure 3 presents a high-resolution image of a single QD inside the monocrystalline matrix. The measure of the plane distances result from d p ≈ 8.21 Å, which corresponds to the (100) TiSi 2 planes.

The EDS chemical mapping of the sample is presented in figure 4. Two main observations can be deduced from these images. First, looking at the Ti signal in figure 4(d), one can easily see that the QDs are rich in Ti atoms, confirming the measurements discussed above. The second important piece of information is obtained by analyzing the dopant (phosphorus) and the signal (figure 4(e)). Due to a chemical affinity between the two materials, an accumulation of phosphorus is observed in the zones corresponding to the QD. A similar result was observed in [8]. In this reference, however, the inclusions were Mo-based QDs. A punctual EDS analysis inside the SiGe embedding layer resulted in a Ge content of 12.7% (atomic fraction). The same analysis was performed inside a QD where a Ti content of 31% was obtained, confirming that the QD phase is TiSi 2 . Comparing figures 4(a), (d), and (e), a dopant accumulation near the QD is observed. This accumulation, which was previously reported in the literature for bulk phosphorous-doped SiGe samples with Mobased inclusions [8], stems from the high affinity between the metal silicide and the dopant atoms, which causes a precipitation around the QD.

It has also been shown that this effect has a huge impact on the material's power factor [9]. The authors observed an increase in the material's electrical conductivity due to an increase in the carriers' mobility, without any changes in the Seebeck coefficient. This effect, called modulation doping, is obtained by spatially separating carriers from their parent atoms.

Polycrystalline sample

Considering the polycrystalline n-doped Ti-based QDSL, a global cross-sectional TEM image is presented in figure 5.

Comparing the two distinct zones, '1' and '2', in figure 5(a), there is a difference in the QD density between the two zones, with zone 2 being richer in QDs than zone 1. Zone 2 corresponds to an individual grain, and the particles tend to agglomerate in the grain boundaries. The underlying reason for the preferential growth near the grain boundaries is that these highly disordered sites lower the energy for the nucleation of the Ti phase. By statistically measuring the particle diameter and the thickness of the layers of this sample, a mean QD diameter of d = 42 nm, a thickness of e = 54 nm, and a QD density of 2.7% are obtained. Those values are in good agreement with the theoretical optimized values (see section 1).

As previously stated, one can see that the particles of the lower layers corresponding to the first stages of the QDSL growth are smaller than the upper QDs. These two features indicate that a coalescence effect occurs in the polycrystalline sample during the growth, but at a smaller scale than in the monocrystalline sample. The mean diameter in this case (42 nm) is smaller than that of the monocrystalline QDSL (66 nm). As a result, the grain boundaries act as a barrier for the diffusion of the QDs inside the matrix, reducing the coalescence effect.

Figure 6 presents a high-resolution image of an individual QD inside the polycrystalline sample. In this image, two sets of planes are identified, corresponding to the distances of the (110) and (210) planes of the Ti 3 Si tetragonal phase. A different phase is obtained when a polycrystalline material is grown. In this case, the obtained phase (Ti 3 Si) is much richer in Ti than the phase (TiSi 2 ) of the monocrystalline sample, suggesting that the reaction between Si and Ti is slower.

EDS chemical mapping of this sample is presented in figure 7. The information obtained from these analyses is very similar to the information obtained for the monocrystalline sample (i.e., the good correspondence between the quantum dots, the Ti, and the P dopant signals). Quantitative EDS analysis of the matrix provides a Ge content of 12%, which is similar to the monocrystalline sample. By performing a quantitative analysis of a single QD, a Ti content of 70% was obtained, which is close to the value corresponding to the Ti 3 Si phase (Ti = 75%).

Finally, by comparing figures 7(a), (d), and (e), the same dopant accumulation near the QD is observed, which also proves a modulation doping effect.

QDSL thermoelectric characterizations

In this section, we present the measurement of the QDSL thermoelectric power factor and thermal conductivity.

Characterization methods

The thermoelectric power factor is usually defined as the product of the electrical conductivity and the Seebeck coefficient square. An ULVAC ZEM-3 device is used to measure both parameters at temperatures ranging from room temperature to 200 °C.

Usually this device is used to measure the TE properties of bulk materials. For thin-film samples, one must adapt their structures to be compatible with this method. A schematic representation of the system is shown in figure 8.

As a first step, we evaporate two rectangular strips of Al/ Au onto the thin film's surface. The 200 nm Al metal deposition by evaporation is chosen to provide good ohmic contact with the SiGe-based material. An additional 100 nmthick Au layer is deposited using the same equipment to protect the surface against oxidation.

Then, the sample is mounted between the electrodes (figure 8(a)). The upper electrode serves as a heating element. A polyimide electrical insulating film is inserted between the sample edge and the electrode, and a copper wire is positioned to electrically connect the electrodes and the Al/Au strips on the sample's surface. The Si (or SOI) substrate does not interfere with the electrical measurements, and the electrical contacts are established only between the electrodes and the thin film.

The equipment works simultaneously in two modes. The first mode measures the electrical conductivity, and the second measures the Seebeck coefficient. The first mode is represented in figure 8(b) and can be compared to a four-point probe method. An electrical current, I, is imposed through the sample and the resulting voltage, V, is measured using the thermocouples as electrodes. Using Ohm's law, the thin film's electrical resistance is obtained. The electrical conductivity is then calculated using the sample length l, the width w, and the thin film's thickness e, according to equation (2):

l I w e V ( )/( ). (2) σ = × × ×
The Seebeck coefficient-measurement setup is presented in figure 8(c). The thermocouples in the center of the sample are used to measure the temperature difference, ΔT. Then the driver software switches the electrode mode to measure the voltage created by the temperature gradient. The carriers' concentration and mobility are measured using a Hall effect method.

Note that these techniques measure the properties along the in-plane direction (i.e., parallel to the thin film's surface).

The thermal conductivity of the samples was measured at room temperature by time-domain thermoreflectance (TDTR). This optical technique consists of exciting the surface of the samples with a femtosecond laser pulse, creating a transient heat source. Another laser pulse monitors the variation of the sample's reflectivity. Prior to any measurement, the samples are capped with a 70 nm aluminum thin film that acts as a transducer. Then, by using a three-dimensional heat-diffusion model, it is possible to fit the experimental data and extract the material's thermal conductivity [10,11]. Unlike the electrical conductivity and the power factor, the TDTR technique measures the cross-plane thermal conductivity component of thin films.

Note that an SiGe reference sample has been grown with the same Ge content (13%) and electrical conductivity as the corresponding mono-and polycrystalline QDSLs. These references are made to compare and evaluate the role of the QDs' inclusion on the material's Seebeck coefficient and thermal conductivity.

Thermoelectric parameters

The evolution of the Seebeck coefficient, electrical resistivity, and power factor of the QDSLs and their corresponding SiGe reference samples as a function of the temperature are presented in figures 9 and 10 for the monocrystalline and polycrystalline structures, respectively. Table 1 presents the carriers' concentration, n, mobility, μ, and thermal conductivity, λ, of the monocrystalline and polycrystalline QDSLs and their SiGe references at room temperature. In figures 9(a) and 10(a), the behavior of the electrical resistivity is, as expected, the same for the QDSL and the reference sample. For the monocrystalline samples, the electrical resistivity values increase with the temperature. The main reason for this is the reduction of the charge carrier's mobility at higher temperatures due to the increase of carrier's scattering by phonons, which is coherent with the metallic characteristic of these high-doped semiconductors. For the polycrystalline samples, the electrical resistivity exhibits tiny changes with the temperature. In that case, the scattering on grain boundaries is the dominant scattering effect [4].

Moreover, as shown in table 1, this effect is higher than any other scattering mode, so the observed mobility values are much lower (18 and 13 cm 2 V -1 s -1 ) for the polycrystalline materials than for the monocrystalline materials (73 and 56 cm 2 V -1 s -1 ). As grain boundaries can act as trapping sites for the carriers, their accumulation results in the creation of a potential energy barrier for the carrier transport [12]. By increasing the material's temperature, more energy is available for the carriers to cross the potential barrier, and the mobility increases.

Moreover, the values of the room-temperature electrical resistivity are considerably higher for the polycrystalline samples (around 1 × 10 -2 Ω cm) compared to the monocrystalline samples (around 1 × 10 -3 Ω cm), due to the reduction of the electronic carriers' mobility (see table 1).

A considerable increase in the Seebeck coefficient is observed for the QDSL samples. For the monocrystalline samples, the Seebeck coefficient is around 100 μV K -1 and 190 μV K -1 at 50 °C for the reference and the QDSL, respectively, resulting in a 90% increase with the inclusion of the QDs. For the polycrystalline samples, the Seebeck coefficient is around 130 μV K -1 and 230 μV K -1 at 50 °C for the reference and the QDSL, respectively, which is a 77% increase as a result of the inclusion of QDs.

The increase in the Seebeck coefficient, combined with a similar electrical resistivity, results in a large increase in the QDSLs' power factor, as shown in figures 9(b) and 10(b). These results clearly demonstrate the influence of the inclusion of Ti-based QDs on the power factor's improvement.

Moreover, the increase in the power factor is obtained here by an increase in the Seebeck coefficient and a stable electrical resistivity, which is the opposite of how it was obtained in the literature [9]. But as explained previously, our references samples have been voluntarily made with the same electrical conductivity as the QDSL to evaluate the effect of including QDs on the material's Seebeck coefficient. This prevents a strict comparison between these works.

Thermal conductivities obtained by TDTR are also presented in table 1. For the monocrystalline samples, a decrease in thermal conductivity is obtained for the QDSL (6.8 W m -1 K -1 ) compared to the reference (8.5 W m -1 K -1 ). This result shows the effect of nanostructuration by including silicide QDs in the reduction of the material's thermal conductivity. Reference values of 6.1 W m -1 K -1 for a monocrystalline Si 0.85 Ge 0.15 thin film and about 8 W m -1 K -1 for a Si 0.5 Ge 0.5 monocrystalline bulk sample are in good agreement with the values reported in the literature [4,7].

The main difference in the behavior of the mono-and polycrystalline QDSLs can be seen in the thermal conductivity values. In the case of the polycrystalline samples, no change is observed when QDs are included inside the SiGe matrix because the grain boundaries are the dominant mechanism for phonon diffusion, masking the contribution of the phonons' diffusion with the QDs. The thermal conductivity values of the mono-and polycrystalline reference samples are 8.5 and 4.2 W m -1 K -1 , respectively. By taking into account that both SiGe references have a aimilar Ge content and similar carrier concentration, the explanation of the grain boundary effect acting as the main and dominating diffusion mechanism is demonstrated.

As power factors have been measured in the in-plane direction while thermal conductivities have been measured in the cross-plane direction, it is not possible to calculate the ZT values of these anisotropic QDSLs. Nevertheless, a significant (cm 2 V -1 s -1 ) (Wm -1 K -1 ) (10 19 cm -3 ) (cm 2 V -1 s -1 ) (Wm -1 K -1 ) QDSL 3.2 ± 0.2 73 ± 5 6.8 ± 0. improvement in both the power factors and thermal conductivities has been demonstrated, even in different directions, proving the beneficial influence of nanostructuration on the thermoelectric properties.

Conclusions

We present here the first instance of the growth of n-doped monocrystalline and polycrystalline QDSLs using the CVD technique. TEM analyses have confirmed the presence of Ti-based silicide inclusions inside the matrix, and have shown the specific behavior of the QDs during the growth, with a coalescence phenomenon, for the polycrystalline QDSLs.

Additionally, both the mono-and polycrystalline QDSLs exhibit a significant increase in their power factors after the inclusion of the QDs up to four times higher for the monocrystalline sample compared to the SiGe reference, and 2.3 times higher for the polycrystalline sample. Finally, the monocrystalline QDSLs also present a 20% reduction in their thermal conductivity due to the inclusion of QDs.

These results show that a great improvement of the TE materials' properties is obtained by including QDs in an SiGe matrix, making these materials very promising for energyharvesting or cooling applications.

Figure 1 .

 1 Figure 1. Schematic drawing of the successive steps to grow a QDSL by CVD. (a) Metal nanoislands are grown on a SiGe thin film. (b) A SiGe layer is deposited to embed the islands and form the QDs. Steps (a) and (b) are repeated to produce a QDSL, represented in (c).

  (b), the QD diameter, d, and the layer thickness, e, are measured. A statistical measurement of this image provides the mean values of d = 66 nm, e = 70 nm, and a 2.3% QD density, which are close to the theoretical optimized values (see section 1).

Figure 2 .

 2 Figure 2. Cross sectional STEM/HAADF images of the monocrystalline sample. (a) A global view of the QDSL showing the Ti based QD, and (b) a higher magnification showing a QD with diameter, d = 66 nm, and layer thickness, e = 65 nm.

Figure 3 .

 3 Figure 3. High resolution STEM/HAADF image of a QD inside the monocrystalline sample. The plane distances correspond to the (100) planes of the TiSi 2 phase.

Figure 4 .

 4 Figure 4. EDS chemical mapping of the monocrystalline sample: (a) STEM/HAADF image, (b) silicon signal, (c) germanium signal, (d) titanium signal, and (e) phosphorus signal. Note the similarity between the QDs, the titanium, and the phosphorus signals.

Figure 5 .

 5 Figure 5. Cross sectional STEM/HAADF image of the polycrystalline sample. (a) A global view of the QDSL, showing the Ti based QD and zones 1 and 2 with different QD densities. (b) A higher magnification of the image, showing a SiGe grain and a QD with diameter, d = 44 nm, and layer thickness, e = 50 nm.

Figure 6 .

 6 Figure 6. High resolution TEM image of a QD inside the polycrystalline sample. The plane distances correspond to the (110) and (210) planes of the Ti 3 Si tetragonal phase.

Figure 7 .

 7 Figure 7. EDS chemical mapping of the polycrystalline sample: (a) STEM/HAADF image, (b) silicon signal, (c) germanium signal, (d) phosphorus signal, and (e) titanium signal. Note the accumulation of the phosphorus dopant in the QD zone.

8 .

 8 Representation of the ZEM 3 measurement system, adapted to characterize the thin films. (a) Schematic view of the sample inside the system. (b) Representation of the electrical conductivity measurement. (c) Representation of the Seebeck coefficient measurement.

Figure 9 .

 9 Figure 9. Seebeck coefficient, electrical resistivity (a), and power factor (b) of the monocrystalline QDSL (blue color) and the corresponding SiGe reference (red color) as a function of the temperature.

Figure 10 .

 10 Figure 10. Seebeck coefficient, electrical resistivity (a), and power factor (b) of the polycrystalline QDSL (blue color) and the corresponding SiGe reference (red color) as a function of the temperature.

Table 1 .

 1 Comparison of the carriers' concentration, n, mobility, μ, and thermal conductivity, λ, between the monocrystalline and polycrystalline QDSLs and their corresponding SiGe references at room temperature.


Keywords: mobile network, ad hoc network, network architecture, address configuration, routing, IP networks, NEMO, MANET réseaux mobiles, réseaux ad-hoc, architecture de réseau, configuration d'adresses, protocole de routage, réseaux IP, NEMO, MANET

details the consequences of this misperception -breaking compatibility with existing applications and protocols -and offers an architectural model for MANETs, which integrates MANETs into the IP networking architecture and encapsulates the MANET specific behavior in a way transparent to existing applications and protocols. Finally, this memorandum shows how the presented MANET architectural model fits with MANET deployment scenarios, including "simple MANETs" and management of nested NEMO networks.

Introduction & Background

A typical text on Mobile Ad hoc NETworks (MANETS) will describe such networks as simply being "a collection of mobile nodes, communicating among themselves over wireless links and thereby forming a dynamic, arbitrary graph" -listing wireless characteristics and graph dynamics as the main challenges for designing protocols and applications for this network.

While capturing important characteristics, this description does not make explicit how MANETs map into the Internet architecture -and does therefore not allow evaluation of existing IP protocols and their applicability on MANETs. Similarly, the lack of a clear architectural description within the context of the Internet has impeded the evaluation of the applicability of MANETs within the Internet.

This fact became explicit during the chartering of the IETF AUTOCONF working group [8]: in simple terms, the goal of the AUTOCONF working group is to provide automatic address configuration for MANET nodes. Most researchers and engineers familiar with MANETs shared the understanding that existing autoconfiguration approaches did not apply. Describing why and how was, absent a clear and agreed upon architectural model of MANETs, difficult -as was communication to experts outside the MANET community.

The issue arose again within the context of routing and route optimisation within nested NEMO networks, where a clear architectural description of MANETs lead to a poor general understanding of how MANETs might be a candidate technology.

The purpose of this memorandum is to document the MANET architecture within the general Internet and IP architecture.

Memorandum Outline

The remainder of this memorandum are organised as follows: section 2 provides an overview of the classic IP link and network model, in particular the assumptions made by IP regarding subnets and links. Section 3 then elaborates important characteristics regarding MANET interfaces, comparing and contrasting with the IP assumptions of section 2. This is followed by section 4, in which a common misperception of the MANET architecture is elaborated, and where the shortcomings of this architectural misperception are presented.

Section 5 presents a MANET architectural model which integrates MANETs as a natural part of the Internet and the IP architecture -fitting the MANET characteristics (section 3) to the classic IP link and network model (section 2). Section 6 summarise the characteristics of the MANET architecture model. Given this architectural model for MANETs, section 7 describes the morphology of MANETs, in particular how one would use the model for configuring a "classic" MANET which respects the IP architecture. Section 8 discusses how MANETs can be employed as a component in the case of managing nested NEMO networks.

As pointed out by [4], network protocols and applications are designed with specific assumptions of the nature of an IP link. Considering figure 1, these assumptions can be summarized as follows:

all hosts (H) with network interfaces configured with addresses from within the same prefix p::, and with the same prefix p:: assigned to the interfaces, can communicate directly with one another -i.e.:

-IP datagrams are not forwarded at the network layer when communicating between interfaces which are configured with addresses from within the same prefix; hence -TTL/hop-limit in IP datagrams are not decremented when communicating between interfaces which are configured with addresses from within the same prefix, and;

-IP datagrams with a TTL/hop-limit of 1 are (modulo data loss) delivered to all interfaces within the same subnet.

link-local multicasts and broadcasts are received by all interfaces configured with addresses from within the same prefix without forwarding.

An even shorter summary of the "classic IP link model" is to say that "an IP link looks like an Ethernet".

It follows from the above that the notion of "IP link" is tied with the notion of an "IP Subnet" (IPv4) or a prefix (IPv6), in that all interfaces which are configured with the same subnet address or prefix are considered to be on the same IP link and thus that for communication between nodes on the same subnet, no forwarding is required and no decrement of TTL/hop-limit is performed.

Interfaces within the same prefix or, for IPv4, within the same subnet, are within the classic IP link model assumed to also be attached to the same classic IP link as described INRIA above. For completeness, it should be mentioned that the inverse is not necessarily true: in some network configurations, interfaces connected to the same classic IP link may be configured within different prefixes or subnets.

MANET Interface Characteristics

MANET nodes are equipped with MANET interfaces, which have different characteristics than the interfaces described for the classic IP Link and Network Model in section 2. These characteristics are briefly summarised in this section, with the purpose of exemplifying the difference with "Ethernet-like" interfaces. A MANET version of figure 1 looks as in figure 3.

N1 N2 N3 N4 N5

Semi Broadcast Interfaces

Each MANET interface is a broadcast interface, typically, but not necessarily, wireless, which is able to establish a direct L2 connection with only those nodes which are within its coverage area. In figure 3, this coverage area is approximated by a simple disc of fixed radius -in the real world, both the shape and size of the coverage area is variable as a function of the interface, interference from the environment etc. Referring to figure 3 if, for example, if N3 transmits, then this transmission may be received by N2 and N4, but not by N1 and N5. This implies that, e.g., N3 and N4 -despite being neighbours and on the same "link" -do not share the same view of which other nodes are neighbours and on the same "link": N3 considers that it is on the same "link" as N2 and N4, whereas N4 considers itself to be on the same "link" as N3 and N5. This sometimes leads to describing MANET interfaces as "semi-broadcast interfaces", with non-transitive neighbour relationships: neighbouring nodes may experience distinctly different neighbourhoods.

Shared Bandwidth

Depending on the radio technology used, MANET interfaces may interfere with each other -this is for example the case with the commonly used IEEE 802.11 interfaces. In figure 3, if N3 transmits over its MANET interface, then this may cause N2 and N4 to be unable to transmit concurrently over their respective MANET interfaces. The direct consequence INRIA hereof is, that available bandwidth is shared among the MANET interfaces within the same coverage area. A further consideration is, that a wireless interface has an "interference area" which may be greater than its coverage area, i.e. a transmission by N3 in figure 3 will, as indicated above, be correctly received by the interfaces N2 and N4. At the same time, however, this transmission may be propagating to interfaces of N1 and N5 where, while the transmission can not be correctly decoded, it can be detected, and cause interference with other transmissions which could otherwise be correctly received over the MANET interfaces of N1 and N5 (such as transmissions from N0 and N6).

Hidden Terminals

A property of MANETs which is commonly brought forward is the "hidden terminal problem": if N3 through some protocol agrees with its neighbours (N2 and N4) that it will, for the moment, have exclusive access to the wireless media via its MANET interface, then N3 may go ahead and make a transmission. However, if at the same time N1 also transmits over its MANET interface, then the transmissions of the MANET interfaces of N1 and N3 may appear concurrently at the MANET interface of N2 -potentially interfering and causing N2 to receive neither of the transmissions. Denoted a "collision", the possibility and probability of this occurring depends on the L2 (data link layer) mechanisms in place -suffice to observe that the such collisions can and do occur when using some common wireless interfaces such as IEEE 802.11.

The term "hidden terminal" originates from the fact that while the node wishing exclusive access to the wireless media may negotiate this with its direct neighbours (in our case N2 and N4), whereas nodes out of direct radio range (in our case N1 and N5) are "hidden".

Asymmetric Connectivity

Considering figure 1, an axiomatic assumption is that neighbour relationships are symmetric: if communication from one interface to another interface is possible in one hop, then communication in the inverse direction is also possible -in other words, connectivity between neighbour interfaces is symmetric. Considering the small MANET in figure 4: for some reason (powerful transmitter, large antenna, ...) the MANET interface of N1 has a large enough coverage area that its transmissions can be received by the MANET interface N2. The MANET interface of N2, on the other hand, has a much smaller coverage radius, such that transmissions from the MANET interface of N2 do not arrive at the MANET interface of N1. Thus an asymmetric -or more precisely, an unidirectional -connectivity between the MANET interface of N1 and the MANET interface of N2 exists: N2 sees N1 as a neighbour (since the MANET interface N2 can receive transmissions from the MANET interface of N1), whereas N1 does not see N2 as a neighbour (since the MANET interface of N1 can not receive transmissions from the MANET interface of N2). Thus, MANET neighbour relationships are non-reflective.

N1 N2

Figure 4: MANET: neighbour asymmetry.

Neighbourhood & Network Membership

Returning to the initial description of a MANET in the introduction, MANET interfaces form, "a dynamic, arbitrary graph" among themselves. This indicates that the neighbourhood of a MANET interface is dynamic and varies over time -either due to node mobility or due to environmental factors which impact the area of coverage of a MANET interface. On a larger scale even the MANET membership may be time varying, with MANET interfaces appearing and disappearing over time, and for the same reasons.

INRIA 4 Common MANET Misperception

Considering the classic IP link model described in section 2, a common misperception is that "a MANET should emulate an Ethernet at L3", and that the nodes in a MANET are "hosts". This has lead to MANET nodes being perceived and configured as indicated in figure 5 as hosts in an Ethernet: the MANET interface is assigned an IP address and a subnet prefix p:: -a prefix which is shared among all the nodes in the MANET as indicated in figure 6.

R + H MANET interface

MANET "node" Configuring a MANET with a single subnet prefix shared among the MANET nodes implies that all MANET nodes would be considered as belonging to the same subnet -and as such on the same IP link. However with the MANET forming a multi-hop L3 network, and given the characteristics outlined in section 3 the protocol and application assumptions for IP links listed in section 2 do not hold: for interfaces within the MANET and with the same prefix to communicate, L3 forwarding of IP datagrams may occur, and with such forwarding, TTL/hop-limit are decremented; link-local multicast or broadcasts either do not reach all nodes within the subnet -or if they are to reach all nodes within the subnet, they are to be forwarded by intermediate nodes In short, considering and configuring MANET nodes as if the MANET forms a single subnet breaks the classic IP link model and the applications which assume the characteristics of the classic IP link model. [4] explores this in more detail.

IP address + MANET Subnet Prefix

Routing Incompatibility

A perhaps surprising example of an application, which breaks under this common MANET misperception, is routing: if a multi-hop MANET is configured as described in this section, with all nodes within the MANET assumed to be also in the same subnet, then forwarding of IP datagrams within the MANET will prompt intermediate nodes to produce ICMP redirects. This is appropriate since IP datagrams delivered within a subnet are not supposed to be forwarded by a router since a direct link between any two nodes within a subnet is supposed to exist, according to the classic IP link model described in section 2.

A rough work-around, often proposed in order to "mask" this problem, is to disable ICMP redirect.

Incompatibility with Other Protocols and Applications?

Disabling ICMP redirects to make routing operate is disabling the symptom of an incorrect network model, for a single application (routing) only, and leads to the specific and reasonable question if other applications and protocols require similar tweaks (if so, which applications/protocols and which tweaks?). Even more general: one could ask if MANETs even do belong in the IP world? The answer is yes, MANETs do belong in the IP worldhowever it also means that the architectural view, presented in this section, is inappropriate and indeed a common misperception of MANETs, which does not take into consideration their integration within the IP architecture.

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A MANET Architectural Model

This section presents an architectural model for MANETs which preserves the integrity of the IP architecture while allowing for the particularities of MANETs.

MANET Node Morphology

This architectural model considers MANET nodes as routers with hosts attached, as illustrated in figure 7. These attached hosts may be "external" (i.e. attached to the router via other network interfaces) or "internal" -however the important observation to make is, that the links between these hosts and the router are classic IP links, behaving as described in section 2. This implies that, from the point of view of the hosts, and the applications running on these hosts, connectivity is via a classic IP link. Hosts, and their applications, are not exposed to the specific characteristics of the MANET interfaces and are connected to the MANET via a router, which has one or more MANET interfaces. This is symmetric with how hosts on an Ethernet, such as illustrated in figure 1 are not exposed to the intricacies of what type of connectivity the router has beyond the Ethernet. Since the hosts in figure 7 are connected to a classic IP link, these hosts are configured and behave as hosts in any other network, and the links to which they are connected have properties identical to those of any other classic IP link.

Addresses and Prefixes

If the MANET router is delegated a prefix p::, this prefix can be assigned to the classic IP link(s), and hosts can be assigned addresses from within this prefix, and configured with this prefix as illustrated in figure 8. Specifically, the MANET interface(s) of the router are not configured with this prefix, for the reasons explained in section 4: the MANET interface(s) is not on the same "link" as the other interfaces with addresses from within this prefix, and so direct communication without crossing a router is not possible. The configuration of MANET interfaces is detailed below. MANET specific behaviors are exclusively exposed to the MANET interface(s) of the routers. This includes MANET routing protocols and interface and link characteristics (asymmetric neighbourhoods, semi-broadcast interfaces, fuzzy neighbor relationships, topology dynamics etc.) The following characteristics deserve particular mention, since they distinguish MANET interfaces and the MANET link model from the classic IP link model:

MANET Interface Configuration & Properties

Unique Prefixes

MANET interfaces must be configured with unique prefixes, i.e. such that no two MANET interfaces are configured such that they appear within the same IP subnet. Some common ways to achieve this are:

unnumbered interfaces (IPv4) [START_REF] Baker | RFC1812: Requirements for IP Version 4 Routers[END_REF];

Link-Local Addresses (IPv6);

/128 (IPv6) or /32 (IPv4) prefixes.

However it is worth noting that prefix lengths shorter than /128 (IPv6) or /32 (IPv4) are possible on the MANET interface, so long as the prefixes are unique to a single MANET interface.

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Link Local Mulitcast/broadcast Scope On a MANET interface, a Link Local multicast or broadcasts reach MANET interfaces of neighbor nodes only, regardless of their configured addresses. A Link Local multicast or broadcast on a MANET interface is, thus, a "neighborcast", and is not forwarded nor assumed to be received by all nodes within a MANET.

MANET Network View

Following the architecture described in the above, a configured MANET with routers and hosts, looks as in figure 9: the inner white cloud represents where MANET interfaces and links form a MANET -and the outer gray cloud represents where the classic IP link model as described in section 2 is assumed.

Properties of Proposed Architectural Model

The MANET architecture model presented in this memorandum makes a clear separation between the roles of router and host in a MANET, recognizing that:

MANET interfaces are seen only by the router, assumed to be MANET aware and running appropriate protocols and applications;

MANET interfaces forming a multi-hop MANET area may use a site (not subnet) prefix (aggregation, ...); hosts/subnets on non-MANET interfaces assume a classic IP link model; applications on hosts see classic IP interfaces connected to a classic IP link, and therefore;

applications on hosts and protocols assuming classic IP interfaces can run unmodified.

Refering to figure 9, the scope of MANET specific protocols is, thus, the inner white cloud. This thus scopes routing protocols such as those developed by the IETF MANET working group [7] and autoconfiguration protocols developed by the IETF AUTOCONF working group [8] to routing and configuring MANET interfaces on MANET routers.

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MANET Configurations

The MANET architectural model outlined in section 5 does not conflict how MANETs have been perceived and deployed. Rather, it gives a way of thinking about MANETs corresponding to the IP architecture. This section will exemplify how "common" MANET deployments fit with this architectural model. Notice that this section contains examples which correspond to the architectural model, but does not attempt to exhaustively enumerate all possible deployments or scenario nor to capture all possible requirements to MANETs.

A MANET with a Single Internal Host

A source for the misperception in section 4 is a common configuration of MANET nodes, where each node has one MANET interface and one internal host, as in figure 10. For this example, addresses within the MANET are extracted from a single common MANET prefix -e.g. 192.168.0.0/16. The interface of the host must be configured, and see a classic IP link as described in section 2. The interface of this host is the only interface on the link (other than that of the router), and can be assigned an IP address of the form 192.168.1.1/32 to, as in figure 11.

T. Clausen

This corresponds to the router having been delegated the prefix 192.168.1.1/32 -IP addresses from within that prefix are then distributed to hosts connected over a classic IP link.

The MANET interface must be configured according to the requirements in section 5. One way of satisfying the requirements set forth in that section is through assigning the same address/prefix to the MANET interface as to the internal host. Traffic to the router will typically be addressed to a well-known multicast address, thus the router can distinguish between traffic to itself and traffic to the host -similar to unnumbered interfaces.

A common misperception is to consider all MANET nodes as belonging to the same subnet (192.168.0.0/16) and configuring each MANET interface with an address/prefix such as 192.168.0.0/16, as in figure 12 (b). This is wrong, as described in section 4. Figure 13 illustrates a cloud of simple MANET nodes, each correctly configured with IPv4 address and a /32 prefix length on their MANET interfaces. In this case, a MANET node consists of a router and a set of hosts, attached to the router via a classic IP link. As in the previous example, the MANET interface, and the interfaces on the classic IP link are to be configured with addresses/prefixes from MANET prefix of 192.168.0.0/16. Each MANET router is delegated a prefix, e.g. 192.168.1.0/24, which is assigned to the classic IP link. The interfaces (of hosts and of the router) connected to that link are configured, using any standard mechanism such as [START_REF] Narten | RFC2462: IPv6 Stateless Address Autoconfiguration[END_REF], with an IP address from that prefix and a /24 prefix. Again, the hosts are exposed to a classic IP link, retaining compatibility with existing applications and protocols.

The MANET interface is configured as an unnumbered interface, with a prefix length of /32, borrowing the IP address from the other (non-MANET) interface of the router. This is illustrated in figure 14.

Nested Mobile Networks

The NEMO basic support specification [3] describes how a mobile network -a mobile router with attached hosts -can change its point of attachment to the Internet, employing MobileIP-like mechanisms to remain reachable: a care-of-prefix1 is acquired at the current point of attachment, signalled to a home agent, and used by the home agent to tunnel traffic destined for hosts in the mobile network to this new point of attachment.

A mobile router may attach to any router, including another mobile router, forming networks of mobile routers to an arbitrary depth, and may change their point of attachment at any given time. Commonly, the terms "nested mobile network" or "nested NEMO" are used for this situation. A nested mobile network, thus, looks as illustrated in figure 15.

Issues & Tasks

[3] does not stipulate how nested mobile networks are structured or managed, which entails that sub-optimal paths and loops can occur [5], [START_REF] Clausen | Route Optimisation in Nested Mobile Networks (NEMO) using OLSR[END_REF]. Alleviating these, each mobile router (MR) must:

maintain loop-free paths to Internet Gateway(s) (IGW); IGWs must maintain loop-free paths to the MRs; select (according to some metrics) one or more IGWs, from which it will acquire a care-of-prefix; maintain loop-free paths to other MRs, whereby traffic between nodes within the nested mobile network can avoid crossing through the Internet.

Relationship to MANETs

MANET routers (see section 5 and mobile routers in a nested mobile network must, both:

form time-varying connections with other MANET nodes / mobile routers; maintain loop-free paths to other MANET nodes / other mobile routers and IGWs; acquire prefixes in order to correctly configure interfaces and attached hosts.

MANET routing protocols are developed by [7], satisfying the first two of these items, thereby providing paths between MRs and IGWs and between MRs themselves.

The AUTOCONF activity [8] is chartered to develop solutions satisfying the third of the items above, including allowing MRs to acquire prefixes from an IGW. Specifically, to: provide MANET-wide unique prefixes to each MANET node / mobile router; if one or more IGW is present, provide unique global prefixes to each MANET node / mobile router; detect and resolve if non-unique prefixes are assigned to MANET nodes / mobile routers (e.g. as a result of a network partitioning/merger).

MANET Supported Route Optimisation in Nested Mobile Networks

Assuming a nested mobile network, where each MR has a home prefix, as in figure 16a.

Each MR acquires a care-of-prefix from the IGW. This care-of-prefix is topologically correct with respect to the IGW, but not necessarily hierarchical within the nested mobile network, as in figure 16b. MRs will perform binding updates to their home agents using the care-of-prefix as obtained from the IGW. Thus, the MR appears directly attached to the IGW, and nested redirects (as in figure 17a) for communication from the Internet are avoided. It is worth noting that this exactly as described in section 7.2, where each MANET node is assigned a unique prefix from within a MANET wide prefix.

By using a MANET routing protocol and an AUTOCONF [8] autoconfiguration mechanism, route optimisation for communication over the Internet is obtained.

MRs will run a MANET routing protocol, which will advertise both their home prefix and the care-of-prefix. Hosts in the nested mobile network are, via their MRs, able to find paths between each other using their home addresses, without passing through the IGW and the Internet (as in figure 17b). Figure 16: Addresses in Nested Mobile Network: (a) MRs with their home prefixes (hp1:: ...) and an IGW with global prefix (q::) (b) MRs with home prefixes(hp:: ...) and care-of-prefixes (q:1::, ...) assigned by the IGW.

By using a MANET routing protocol and an AUTOCONF [8] autoconfiguration mechanism, route optimisation for communication within the nested mobile network is obtained.

Conclusion

This memorandum has described a coherent MANET architectural model, which conforms to the architectural model of the Internet. In particular, this model respects the addressing and prefix architecture of the Internet, preserves the usual semantics of a subnet as related to a link, and thereby preserves compatibility with classic applications and protocols running on hosts or between hosts and routers in the Internet. MANET specific issues are isolated to a MANET interface, and are therefore exposed only to protocols dedicated for managing MANETs. The architectural model describes the MANET specific issues, which must be taken into consideration when designing such protocols for MANET management.

Furthermore, this memorandum has shown how the architectural model is compatible with different MANET deployments and with other Internet protocols. In particular, this memorandum has shown how nested mobile networks are a typical example of a MANET, where MANET protocols solve problems such as route optimisation. This, both for communication between hosts on the internet and in the nested mobile network, and for communication between hosts within the nested mobile network.

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Figure 1 :

 1 Figure 1: Classic IP Link Model: hosts (H) connected to the same link have assigned IP addresses from a common prefix, possibly assigned by a router (R).

Figure 2 :

 2 Figure 2: MANET: nodes (N) with MANET interfaces. The light grey area indicates the coverage area of each MANET interface.

Figure 3 :

 3 Figure 3: MANET: nodes (N) with MANET interfaces. The light grey area indicates the coverage area of each MANET interface. The dark grey circle indicates the interference area of the MANET interface of N3.

Figure 5 :

 5 Figure 5: Common Misperception of MANET Nodes: viewing MANET nodes as regular hosts in a subnet, with an IP address and a subnet prefix assigned to their MANET interface.

  with addresses assigned from and configured with prefix p::

Figure 6 :

 6 Figure 6: Common misperception of a MANET: viewing the MANET as a classic IP subnet as in figure1such that all nodes participate in the same "subnet", and thus share the same subnet prefix.

Figure 7 :

 7 Figure 7: MANET node model: the router (R) has on the top a MANET interface, and is connected, on the bottom, to hosts (H) via classic IP links.

Figure 8 :

 8 Figure 8: MANET node and prefixes: the MANET router (R) is delegated a prefix p::, which it assigns to the classic IP links to which the hosts (H) are attached.

Figure 9 :

 9 Figure 9: MANET Network Model: the inner white cloud is where MANET interfaces and links for a MANET are found and MANET specific protocols apply. The outer gray cloud represents where the classic IP link model (and regular applications/protocols) applies.

Figure 10 :

 10 Figure 10: A Simple MANET Node: one MANET interface and one internal host (H).

Figure 11 :

 11 Figure 11: A Simple MANET Node: one MANET interface and one internal host (H), with the interface of the host configured with an IP address and an "all-ones" netmask.

Figure 12 :

 12 Figure 12: Simple MANET Nodes: addresses assigned from the 192.168.0.0/16 prefix. Left: correct configuration wrt. the architectural model (section 5). Right: incorrect configuration, leading to the "common MANET misperception".

Figure 14 :

 14 Figure 14: MANET Node with Multiple Hosts: one MANET interface and multiple attached hosts (H).

Figure 15 :

 15 Figure 15: Nested Mobile Network: mobile routers (R) with attached hosts (H) connected to the Internet via an Internet Gateway (IGW).

Figure 17 :

 17 Figure 17: Sub-optimal Routes in Mobile Networks: (a) traffic from the Internet to a host not directly connected to the IGW is redirected via two home agents. (b) traffic from one host in the nested mobile network to another host in the nested mobile network is forced through the Internet.

  Figure 13: Simple MANET: addresses assigned from, 192.168.0.0/16, each MANET interface configured with a /32.

	7.2 A MANET with Attached Hosts
	MANET prefix:
	192.168.0.0/16	9 2 .1 8 . 6 2 /3 .1 1 1	192.168.	8.	19 1. 2/ 2. 16 32
			1.3/32
		1 .4 /3 2 1 9 2 .1 6 8 .			19 2. 16 8. 1. 5/ 32
					INRIA

RR n°6145

Strictly speaking, NEMO requires that a care-of address be acquired, yet an address is a special instance of a prefix where the prefix length is equal to the address length. Since the nature of addresses and prefixes is otherwise the same, and since mechanisms for assigning/acquiring addresses are a subset of those assigning/acquiring prefixes, this memo will employ the term care-of-prefix.INRIA

http://www.inria.fr ISSN 0249-6399

Acknowledgements

The MANET architecture model presented is the result of exhaustive discussions with the participants in the IETF MANET and AUTOCONF working groups.

Special thanks to Dave Thaler (the author of [4], from which much of this discussion is derived) and Ian Chakeres for discussion and inspiration to the document in general and to section 3 in particular.

Thanks to Jari Akko, Thomas Nartan, Bob Hinden and Joe Macker for their significant contributions in formulating and refining this model. Also thanks to Christopher Dearlove, Allan Cullen, Andreas Enge, Ulrich Herberg, Philippe Jacquet, Cedric Adjih and Emmanuel Baccelli for their reviews and comments.