Frédéric Mazenc

Christophe Prieur

STRICT LYAPUNOV FUNCTIONS FOR SEMILINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

Keywords: 2000 Mathematics Subject Classification. Primary: 93D05, 35K55; Secondary: 93C10 Strictification, Lyapunov function, semilinear parabolic equation

For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

1. Introduction. Lyapunov function based techniques are central in the study of partial differential equations (PDEs). The techniques are useful for the stability analysis of systems of many different families (although other approaches can be used too, especially when parabolic PDEs are studied; see in particular the contributions [START_REF] Dramé | Asymptotic behavior and stability for solutions of a biochemical reactor distributed parameter model[END_REF][START_REF] Iftime | Optimal control of switched distributed parameter systems with spatially scheduled actuators[END_REF][START_REF] Matignon | Asymptotic stability of linear conservative systems when coupled with diffusive systems[END_REF]).

Amongst the remarkable results for PDEs which extensively use Lyapunov functions, it is worth mentioning the following. In [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF] a Lyapunov function is used to establish the existence of a global solution to the celebrated heat equation. In [START_REF] Krstic | Adaptive boundary control for unstable parabolic PDEs. I. Lyapunov design[END_REF], Lyapunov functions are designed for the heat equation with unknown destabilizing parameters (see also [START_REF] Smyshlyaev | Adaptive boundary control for unstable parabolic PDEs. II. Estimation-based designs[END_REF][START_REF] Smyshlyaev | Adaptive boundary control for unstable parabolic PDEs. III. Output feedback examples with swapping identifiers[END_REF] for further results on the design of output stabilizers). Lyapunov functions have been also used to establish controllability results for semilinear heat equations. For example in [START_REF] Coron | Global steady-state controllability of one-dimensional semilinear heat equations[END_REF] the computation of a Lyapunov function, in combination with the quasi-static deformation method, is a key ingredient of the proof of the global controllability of this equation. In all the preceding papers, parabolic PDEs are considered, but Lyapunov functions can also be useful for other kinds of dynamics. For instance, in [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF] the stabilization of a linear dynamic equation modeling a rotating beam is achieved through a control design relying on weak Lyapunov function, i.e. a Lyapunov function whose derivative along the trajectories of the system which is considered is non-positive (but not necessarily negative definite), and in [START_REF] Coron | Global steady-state stabilization and controllability of 1D semilinear wave equations[END_REF] the controllability of the wave equation is demonstrated via a Lyapunov function. Besides, the knowledge of Lyapunov functions can be useful for the stability analysis of nonlinear hyperbolic systems (see the recent work [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]) or even for designing boundary controls which stabilize a system of conservation laws (see [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]).

To demonstrate asymptotic stability through the knowledge of a weak Lyapunov function, the celebrated LaSalle invariance principle has to be invoked (see e.g. [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF][START_REF] Luo | Stability and stabilization of infinite dimensional systems with applications[END_REF][START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF]). It requires to demonstrate a precompactness property for the solutions, which may be difficult to prove (and is not even always satisfied, as illustrated by the hyperbolic systems considered in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]). This technical step is not needed when is available a strict Lyapunov function i.e. a Lyapunov function whose derivative along the trajectories of a system which is considered is negative definite. Thus designing such a Lyapunov function is a way to overcome this technical difficulty. This is not the unique motivation for designing strict Lyapunov functions. From the knowledge of the explicit expression of a strict Lyapunov function, one can estimate the robustness of the stability of a system with respect to the presence of uncertainties and one can analyze the sensitivity of the solutions with respect to external disturbances.

The present paper is devoted to new techniques of constructions of strict Lyapunov functions for parabolic PDEs. For particular families of PDEs with diffusion and convection terms and specific boundary conditions, we modify weak Lyapunov functions, which are readily available, to obtain strict Lyapunov functions given by explicit formulas. The resulting functions have rather simple explicit expressions. The underlying concept of strictification used in our paper is the same as the one exposed in [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF], (see also [START_REF] Mazenc | A simplified design for strict Lyapunov functions under Matrosov conditions[END_REF][START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF]). However, due to the specificity of PDEs, the techniques of construction that we shall present are by no means a direct application of any constructions available for ordinary differential equations.

In the second part of our work, we design strict Lyapunov functions to establish robustness properties of Input-to-State (ISS) type for a family of PDEs with disturbances which are globally asymptotically stable in the absence of disturbances. Although the ISS notion is very popular in the area of the dynamical systems of finite dimension (see e.g. the recent survey [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]) and, for a few years, begins to be used in the domain of the systems with delay (see for instance [START_REF] Karafyllis | Input-to-output stability for systems described by retarded functional differential equations[END_REF][START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF][START_REF] Pepe | Input-to-state stabilization of stabilizable, time-delay, control-affine, nonlinear systems[END_REF][START_REF] Pepe | On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign[END_REF]), the present work is, to the best of our knowledge, the first one which uses it to characterize a robustness property of a PDE.

Our paper is organized as follows. Basic definitions and notations are introduced in Section 2. Constructions of Lyapunov functions under various sets of assumptions are performed in Section 3. In Section 4 the analysis of the robustness of a family of PDEs with uncertainties is carried out by means of the design of a so called ISS Lyapunov function. Two examples in Section 5 illustrate respectively the design of a strict Lyapunov function of an ISS Lyapunov function. Section 6 contains an example of linear system which is globally asymptotically stable and for which there does not exist any ISS Lyapunov function. A technical result is proven in Section 7. Concluding remarks in Section 8 end the work.

Notation. Throughout the paper, the argument of the functions will be omitted or simplified when no confusion can arise from the context. A function α : [0, +∞) → [0, +∞) is said to be of class K ∞ if it is continuous, zero at zero, increasing and unbounded. Given a continuously differentiable function A : R n → R, ∂A ∂Ξ (Ξ) stands for the vector ( ∂A ∂ξ1 (Ξ)..... ∂A ∂ξn (Ξ)) ∈ R n , whereas

∂ 2 A ∂Ξ 2 (Ξ) is the square matrix function (a i,j (Ξ)) ∈ R n×n with a i,j (Ξ) = ∂ 2 A ∂ξi∂ξj (Ξ).
The Euclidean inner product of two vectors x and y will be denoted by x.y, the induced norm will be denoted by | • |. Given a matrix A, its induced matrix norm will be denoted simply by |A|, and

Sym(A) = 1 2 (A + A ⊤ ) stands for the symmetric part of A. The norm | • | L 2 (0,L) is defined by: |φ| L 2 (0,L) = L 0 |φ(z)| 2 dz. Finally, we denote C L = C 2 ([0, L], R n ), the set of all twice-differentiable R n -valued functions defined on a given interval [0, L].
2. Basic definitions and notions. Throughout our work, we will consider partial differential equations of the form

∂X ∂t (z, t) = ∂ 2 X ∂z 2 (z, t) + ∆(z, t) ∂X ∂z (z, t) +f (X(z, t)) + u(z, t) , (1)
with z ∈ [0, L] and X(., t) ∈ C L for all t ≥ 0, where ∆ : [0, L] × [0, +∞) → R is continuous and bounded in norm, where f : R n → R n is a continuously differentiable function, and where u : [0, L] × [0, +∞) → R n is a continuous function (which typically is unknown and represents disturbances).

Let us introduce the notions of weak and strict Lyapunov functions that we will consider in this paper (see e.g. [START_REF] Luo | Stability and stabilization of infinite dimensional systems with applications[END_REF]Def. 3.62]). Definition 2.1. Let µ : C L → R be a continuously differentiable function. The function µ is said to be a weak Lyapunov function for (1), if there are two functions κ S and κ M of class K ∞ such that, for all functions φ ∈ C L ,

κ S |φ| L 2 (0,L) ≤ µ(φ) ≤ L 0 κ M (|φ(z)|) dz (2)
and, when u is identically equal to zero, for all solutions of (1), for all t ≥ 0, dµ(X(., t)) dt ≤ 0 .

The function µ is said to be a strict Lyapunov function for (1) if, additionally, u is identically equal to zero, there exits λ 1 > 0 such that, for all solutions of (1), for all t ≥ 0, dµ(X(., t)) dt ≤ -λ 1 µ(X(., t)) .

The function µ is said to be an ISS Lyapunov function for (1) if, additionally, there exist λ 1 > 0 and a function λ 2 of class K such that, for all continuous functions u, for all solutions of (1), and for all t ≥ 0, dµ(X(., t)) dt ≤ -λ 1 µ(X(., t))

+ L 0 λ 2 (|u(z, t)|)dz .
Remark 1. 1. For conciseness, we will often use the notation μ instead of dµ(X(.,t)) dt . 2. When a strict Lyapunov function exists and when u is identically equal to zero, then the value of a strict Lyapunov function for (1) along the solutions of (1) exponentially decays to zero and therefore each solution X(z, t) satisfies lim t→+∞ |X(., t)| L 2 (0,L) = 0. When in addition, there exists a function κ L of class

K ∞ , such that, for all functions φ ∈ C L , µ(φ) ≤ κ L |φ| L 2 (0,L) , (3)
then the system (1) is globally asymptotically stable for the topology of the L 2 norm. We note that the L 2 convergence of X(., t) to zero does not imply that for all fixed z ∈ (0, L), lim t→+∞ X(z, t) = 0.

3. Let us recall that, when is known a weak Lyapunov function, asymptotic stability can be often established via the celebrated LaSalle invariance principle applies (see [START_REF] Luo | Stability and stabilization of infinite dimensional systems with applications[END_REF]Theorem 3.64] among other references).

4. When the system (1) admits an ISS Lyapunov function µ, then, one can check through elementary calculations 1 that, for all solutions of (1) and for all instants t ≥ t 0 , the inequality

|X(., t)| L 2 (0,L) ≤ κ -1 S 2e -λ1(t-t0) L 0 κ M (|X(z, t 0 )|)dz +κ -1 S 2 λ1 sup ℓ∈[t0,t] L 0 λ 2 (|u(z, ℓ)|)dz
holds. This inequality is the analogue for the PDE (1) of the ISS inequalities for ordinary differential equations. It gives an estimate of the influence of the disturbance u on the solutions of the system (1).

•

3.

Constructions of Lyapunov functions. In this section, we give several constructions of Lyapunov functions for the system

∂X ∂t (z, t) = ∂ 2 X ∂z 2 (z, t) + f (X(z, t)) (4)
with z ∈ [0, L], X(z, t) ∈ R n and where f is a nonlinear function of class C 1 .

3.1. Weak Lyapunov function for the system (4). To prepare the construction of strict Lyapunov functions of the forthcoming sections, we recall how a weak Lyapunov function can be constructed for the system (4) under the following assumptions:

Assumption 1. There is a symmetric positive definite matrix Q such that the function W 1 defined by, for all

Ξ ∈ R n , W 1 (Ξ) := -∂V ∂Ξ (Ξ)f (Ξ) , (5)
with

V (Ξ) = 1 2 Ξ ⊤ QΞ, is nonnegative. Assumption 2.
The boundary conditions are such that, for all t ≥ 0,

either |X(L, t)| ∂X ∂z (L, t) = |X(0, t)| ∂X ∂z (0, t) = 0 , or X(L, t) = X(0, t) and ∂X ∂z (L, t) = ∂X ∂z (0, t) . (6)
1 For instance this inequality follows from the fact that we have, for all κ of class K∞, and for all positive values a and b,

κ(a + b) ≤ κ(2a) + κ(2b) ,
and from the fact that the function κ -1 S is of class K∞.

The problem of the proof of existence of solutions of (4) under Assumptions 1 and 2 is an important issue that has been tackled in the literature depending on the regularity of the function f . Consider e.g. [START_REF] Taylor | Partial differential equations[END_REF]Chap. 15] for local (in time) existence of solution for sufficiently small (with respect to the existence time) and smooth function f . The global (in time) existence of solutions holds as soon as f is globally Lipschitz (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Chap. 6] among other references). When f is superlinear, the finite escape time phenomenon may occur (see for instance [START_REF] Bebernes | Mathematical problems from combustion theory[END_REF]Chap. 5] or [START_REF] Zaag | Stability of the blow-up profile for equations of the type ut = ∆u + |u| p-1 u[END_REF]). In this paper, we do not consider this issue and the results presented here are valid, as long as there exists a solution.

Some comments on Assumptions 1 and 2 follow.

Remark 2. 1. Assumption 1 is equivalent to claiming that V is a weak Lyapunov function for the ordinary differential equation

Ξ = f (Ξ) (7)
with Ξ ∈ R n . Therefore it implies that this system is globally stable.

2. Assumption 2 is satisfied in particular if the Dirichlet or Neumann conditions or the periodic conditions, i.e. X(0, t) = X(L, t) and ∂X ∂z (0, t) = ∂X ∂z (L, t) for all t ≥ 0 (see [START_REF] Chen | Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations[END_REF]), are satisfied.

3. Since Q is positive definite, there exist two positive real values q 1 and q 2 such that, for all

Ξ ∈ R n , q 1 |Ξ| 2 ≤ V (Ξ) ≤ q 2 |Ξ| 2 . ( 8
)
The constants q 1 and q 2 will be used in the constructions of strict Lyapunov functions we shall perform later.

•

The construction we perform below is given for instance in [START_REF] Krstic | Adaptive boundary control for unstable parabolic PDEs. I. Lyapunov design[END_REF][START_REF] Coron | Global steady-state controllability of one-dimensional semilinear heat equations[END_REF].

Lemma 3.1. Under Assumptions 1 and 2, the function

U (φ) = L 0 V (φ(z))dz (9)
is a weak Lyapunov function whose derivative along the solutions of (4) satisfies

U = - L 0 ∂X ⊤ ∂z (z, t)Q ∂X ∂z (z, t)dz - L 0 W 1 (X(z, t))dz . (10)
Proof. From ( 8), it follows that, for all φ ∈ L 2 (0, L),

q 1 |φ| 2 L 2 (0,L) ≤ U (φ) ≤ q 2 |φ| 2 L 2 (0,L) . Since U = L 0 X ⊤ (z, t)Q ∂X ∂t (z, t)dz ,
it follows that, along a solution of (4),

U = L 0 X ⊤ (z, t)Q ∂ 2 X ∂z 2 (z, t) + f (X(z, t)) dz .
Thus, from [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF] in Assumption 1, we have

U = L 0 X ⊤ (z, t)Q ∂ 2 X ∂z 2 (z, t)dz - L 0 W 1 (X(z, t))dz . (11)
By integrating by part the first integral in the latter equality, we obtain

U = X ⊤ (L, t)Q ∂X ∂z (L, t) -X ⊤ (0, t)Q ∂X ∂z (0, t) - L 0 ∂X ⊤ ∂z (z, t)Q ∂X ∂z (z, t)dz - L 0 W 1 (X(z, t))dz .
From Assumption 2, we deduce that ( 10) is satisfied. • 3.2. Strict Lyapunov function for the system (4): first result. In this section, we show that the function U given in ( 9) is a strict Lyapunov function for (4) when this system is associated with special families of boundary conditions or when W 1 is larger than a positive definite quadratic function. We state and prove the following result:

Theorem 1. Assume that the system (4) satisfies Assumptions 1 and 2 and that one of the following property is satisfied: (i) there exists a constant α > 0 such that, for all Ξ ∈ R n ,

W 1 (Ξ) ≥ α|Ξ| 2 ,
(ii) X(0, t) = 0 for all t ≥ 0, (iii) X(L, t) = 0 for all t ≥ 0. Then the function U given in ( 9) is a strict Lyapunov function for (4).

Proof. Let us assume that the property (i) holds. Then it follows straightforwardly from ( 8) and ( 10) that

U ≤ - α q 2 L 0 V (X(z, t))dz
and U is a strict Lyapunov function (see Definition 2.1).

We consider now the cases (ii) and (iii). Let us recall the Poincaré inequality (see [START_REF] Krstic | Boundary control of PDEs[END_REF]Lemma 2.1]). Lemma 3.2. For any function w, continuously differentiable on [0, 1], and for c = 0 or c = 1,

1 0 |w(z)| 2 dz ≤ 2w 2 (c) + 4 1 0 ∂w ∂z (z) 2 dz . (12)
From this lemma, we deduce that for all L ≥ 0 and for c = 0 or L, and for any function w, continuously differentiable on [0, L], the inequality

L 0 |w(z)| 2 dz ≤ 2Lw 2 (c) + 4L 2 L 0 ∂w ∂z (z) 2 dz , (13)
is valid. We deduce that when X(0, t) = 0 for all t ≥ 0 or X(L, t) = 0 for all t ≥ 0 then, for all t ≥ 0, the inequality

L 0 |X(z, t)| 2 dz ≤ 2L 2 q1 L 0 ∂X ⊤ ∂z (z, t)Q ∂X ∂z (z, t)dz
where q 1 is the constant in (8), is satisfied. Combining this inequality with [START_REF] Iftime | Optimal control of switched distributed parameter systems with spatially scheduled actuators[END_REF] yields

U ≤ -q1 2L 2 L 0 |X(z, t)| 2 dz .
Using (8) again, we can conclude that U is a strict Lyapunov function for the system (4). • 3.3. Strict Lyapunov function for the system (4): second result. One can check easily that Assumptions 1 and 2 alone do not ensure that the system (4) admits the zero solution as an asymptotically stable solution. 2 Therefore an extra assumption must be introduced to guarantee that a strict Lyapunov function exists. In Section 3.2 we have exhibited simple conditions which ensure that U is a strict Lyapunov function. In this section, we introduce a new assumption, less restrictive than the condition (i) of Theorem 1, which ensures that a strict Lyapunov function different from U can be constructed.

M (0) = 0 , ∂M ∂Ξ (0) = 0 , (14)
∂M ∂Ξ (Ξ)f (Ξ) ≤ -W 2 (Ξ) , ∀Ξ ∈ R n , (15)
∂ 2 M ∂Ξ 2 (Ξ) ≤ q 1 2 , ∀Ξ ∈ R n , (16)
and there exists a constant q 3 > 0 such that W 1 + W 2 is positive definite and

W 1 (Ξ) + W 2 (Ξ) ≥ q 3 |Ξ| 2 , ∀Ξ ∈ R n : |Ξ| ≤ 1 , (17)
where W 1 is the function defined in [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF].

We are ready to state and prove the following result:

Theorem 2. Under Assumptions 1 to 3, there exists a function k of class K ∞ , of class C 2 such that k ′ is positive, k ′′ is nonnegative and the function

U (φ) = L 0 k(V (φ(z)) + M (φ(z)))dz ( 18
)
is a strict Lyapunov function for (4).

Remark 3. Assumption 3 seems to be restrictive. In fact, it can be significantly relaxed. Indeed, if the system

Ξ = f (Ξ) ( 19
)
is locally exponentially stable and satisfies one of Matrosov's conditions which ensure that a strict Lyapunov function can be constructed then one can construct a function M which satisfies Assumption 3. For constructions of strict Lyapunov functions under Matrosov's conditions, the reader is referred to [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF].

•
Proof. Let us consider the functional U defined in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]. Since we impose a priori on k to be of class K ∞ and ( 16) holds, we deduce easily that inequalities of the type (2) are satisfied.

2 More precisely, we can construct examples of systems (4) which are not asymptotically stable when Assumption 1 is satisfied and Assumption 2 holds with the Neumann boundary conditions. For example the system ∂X ∂t = ∂ 2 X ∂z 2 where X ∈ R with Neumann boundary conditions at z = 0 and z = L admits all constant functions as solutions and thus it is not asymptotically stable in L 2 norm.

Next, let us evaluate the time derivative of U along the solutions of (4). With the notation S = V + M , we have

U = L 0 k ′ (S(X(z, t))) ∂S ∂Ξ (X(z, t)) ∂X ∂t (z, t)dz = T 1 (X(., t)) + T 2 (X(., t)) (20)
with

T 1 (φ) = L 0 k ′ (S(φ(z))) ∂S ∂Ξ (φ(z))f (φ(z))dz , T 2 (φ) = L 0 k ′ (S(φ(z))) ∂S ∂Ξ (φ(z)) ∂ 2 φ ∂z 2 (z)dz . ( 21
) Since ∂S ∂Ξ (Ξ)f (Ξ) = ∂V ∂Ξ (Ξ)f (Ξ) + ∂M ∂Ξ (Ξ)f (Ξ) ,
we deduce from Assumptions 1 and 3 that

T 1 (φ) ≤ - L 0 k ′ (S(φ(z)))[W 1 (φ(z)) + W 2 (φ(z))]dz . (22)
Now, we consider T 2 . By integrating by part, we obtain

T 2 (φ) = T 3 (φ) - L 0 ∂H(φ(z)) ∂z ∂φ ∂z (z)dz (23)
with

T 3 (φ) = k ′ (S(φ(L))) ∂S ∂Ξ (φ(L)) ∂φ ∂z (L) -k ′ (S(φ(0))) ∂S ∂Ξ (φ(0)) ∂φ ∂z (0) (24)
and

H(Ξ) = k ′ (S(Ξ)) ∂S ∂Ξ (Ξ) . (25)
Since

∂H(φ(z)) ∂z = k ′′ (S(φ(z))) ∂S ∂Ξ (φ(z)) ∂φ ∂z (z) ∂S ∂Ξ (φ(z)) +k ′ (S(φ(z))) ∂φ ⊤ ∂z (z) ∂ 2 S ∂Ξ 2 (φ(z)) , (26)
we deduce from (23) that

T 2 (φ) = T 3 (φ) -T 4 (φ) -T 5 (φ) (27)
with

T 4 (φ) = L 0 k ′′ (S(φ(z))) ∂S ∂Ξ (φ(z)) ∂φ ∂z (z) 2 dz , T 5 (φ) = L 0 k ′ (S(φ(z))) ∂φ ⊤ ∂z (z) ∂ 2 S ∂Ξ 2 (φ(z)) ∂φ ∂z (z)dz . (28)
Since we impose on k to be such that k ′′ is nonnegative, we immediately deduce that T 2 (φ) ≤ T 3 (φ) -T 5 (φ) . (29) Now, observe that

∂ 2 S ∂Ξ 2 (φ(z)) = Q + ∂ 2 M ∂Ξ 2 (φ(z)) . (30)
This equality, inequalities (8), Assumption 3 and the fact that we impose on k to be such that k ′ > 0 ensure that

T 5 (φ) ≥ 2q 1 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz - q 1 2 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz = 3q1 2 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz .
It follows that

T 2 (φ) ≤ T 3 (φ) -3q1 2 L 0 k ′ (S(φ(z))) ∂φ ∂z (z) 2 dz . (31)
Hence ( 20), [START_REF] Pepe | Input-to-state stabilization of stabilizable, time-delay, control-affine, nonlinear systems[END_REF], and (31) give

U ≤ - L 0 k ′ (S(X(z, t)))W 3 (X(z, t))dz +T 3 (X(., t)) -3q1 2 L 0 k ′ (S(X(z, t))) ∂X ∂z (z, t) 2 dz with W 3 = W 1 + W 2 .
Assumption 2 ensures that, for all t ≥ 0, T 3 (X(., t)) = 0. We deduce that U ≤ -

L 0 k ′ (S(X(z, t)))W 3 (X(z, t))dz .
By [START_REF] Mazenc | A simplified design for strict Lyapunov functions under Matrosov conditions[END_REF] and the inequality S(Ξ) ≥ q 1 |Ξ| 2 , we can prove the following Lemma 3.3. There exist a constant C > 0, a twice continuously differentiable function k of class K ∞ such that k ′ is positive and k ′′ is nonnegative and

k ′ (S(Ξ))W 3 (Ξ) ≥ Ck(S(Ξ)) (32)
for all Ξ ∈ R n .

The proof of this lemma is postponed to Appendix 7. Selecting the function k provided by Lemma 3.3, we obtain U ≤ -CU (X(., t)) .

It follows that U is a strict Lyapunov function for (4). • 4. ISS property for a family of PDEs. In the previous section, we have constructed Lyapunov functions for PDEs without uncertainties and without convection term. In this section, we show how our technique of construction can be used to estimate the impact of uncertainties on the solutions of PDEs with a convection term and uncertainties of the form

∂X ∂t (z, t) = ∂ 2 X ∂z 2 (z, t) + [D 1 + v(z, t)] ∂X ∂z (z, t) +f (X(z, t)) + u(z, t) , (33)
where D 1 is a constant matrix, v is an unknown continuous matrix function and u is an unknown continuous function.

Remark 4. For a linear finite dimensional system

ẋ = Ax + Bu , (34)
where A and B are constant matrices respectively in R n×n and in R n×1 , it is wellknown that if ẋ = Ax, is asymptotically stable then bounded inputs result in bounded solutions. However, for nonlinear finite dimensional systems, global asymptotic stability does not imply input/state stability of any sort. See [28, Section 2.6] for a simple scalar example, which is globally asymptotically stable but which has solutions with a finite time explosion for a suitable constant input. Also for linear infinite dimensional system, asymptotic stability does not imply input-to-state stability. More precisely, we exhibit in Section 6 an example of linear system which is globally asymptotically stable without any input, but which can have unbounded solutions in the presence of a bounded input (and thus for this system there does not exist any ISS Lyapunov function).

•

To cope with the presence of a convection term and the uncertainty v, we introduce the following assumption: Assumption 4. There exists a nonnegative real number δ such that

|v(z, t)| ≤ δ |Q| , ∀z ∈ [0, L] , t ≥ 0 , ( 35
)
where Q is the the symmetric positive definite matrix in Assumption 1. Moreover, the matrix QD 1 is symmetric.

Moreover, we replace Assumption 3 by a more restrictive assumption:

Assumption 5. There exists a nonnegative function M : R n → R such that, for all Ξ ∈ R n ,

M (0) = 0 , ∂M ∂Ξ (Ξ)f (Ξ) = -W 2 (Ξ) , ( 36
)
where W 2 is a nonnegative function and there exist c a > 0, c b > 0 and c c > 0 such that, for all Ξ ∈ R n , the inequalities

∂M ∂Ξ (Ξ) ≤ c a |Ξ| , ∂ 2 M ∂Ξ 2 (Ξ) ≤ c b , (37)
|Ξ| 2 ≤ c c [W 1 (Ξ) + W 2 (Ξ)] , (38)
where W 1 is the function defined in [START_REF] Coron | Stabilization of a rotating body beam without damping[END_REF], are satisfied.

Remark 5. If f is linear and Ξ = f (Ξ) is exponentially stable, then Assumption 5 is satisfied with a positive definite quadratic function as function M .

•

We are ready to state and prove the main result of the section Theorem 3. Assume that the system (33) satisfies Assumptions 1, 4 and 5 and is associated with boundary conditions satisfying

X(L, t) = X(0, t) and ∂X ∂z (L, t) = ∂X ∂z (0, t) , ∀t ≥ 0 . ( 39
)
Then the function

U (φ) = L 0 [KV (φ(z)) + M (φ(z))]dz ( 40
)
with

K = max 1, 2c b q 1 , 8c c c 2 a (|D 1 | + 1) 2 q 1 (41)
satisfies, along the trajectories of (33),

U ≤ -λ 1 U (X(z, t)) + λ 2 L 0 |u(z, t)| 2 dz (42)
for some positive constants λ 1 , λ 2 , provided that δ in Assumption 4 satisfies

δ ≤ min |Q|, √ q 1 2 √ 2c c K . ( 43
)
Remark 6. Using Assumption 5, one can prove easily that the ISS Lyapunov function U defined in ( 40) is upper and lower bounded by a positive definite quadratic function. We deduce easily that (42) leads to an ISS inequality of the type

|X(., t)| L 2 (0,L) ≤ Λ 1 e -λ 1 2 (t-t0) |X(., t 0 )| L 2 (0,L) +Λ 2 sup m∈[t0,t] L 0 |u(z, m)| 2 dz , ( 44
)
where Λ 1 , Λ 2 are positive real numbers. •

Proof. Since K ≥ 1 and M is nonnegative, we deduce easily that inequalities of the type (2) are satisfied by U defined in (40).

Considering again the function U defined by ( 9), we obtain that, along the solutions of (33), the equality

U = L 0 X ⊤ (z, t)Q ∂ 2 X ∂z 2 (z, t) + f (X(z, t)) dz + L 0 X ⊤ (z, t)QD 1 ∂X ∂z (z, t)dz + L 0 X ⊤ (z, t)Qv(z, t) ∂X ∂z (z, t)dz + L 0 X ⊤ (z, t)Qu(z, t)dz (45)
is satisfied. Since Assumption 4 guarantees that QD 1 = Sym(QD 1 ), where Sym is the function defined in the introduction, and since the equality

L 0 φ ⊤ (z)Sym(QD 1 ) ∂φ ∂z (z)dz = 1 2 [φ ⊤ (L)Sym(QD 1 )φ(L) -φ ⊤ (0)Sym(QD 1 )φ(0)]
is satisfied for all φ ∈ C L , Assumptions 1, 4 and the condition (39) yield

U ≤ L 0 X ⊤ (z, t)Q ∂ 2 X ∂z 2 (z, t)dz - L 0 W 1 (X(z, t))dz +δ L 0 |X(z, t)| ∂X ∂z (z, t) dz + |Q| L 0 |X(z, t)||u(z, t)|dz .
Next, arguing as we did from the inequality [START_REF] Karafyllis | Input-to-output stability for systems described by retarded functional differential equations[END_REF] to the end of the proof of Lemma 3.1, we obtain:

U ≤ -2q 1 L 0 ∂X ∂z (z, t) 2 dz - L 0 W 1 (X(z, t))dz +δ L 0 |X(z, t)| ∂X ∂z (z, t) dz + |Q| L 0 |X(z, t)||u(z, t)|dz . (46)
Next, we evaluate the derivative of M along the trajectories of (33)

Ṁ = L 0 ∂M ∂Ξ (X(z, t)) ∂ 2 X ∂z 2 (z, t) + f (X(z, t)) dz + L 0 ∂M ∂Ξ (X(z, t))D 1 ∂X ∂z (z, t)dz + L 0 ∂M ∂Ξ (X(z, t))v(z, t) ∂X ∂z (z, t)dz + L 0 ∂M ∂Ξ (X(z, t))u(z, t)dz .
Using Assumption 5, we deduce that

Ṁ ≤ - L 0 W 2 (X(z, t))dz + S 1 (X(., t)) +c a (|D 1 | + δ |Q| ) L 0 |X(z, t)| ∂X ∂z (z, t) dz +c a L 0 |X(z, t)||u(z, t)|dz with S 1 (φ) = L 0 ∂M ∂Ξ (φ(z)) ∂ 2 φ ∂z 2 (z)dz
. By integrating by part, we obtain

S 1 (φ) = ∂M ∂Ξ (φ(L)) ∂φ ∂z (L) -∂M ∂Ξ (φ(0)) ∂φ ∂z (0) - L 0 ∂φ ⊤ ∂z (z) ∂ 2 M ∂Ξ 2 (φ(z)) ∂φ ∂z (z)dz .
From (39) and Assumption 5, we deduce that

S 1 (X(., t)) ≤ c b L 0 ∂X ∂z (z, t) 2 dz .
It follows that

Ṁ ≤ - L 0 W 2 (X(z, t))dz + c b L 0 ∂X ∂z (z, t) 2 dz +c a (|D 1 | + δ |Q| ) L 0 |X(z, t)| ∂X ∂z (z, t) dz +c a L 0 |X(z, t)||u(z, t)|dz . (47)
From ( 46) and (47), we deduce that

U ≤ (-2q 1 K + c b ) L 0 ∂X ∂z (z, t) 2 dz -S 2 (X(., t)) + S 3 (X(., t)) + c d L 0 |X(z, t)||u(z, t)|dz (48)
with

c d = K|Q| + c a , S 2 (φ) = L 0 [KW 1 (φ(z)) + W 2 (φ(z))]dz , S 3 (φ) = [Kδ + c a (|D 1 | + δ)] L 0 |φ(z)| ∂φ ∂z (z) dz .
Since K ≥ 1, we deduce from Assumption 5 that

S 2 (φ) ≥ 1 c c L 0 |φ(z)| 2 dz . (49)
From Young's inequality, we deduce that

S 3 (φ) ≤ 1 2 q 1 K L 0 ∂φ ∂z (z) 2 dz + [Kδ + c a (|D 1 | + δ |Q| )] 2 2q 1 K L 0 |φ(z)| 2 dz , (50) L 0 |X(z, t)||u(z, t)|dz ≤ 1 2ccc d L 0 |X(z, t)| 2 dz + c c c d 2 L 0 |u(z, t)| 2 dz . (51)
Combining ( 48), ( 49), (50), (51) and the inequality [Kδ +c a (|D

1 |+ δ |Q| )] 2 ≤ 2K 2 δ 2 + 2c 2 a (|D 1 | + δ |Q| ) 2 , we obtain U ≤ (-3 2 q 1 K + c b ) L 0 ∂X ∂z (z, t) 2 dz + K 2 δ 2 + c 2 a (|D 1 | + δ |Q| ) 2 q 1 K - 1 2c c L 0 |X(z, t)| 2 dz + c c c 2 d 2 L 0 |u(z, t)| 2 dz .
We deduce easily that if (43) is satisfied, then 2 . This inequality and (41) imply that

K 2 δ 2 ≤ q1 8cc K and c 2 a (|D 1 | + δ |Q| ) 2 ≤ c 2 a (|D 1 | + 1) 2 . Therefore we have K 2 δ 2 + c 2 a (|D 1 | + δ |Q| ) 2 ≤ q1 8cc K + c 2 a (|D 1 | + 1)
K 2 δ 2 + c 2 a (|D 1 | + δ |Q| ) 2 ≤ q1 8cc K + q1 8cc K.
Next, from this inequality and (41), we deduce easily that

U ≤ -q 1 K L 0 ∂X ∂z (z, t) 2 dz - 1 4c c L 0 |X(z, t)| 2 dz + ccc 2 d 2 L 0 |u(z, t)| 2 dz .
From Assumption 5, we deduce that, for all

Ξ ∈ R n , M (Ξ) ≤ c a |Ξ| 2 . It follows that (42) is satisfied with λ 1 = 1 4cc(Kq2+ca) , and λ 2 = ccc 2 d 2 .
This concludes the proof. •

Examples.

5.1. First example. In this section, we illustrate Theorem 1 through the following linear parabolic system, for all z ∈ (0, L) and for all t ≥ 0,

∂x1 ∂t (z, t) = ∂ 2 x1 ∂z 2 (z, t) + x 2 (z, t) ∂x2 ∂t (z, t) = ∂ 2 x2 ∂z 2 (z, t) -x 1 (z, t) (52)
with X = (x 1 , x 2 ) ⊤ ∈ R 2 and with the Dirichlet boundary conditions: x 1 (0, t) = x 1 (L, t) = x 2 (0, t) = x 2 (L, t) for all t ≥ 0. Equation (52) is a system of two heat equations with a coupling term which does not introduce any stability. Indeed the coupling between both dynamics is given by f (X) = 0 1 -1 0 X and the matrix 0 1 -1 0 is stable but not asymptotically stable. Hence the stability that will be observed is a consequence of the presence of the diffusion terms only. Assumptions 1 and 2 hold with the positive definite quadratic function

V (Ξ) = 1 2 [ξ 2 1 + ξ 2 2 ]
and the nonnegative function W 1 (Ξ) = 0, for all Ξ ∈ R 2 . Moreover the condition (i) of Theorem 1 does not hold but conditions (ii) and (iii) are satisfied. Therefore Theorem 1 guarantees that U given in ( 9) is a strict Lyapunov function for (52).

To numerically check the stability, let us discretize the linear parabolic equation (52) using an explicit Euler discretization 3 . We select the parameters of the numerical scheme so that the CFL condition for the stability holds. More precisely, with L = 2π, we divide the space domain [0, L] into 40 intervals of identical length, and choosing 10 as final time, we set a time discretization of 10 -2 . For the initial condition, we select the functions x 1 (z, 0) = sin(z) and x 2 (z, 0) = z(z -L), for all z ∈ [0, L]. The time evolutions of the components x 1 and x 2 of the solution are given in Figures 1 and2 respectively. We can observe that the solution converges as expected to the equilibrium. 5.2. Second example. In this section, we illustrate Theorem 3 through the semilinear parabolic system with a convection term, for all z ∈ (0, 1), and for all t ≥ 0,

       ∂x1 ∂t (z, t) = ∂ 2 x1 ∂z 2 (z, t) -∂x1 ∂z (z, t) +x 2 (z, t)[1 + x 1 (z, t) 2 ] + u 1 (z, t) ∂x2 ∂t (z, t) = ∂ 2 x2 ∂z 2 (z, t) -x 1 (z, t)[1 + x 1 (z, t) 2 ] -x 2 (z, t)[2 + x 1 (z, t) 2 ] + u 2 (z, t) (53)
with X = (x 1 , x 2 ) ⊤ ∈ R 2 and where u 1 and u 2 are continuous real-valued functions. We associate to (53) the boundary conditions (39) particularized to the two-dimensional case. Equation (53) is a system of two heat equations with a convection term in the first. The interest of this example is that its nonlinear terms are not globally Lipschitz, thus, to the best of our knowledge, no global stability result available in the literature applies to this system. Let us check that Theorem 3 applies. One can check readily that Assumptions 1, 4 and 5 are satisfied with the positive definite quadratic functions

V (Ξ) = 1 2 [ξ 2 1 + ξ 2 2 ] , M (Ξ) = ξ 2 1 + ξ 2 2 + ξ 1 ξ 2 ,
(Ξ) = (ξ 2 [1 + ξ 2 1 ] , -ξ 1 [1 + ξ 2 1 ] - ξ 2 [2 + ξ 2 1 ]) ⊤ , we have, for all Ξ ∈ R 2 , ∂V ∂Ξ (Ξ)f (Ξ) = -ξ 2 2 [2 + ξ 2 1 ] , ∂M ∂Ξ (Ξ)f (Ξ) = -(3 + ξ 2 1 )ξ 2 2 -(1 + ξ 2 1 )ξ 2 1 -ξ 1 ξ 2 (2 + ξ 2 1 ) ≤ -ξ 2 2 -1 2 + 3 4 ξ 2 1 ξ 2 1 .
Moreover, through elementary calculations, we obtain the following values:

q 1 = 1 2 , |D 1 | = 1, c a = c b = 3, c c = 2 for
the constants in (41). Therefore Theorem 3 guarantees that, if (39) is satisfied, then the function

U (φ) = 1153 L 0 φ 1 (z) 2 + φ 2 (z) 2 dz + L 0 φ 1 (z)φ 2 (z)dz (54)
is an ISS Lyapunov function for the system (53).

To numerically check the fact that U is a Lyapunov function, let us discretize the semilinear parabolic partial differential equation (53) using an explicit Euler discretization We select the parameters of the numerical scheme so that the CFL condition for the stability holds. More precisely, with L = 2π, we divide the space domain [0, L] into 40 intervals of identical length, and, choosing 10 as final time, we set a time discretization of 10 -2 . For the initial condition, we select the functions x 1 (z, 0) = cos(z) and x 2 (z, 0) = 1, for all z ∈ [0, L]. The numerical simulations are performed with u 1 (z, t) = sin 2 (t), u 2 (z, t) = 0 for all z ∈ [0, L] and t ∈ [0, 5] and u 1 (z, t) = u 2 (z, t) = 0 for all z ∈ [0, L] and for all t ∈ (5, 10]. The time evolutions of the components x 1 and x 2 of the solution are given in Figures 3 and4 respectively. We can observe that the solution converges as expected to the equilibrium after the perturbations vanish (i.e. after t = 5). In this section, we exhibit an example of a linear infinite dimensional system which is globally asymptotically stable in the absence of input, but which may admit unbounded solutions in presence of a bounded input. Consequently, this system is globally asymptotically stable but it doesnot admit any ISS Lyapunov function. This system is composed by an infinite number of scalar ordinary differential equations written by, for each n ∈ N and for all t ≥ 0,

Ẋn (t) = - 1 n + 1 X n (t) + u n (t) . (55)
Given an initial condition in l 2 (N) and for u n (t) = 0 for all t ≥ 0 and for all n ∈ N, each system (55) admits a solution in l 2 (N) defined for all time t ≥ 0. Moreover the system (55) is globally asymptotic stable when, for all n ∈ N and all t ≥ 0, u n (t) = 0. Let us consider the input satisfying, for each n ∈ N, and for all t ≥ 0,

u n (t) = 1 (n + 1) 2 e - 1 (n+1) 2 t .
Note that the function t → (u n (t)) n∈N l 2 (N) , where • l 2 (N) denotes the usual norm in l 2 (N), is bounded. This function is even so that, for all t ≥ 0, (u n (t)) n∈N l 2 (N) ≤ π 2 6 e -t , which implies that (u n (t)) n∈N l 2 (N) converges exponentially to zero. On the other hand, the solution of (55) with, for all n ∈ N, X n (0) = 0 is given by, for all t ≥ 0,

X n (t) = 1 n e - 1 (n+1) 2 t -e -1 n+1 t
for all n ≥ 1. Therefore (X n (t)) n∈N is not in l 2 (N) for t > 0. It follows that the system composed of the systems (55) is not ISS with respect to (u n (t)) n∈N . This allows us to conclude the proof of Lemma 7.1.

• We are now in position to prove Lemma 3.3. Since S(Ξ) = V (Ξ) + M (Ξ), due to (8), ( 14) and ( 16), the inequalities

q 1 |Ξ| 2 ≤ S(Ξ) ≤ q 2 + q 1 2 |Ξ| 2
hold for all Ξ ∈ R n . Next, we observe that, due to (17), W 3 is positive definite and W 3 (Ξ) ≥ q 3 |Ξ| 2 for all Ξ ∈ R n such that |Ξ| ≤ 1. We deduce that there exists a positive, increasing and continuously differentiable function Γ such that, for all Ξ ∈ R n , Γ(q 1 |Ξ| 2 )W 3 (Ξ) ≥ q 2 + q 1 2 |Ξ| 2 .

Therefore, for all Ξ ∈ R n , W 3 (Ξ) ≥

S(Ξ) Γ (S(Ξ))

.

This implies that, for all continuously differentiable functions k of class K ∞ , the inequality

The combination of (56) and (57) implies that (32) is satisfied with C = min 1, 1 Γ(1) . Finally observe that since Γ is increasing and of class C 1 , k ′′ is nonnegative. This concludes the proof of Lemma 3.3.

Conclusion.

For important families of PDEs, we have shown how weak Lyapunov functions can be transformed into strict Lyapunov functions. Robustness properties of ISS type can be inferred from our constructions. In a different context, the stability of hyperbolic systems has been established using the Lyapunov theory. More precisely, based on [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], strictification techniques have been used for a class of time-varying hyperbolic equations in [START_REF] Prieur | ISS Lyapunov functions for time-varying hyperbolic partial differential equations[END_REF]. One natural perspective of the present paper could be to consider time-varying semilinear parabolic equations.

Assumption 3 .

 3 There exist a nonnegative function M : R n → R of class C 2 , and a continuous function W 2 : R n → R such that

Figure 1 .

 1 Figure 1. Component x 1 of the solution of (52) for t ∈ [0, 10]

Figure 2 .

 2 Figure 2. Component x 2 of the solution of (52) for t ∈ [0, 10]

Figure 3 .

 3 Figure 3. Component x 1 of the solution of (53) for t ∈ [0, 10]

Figure 4 .

 4 Figure 4. Component x 2 of the solution of (53) for t ∈ [0, 10]

Figure 5 . 6 .

 56 Figure 5. Time-evolution of the function U given by (54)

7 .Lemma 7 . 1 . 1 P

 7711 Appendix: Proof of Lemma 3.3. Before proving Lemma 3.3, let us state and prove the following technical result: Let P : R → R be a positive and increasing function. Then the function κ(m) = me R m (ℓ)dℓ is such that κ ′ (m)m ≥ min 1, 1 P (1) P (m)κ(m) ,for all m ≥ 0.Proof. An elementary calculation givesκ ′ (m) = [1 + mP (m)] e R m 1 P (ℓ)dℓ . Therefore κ ′ (m)m = [1 + mP (m)] me R m 1 P (ℓ)dℓ = [1 + mP (m)] κ(m) . Therefore, when m ≥ 1, κ ′ (m)m ≥ P (m)κ(m)and, when m ∈ [0, 1], we deduce from the fact that P is positive and increasing that κ ′ (m)m ≥ κ(m) ≥ P (m) P (1) κ(m) .

k

  ′ (S(Ξ))W 3 (Ξ) ≥ k ′ (S(Ξ))S(Ξ) Γ (S(Ξ)) ,(56)is satisfied, for all Ξ ∈ R n . Next, from Lemma 7.1, we deduce that the functionk(m) = me R m 1 Γ(ℓ)dℓ , which is of class K ∞ , twice continuously differentiable such that k ′ is positive, is such that, for all Ξ ∈ R n , k ′ (S(Ξ))S(Ξ) Γ (S(Ξ)) ≥ min 1, 1 Γ(1)k(S(Ξ)) .

The simulation codes for both examples can be downloaded from www.gipsa-lab.fr/ ~christophe.prieur/Codes/2011-Mazenc-Prieur.zip


Essia Hamouda

Nathalie Mitton
email: nathalie.mitton@inria.fr

David Simplot-Ryl
email: david.simplot-ryl@inria.fr

Energy Efficient Mobile Routing in Actuator and Sensor Networks with Connectivity preservation

Keywords: wireless communication, performance optimization, node mobility, connected dominating set

In mobile wireless sensor networks, flows sent from data collecting sensors to a sink could traverse inefficient resource expensive paths. Such paths may have several negative effects such as devices battery depletion that may cause the network to be disconnected and packets to experience arbitrary delays. This is particularly problematic in eventbased sensor networks (deployed in disaster recovery missions) where flows are of great importance. In this paper, we use node mobility to improve energy consumption of computed paths. Mobility is a two-sword edge, however. Moving a node may render the network disconnected and useless. We propose CoMNet (Connectivity preservation Mobile routing protocol for actuator and sensor NETworks), a localized mechanism that modifies the network topology to support resource efficient transmissions. To the best of our knowledge, CoMNet is the first georouting algorithm which considers controlled mobility to improve routing energy consumption while ensuring network connectivity. CoMNet is based on (i) a cost to progress metric which optimizes both sending and moving costs, (ii) the use of a connected dominating set to maintain network connectivity. CoMNet is general enough to be applied to various networks (actuator, sensor). Our simulations show that CoMNet guarantees network connectivity and is effective in achieving high delivery rates and substantial energy savings compared to traditional approaches.

Introduction

Wireless sensor networks are intended to be deployed in hostile environments (battlefield, forest, etc.). Therefore, it is expected that a large number of cheap simple sensor devices will be randomly scattered over a region of interest. These devices are powered by batteries and have limited processing and memory capabilities. Among numerous challenges faced while designing WSN protocols, maintaining connectivity and maximizing the network lifetime stand out as critical considerations. The connectivity condition is generally met by deploying dense homogeneous networks to increase resources per unit area. However, dense networks can have several problems, such as device management and increased transmission interference and contention. Another approach is to use specialized nodes with long-range communication capabilities to maintain a connected network. The second consideration, network lifetime, is directly related to how long the power resources in sensor nodes will last. The network lifetime can be increased by designing and using energy efficient-protocols and algorithms. An example would be a scheduling scheme to make sensors work in batches to extend the network life [START_REF] Gallais | Localized sensor area coverage with low communication overhead[END_REF]. Another solution is to add actuators, i.e. mobile sensor nodes, that can be moved to areas where resources are most needed to efficiently route packets. Actually, it has been shown [START_REF] Wang | Extending the lifetime of wireless sensor networks through mobile relays[END_REF] that deploying resource rich mobile devices in a network can provide the same performances as increasing the network density.

The main motivation of this work is to take advantage of node mobility to extend the life of the network resource and consequently the network itself. Currently, even though the idea of using mobile actuator nodes to improve the performance of the network is well recognized, there is not much work that takes advantage of node mobility to improve routing in wireless networks while ensuring network connectivity. Available solutions adopt existing routing protocols to find an initial route, and iteratively move each node to an arbitrary location on the straight line connecting the source-destination pair. However, adopted node relocation strategies may cause useless zig-zag movements of nodes [START_REF] Goldenberg | Towards mobility as a network control primitive[END_REF] and may disconnect the network [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF]. Moreover, in [START_REF] Goldenberg | Towards mobility as a network control primitive[END_REF][START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF] we argue that the associated energy optimization model is incomplete as it does not incorporate the mobility cost in the routing decision.

In this paper, we propose CoMNet, a Connectivity preservation Mobile routing protocol for actuator and sensor NETworks, an energy efficient positionbased routing protocol which takes advantage of node mobility to minimize energy consumption and ensure network connectivity. CoMNet is based on a cost-over-progress metric where the optimized cost includes both moving and transmission costs. CoMNet has three variants, ORouting on the move, M ove (r) and M ove (DSr) described in more details in Section 4. These variants have the same objective but are intended for different applications. Both M ove (DSr) and ORouting on the move are intended for use in high traffic networks. They assume that the path will be highly used for a long time, so they establish a path on the straight line (the shortest distance) connecting a source-destination pair. Connecting two nodes and aligning relaying nodes on a straight line reduces energy consumption during transmission between the source and destination. Indeed, it has been shown that the straight line is the least energy consuming path connecting a source-destination pair [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF]. ORouting on the move focuses more on optimizing the move distance while M ove (DSr) focuses on optimizing the transmission cost. The third variant M ove (r) assumes that the network traffic between the source and the destination is light thus it is not optimum to move and align nodes. All CoMNet variants have the following properties:

-Localized: A routing decision depends only on local information. A node has to know only its geographical location, the ones of its neighbors and of the final destination.

-Scalable: CoMNet is memoryless. No routing information needs to be stored at a node or in a message.

-Loop free: A message is always sent to a node in the forwarding direction of the destination.

-Energy efficient: In its routing decision, CoMNet considers both the minimum cost of sending a message and moving a node to its new location. At each step, it chooses the least energy consuming solution.

-Guaranteed connectivity: Though nodes may be mobile and may be relocated, CoMNet guarantees that the network connectivity is maintained at all times. This is achieved by first, relying on a connected dominating set (CDS), second, by enforcing that nodes in a CDS to be static and mobile nodes to lay within communication range of at least one dominating node.

This paper is organized as follows. Section 2 reviews literature related to CoMNet. Section 3 presents our model and assumptions. CoMNet principles along with its three variants are described in Section 4 and evaluated in Section 5. We conclude and present future work in Section 6.

Related Work

Before proceeding into the analysis, we present a brief overview of some work of the literature related to position based routing algorithms in static and mobile networks and dominating set algorithms. This is by no means exhaustive and is only indicative of the interest and the applications. For a complete survey, the reader should refer to [START_REF] Frey | Routing in wireless sensor networks[END_REF]. Position based routing in static networks. Position based routing algorithms for static sensor networks have been widely studied in the literature. The basic principle is as follows. Every node is aware of its position, the ones of its neighbors and of the destination. A routing decision is made based only on this local information. In greedy routing [START_REF] Finn | Routing and addressing problems in large metropolitan-scale. Internetworks[END_REF] for instance, the node currently holding a packet forwards it to the neighbor closest to the destination. This method was then extended to energy efficient variants [START_REF] Kuruvila | Progress and location based localized power aware routing for ad hoc sensor wireless networks[END_REF][START_REF] Hamouda Elhafsi | Flattening the gap between sourcedestination paths in energy efficient greedy georouting in wireless sensor networks[END_REF], to guaranteed delivery solutions [START_REF] Bose | Routing with guaranteed delivery in ad-hoc wireless networks[END_REF] and to a combination of both energy efficient and guaranteed delivery approaches [START_REF] Sanchez | Exploiting local knowledge to enhance energyefficient geographic routing[END_REF][START_REF] Elhafsi | End-to-End Energy Efficient Geographic Path Discovery With Guaranteed Delivery in Ad hoc and Sensor Networks[END_REF]. In ORouting [START_REF] Hamouda Elhafsi | Flattening the gap between sourcedestination paths in energy efficient greedy georouting in wireless sensor networks[END_REF], the neighbor closest to the line (SD) connecting the source node S to the destination node is considered as the best candidate relay to minimize energy. Cost Over Progress (COP) based routing [START_REF] Kuruvila | Progress and location based localized power aware routing for ad hoc sensor wireless networks[END_REF] is a localized metric aware greedy routing scheme. A node u forwards a packet to a neighbor v in the forwarding direction of the destination D such that the ratio of the energy consumed for sending a message from u to v (any cost metric can be used) to the progress made (measured as the reduction in distance to D) is minimized. Routing in mobile networks. Little work has been done in routing in mobile sensor networks. We will address this relationship in our analysis section.

Current solutions adopt existing routing protocols to find an initial route, and iteratively move each node to the midpoint of its upstream and downstream nodes on the route. However, these routing protocols may not be efficient. Moving strategy in [START_REF] Goldenberg | Towards mobility as a network control primitive[END_REF] may cause useless zig-zag movements. In MobileCOP [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF], next hop on the path is selected based on COP [START_REF] Kuruvila | Progress and location based localized power aware routing for ad hoc sensor wireless networks[END_REF] metric. Once a path is computed, its nodes are moved and placed equidistantly on the straight line connecting the source to the destination, maintaining the same number of hops as in the computed path. Such move may induce a memory overhead on nodes, since they have to store the path, and a high transmission delay. More importantly, the network may be disconnected (a node may move out of range of its neighbors). This arbitrary move may not be optimum as when a node moves further from its upstream neighbor, the transmission delay will automatically increase. In addition, none of these approaches consider the cost of moving in the routing decision. A closer look to the basic ORouting (described previously) suggests that the protocol minimizes the moving cost of a candidate node to line (SD) and locally decreases the transmission cost. Dominating sets. Dominating sets (DS) are defined as follows. Each node in a graph either belongs to a dominating set or has a neighbor in the DS. The DS is called a Connected Dominating Set if the DS nodes are connected. The problem of computing the smallest CDS is known to be NP-complete even if knowledge of the global topology is available. Dai and Wu [START_REF] Dai | An extended localized algorithm for connected dominating set formation in ad hoc wireless networks[END_REF] introduced a generalized DS concept, where coverage can be provided by an arbitrary number of connected one-hop neighbors. The definition was modified by [START_REF] Simplot-Ryl | Energy efficient backbone construction, broadcasting, and area coverage in sensor networks[END_REF], to avoid message exchange between neighbors, as follows. Node a is covered by its one-hop neighbors b, c, . . . if these neighbors are connected. It is then further simplified in [START_REF] Carle | Energy efficient area monitoring by sensor networks[END_REF] as follows. First, each node checks if it is an intermediate node. Then each intermediate node a constructs a subgraph G ′ of its neighbors with higher key values. If G ′ is empty or disconnected then a belongs to the DS. If G ′ is connected but there exists a neighbor of a which is not a neighbor of any node in G ′ then a is in the CDS. If position information of 1-hop neighbors is available, nodes can decide whether or not to belong to a so defined CDS without exchanging any message with their neighbors. Note that these algorithms are local and do not incur any additional message exchange overhead.

Models

Before introducing CoMNet, we present in this section the assumptions of the protocols and define the cost models involved in its design and functionality. General model assumptions. We consider a sensor network where nodes are randomly scattered, are aware of their geographical location and are able to tune their transmission range between 0 and R (> 0). We also assume that sensor/actuator devices can be either mobile or stationary. The latter assumption can be further relaxed by making all nodes static or all nodes mobile. We denote by N (u) the set of physical neighbors of node u, i.e. the set of nodes in communication range of node u (N (u) = {v | |uv| < R} where |uv| is the Euclidean distance between u and v). Let δ(u) = |N (u)| be the cardinality of N (u), also called the degree of node u. We also define N D (u) the set of neighbors of node u with positive progress toward destination node D:

N D (u) = {v ∈ N (u) ∧ |vD| < |uD|} .
Transmission cost. We denote by C s (.) the cost measured in units of energy consumed to transmit a packet. We use the most common energy model [START_REF] Rodoplu | Minimizing energy mobile wireless networks[END_REF]:

C s (r) = r α + c if r = 0, 0 otherwise, ( 1
)
where r is the distance between a sender and a receiver; c is the overhead (in units of energy) due to signal processing; α is a real constant (> 1) that represents the signal attenuation. The optimal transmission radius, that minimizes the total power consumption for a routing task is r * = α c α-1 assuming that nodes can be placed on a straight line toward the destination [START_REF] Stojmenovic | Power-aware localized routing in wireless networks[END_REF]. Mobility cost. We denote by C m (.) the cost, measured in units of energy consumed, to relocate a node. To the best of our knowledge, there is no accurate model to define such a cost. Therefore, in this work, we use the model adopted in the literature [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF][START_REF] Goldenberg | Towards mobility as a network control primitive[END_REF]:

C m (|vv ′ |) = a|vv ′ | (2)
where v denotes a node at its original position (x v , y v ) (before it is moved). To eliminate confusion, we refer to node v after it has moved as v ′ and its new position is (x ′ v , y ′ v ). |vv ′ | is the Euclidean distance between v and v ′ and a is a constant to be defined.

In this work, we adopt the above cost models because they are widely used and as a proof of concept. However, other cost metrics can be considered.

CoMNet

Motivation

The pitfall of node mobility is the risk of the network to be disconnected once nodes have moved and consequently routing fails. To support our claim, we study the behavior and performance of MobileCOP [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF] in terms of network connectivity. We consider packet transmission between a single source-destination pair. As this method dictates, after every transmission, nodes along the computed paths move. Figure 4(a) displays the percentage of times the network remains connected as a function of the number of routes computed (or number of sourcedestination pair transmissions). Results show that for low node densities (δ ≤ 5), after the first routing task, the network remains connected only in 75% of the cases. After computing 10 consecutive routes, the network remains connected only in 25% of the cases. In fact, node density has to reach δ = 20 to keep a network connected after 10 computed routes. As the number of computed paths increases, to keep the network connected the value of δ has to be large.

CoMNet principles

To prevent the network to disconnect, CoMNet uses the notion of CDS. Furthermore, and contrary to other routing protocols, to minimize overall energy consumption CoMNet not only optimizes transmission energy but also mobility cost. In the following we describe both steps in more details. CoMNet and network connectivity. We claim that as long as every node is connected and more specifically is neighbor to a node in a CDS, the network will never be disconnected. Thus, the first step in CoMNet is to locally compute a CDS, using any one of the methods described in Section 2. At this step, every node is covered by the CDS. In order to keep this property, a node is moved if and only if its final position is covered by the CDS. In CoMNet, nodes that belong to a dominating set never move during a next hop selection. CoMNet and routing cost. To select a forwarding node, CoMNet uses a cost to progress criteria. Current node u chooses node v ∈ N D (u) among its neighbors in the forwarding direction of the destination D. More specifically, the selected node v (v ∈ N D (u)) minimizes the ratio of the global cost (packet transmission cost and node relocation cost) to the progress made towards the destination D. Indeed, v satisfies the following optimization problem:

v = arg min v∈N D (u) C ′ s (|uv|) + C m (|vv ′ |) |uD| -|v ′ D| with C ′ s (|uv|) = C s (|uv|) send before v moves C s (|uv ′ |) + ǫ send after v moves to v'
where v ′ represents the position where v should move (v = v ′ if v is a dominating node). C m and C s are respectively moving and sending cost functions defined by Eq. 2 and 1. In the send after move case, before moving v, a beacon is sent to v to request and advertise its move to v ′ (the cost of sending the beacon is ǫ). Once v moves, u sends its message to v ′ . The transmission cost used depends on the CoMNet variant used which is described in the following paragraphs. The selection of the next hop is summarized in Algorithm 1.

There are major differences between CoMNet and existing solutions. First, CoMNet incorporates the cost of moving in its routing decision and it is memoryless (it does not require nodes to store a computed path to be used at a later time). Second, to preserve network connectivity in spite of node movement, in CoMNet a node moves if and only if it does not belong to the DS and its targeted position is within the communication range of at least one dominating node. Third, CoMNet is general, it can assume that all nodes are mobile or all nodes are static. It can be easily adapted to heterogeneous networks composed of both actuators and sensors, by simply setting v = v ′ for sensor nodes which cannot move. Now we describe CoMNet variants, ORouting on the Move, M ove (DSr) , M ove (r) . The selection principles is the same for all variants. They differ mainly in the computed cost criteria and in the node relocation scheme. We will show that each variant has its own advantages depending on the application.

Algorithm 1 SelectNextHop(u,D) -Run at node u toward destination node D.

1: if ND(u) = ∅ then 2:
Return NULL {Routing has failed.} 3: end if 4: A ← ∅ 5: for all {v ∈ ND(u)} do 6:

v ′ ← NewLocation(u, v, D) {Compute position v ′ . v = v ′ if v ∈ CDS} 7: if {∃a ∈ CDS | |av ′ | < R} then 8: A ← v 9:
end if 10: end for 11:

w ← argmin v∈A Cm(|vv ′ |)+C ′ s (|uv|) |uD|-|v ′ D|
12: Return w {Routing has succeed.}

CoMNet: ORouting on the move

ORouting on the move is based on the plain ORouting [START_REF] Hamouda Elhafsi | Flattening the gap between sourcedestination paths in energy efficient greedy georouting in wireless sensor networks[END_REF] algorithm. It assumes that the traffic sent from the source to the destination is high and a computed path will be highly utilized for a long time. Therefore, the optimal path computed is on a straight line connecting the source to the destination. ORouting on the move further aligns nodes on the straight line in order to reduce the overall path length and consequently the overall routing energy consumed. Source node S selects next hop neighbor A based on the cost-to-progress criteria then takes advantage of mobility and moves the selected node. In Figure 1, which illustrates such a routing, node S has to select the next hop among nodes A 1 , A 2 and A 3 (nodes in forward direction to D). Cost over progress metric is computed for each node ( 2+1 3 = 1 for A 1 , 1+1 1 = 2 for A 2 and 5+2 6 = 1.17 for A 3 ). Having the smallest cost, node A 1 is selected and moved to location A ′ 1 , intersection point of the line connecting the source to the destination and its perpendicular line passing through A 1 . Note that ORouting on the move objective is to minimize the move distance. ORouting on the move is thus well adapted to situations where moving is very costly. In this case, S first needs to send a beacon to node A 1 to request and advertise its move on the line. Once A 1 moves to position A ′ , S forwards the message to A ′ . The sending cost is C ′ s (|SA ′ |) + ǫ and the move cost is C m (|AA ′ |).

CoMNet: Move (DSr)

As in ORouting on the move, M ove (DSr) aims at aligning nodes along a straight line in order to reduce the energy consumed by consecutive transmission from S to D. The objective in this case is to move the selected neighbor to a new position r ′ distant from the source (or relay) node to optimize routing energy consumption. The idea here is to compute a routing path from S to D where all nodes are aligned on line (SD) and where all hop lengths are equal to the optimal transmission distance r * . Since this objective is not always achieved ( |SD| r * is not always an integer), we opted for a hop length r ′ close to the optimal value such that all the nodes in the path are equidistant. The challenge here is to determine the proper value of r ′ . As mentioned in Section 2, since the optimal range r * has a closed form expression, the optimal number of hops n * is expressed as

n * = |SD| r *
. We can thus infer that r ′ is equal to

r ′ = |SD| n * if r * -|SD| n * < |SD| n * -1 -r * |SD| n * -1 otherwise (3)
Note that in the rare case where |SD| = kr * , k a positive integer value, r ′ = r * . An illustration of this method is given in Figure 2 where nodes A i , i ∈ {1, 2, 3} are neighbors of S in the forward direction. Only one of the nodes A i will be selected by S based on the energy cost. In the case presented in Figure 2, A 2 will be selected (cost is 2 + 2 = 4 for A 1 , 2.5 + 1 = 3.5 for A 2 and 5 + 2 = 7 for node A 3 ) and will move to A ′ location such that |SA ′ | = r ′ . Clearly, the progress is a constant for a given neighborhood and does not affect the objective function. It is worth noting that if a node B already lays in position A ′ , it is useless to move A ′ . B will be the selected neighbor. Once the neighbor is selected the packet is transmitted before the node is moved to its new location. The sending cost here is

C ′ s (|AA 2 |).

CoMNet: Move (r)

As in M ove (DSr) , the objective is to select a neighbor and move it to a new position r * distant from the source (relay) in order to decrease the sending cost. But contrary to the previous methods, the new location has to be on the circle C r * of radius r * centered at S. The main objective is to reduce the moving distance while minimizing the sending cost. This variant of CoMNet is used in light traffic network where routes are seldomly used for long transmission and do not need to be fully optimized to align nodes on line (SD). On the other hand, if node A is such that |SA| < r * , A should be relocated on the intersection of C r * and line (AD). If node A is such that |SA| > r * , A should be relocated on the intersection of C r * and line (SD). An illustration of this method is given in Figure 3 where nodes A i , i = 1, 2, 3 are neighbors of S in the forward direction. Only the selected neighbor A i will be moved to node A ′ i location. Node S selects the node which provides the smallest positive COP ratio, i.e. node A 2 (COP for A 1 is 2+0,5 3 = 0, 83, COP for A 2 is 1+1 3 = 0, 66, COP for A 3 is 5+0,5 2,5 = 2, 2). As in M ove (DSr) , the packet is transmitted before the node is moved to its new location. The sending cost is equal to

C ′ s (|AA 1 |).

Analysis and experimental results

We compare the performance of all CoMNet variants ORouting on the move, M ove (DSr) and M ove (r) to MobileCOP. We use WSNet/Worldsens [START_REF] Fraboulet | Worldsens: Development and prototyping tools for application specific wireless sensors networks[END_REF] eventdriven simulator for large scale wireless sensor networks, that assumes 802.11 DCF MAC layer, free space propagation model and packet collisions. For simulation purpose we assume that nodes are uniformly distributed over a 1000×1000 square and can adapt their range between 0 and R = 150. The variants are compared for the same system environment: same samples of node distribution, same source-destination pairs for various network node densities. We consider only connected networks. We compare the variants based on resource consumption and routing success.

Routing success rate

Energy consumption

We evaluate the energy consumption of each algorithm based on the energy models described in Eqs. 1 and 2. To compute C ′ s (.) the energy consumed to send a packet, we use parameter values used in the literature [START_REF] Kuruvila | Progress and location based localized power aware routing for ad hoc sensor wireless networks[END_REF], i.e. c = 3 8 and α = 4, which lead to an optimal transmission range of r * = 100. As stated previously, nodes will be equipped with GPS devices for localization purpose and we assume that the energy consumed for each node to identify its location is the same for all nodes. We also assume that the cost to exchange Hello packets is the same for every node. As such, these consumptions are considered as a constant and will not be included in our energy optimization model. For each source-destination pair our simulator computes 10 consecutive routes between a given source-destination pair and averages the computed statistics of more than 1000 tries. Regarding the mobility model's (Eq. 2) parameter a, since not much research has been done in this area, we run simulations for three different values of constant a, computed as follows.

1. if sending is as costly as moving, C ′ s (.) = C m (.), a is solution to the following equation:

C ′ s (r * ) = C m (r * ).
2. if sending is much more costly than moving, C ′ s (.) >> C m (.), then a is solution to the following equation: C ′ s (r * ) = 10 2 C m (r * ). 3. if moving is much more costly than sending, C ′ s (.) << C m (.), then a is solution to C ′ s (r * ) = 10 -2 C m (r * ).

Energy consumed during consecutive routing. In CoMNet as in Mobile-COP, paths are computed for every packet transmission. As such, the first time a path is computed, nodes are more likely to move than in the second time a path is computed between the same source-destination pair. For this reason, we choose to evaluate the cumulative energy consumed during a fixed number of packet transmissions (or consecutive path computations) to allow a fair comparison of the schemes. Certain protocols do not need to move nodes again in the second route computation.

Figures 5(a), 5(c) and 5(e) display the cumulative energy spent (to move and to send) by each CoMNet variant and MobileCOP, for each mobility model for a node density of 300 nodes and as a function of the number of successive routings. Our extensive simulations show that the algorithms have the same behavior for low network node densities (due to page restrictions supporting data cannot be presented).

In M ove (DSr) , nodes are moved to the desired location every time a path is computed. However, in MobileCOP nodes are moved only after the first path is computed then-after, nodes memorize the path which will be used in future transmissions. Therefore, energy added at subsequent route computation results only from sending cost. Results show that Routing on the move and M ove (r) converge quickly since they use the same path after computing the first 3 consecutive routes from S to D. Since MobileCOP does not take into account the moving energy consumption in its routing decision, the distance moved by each node is the same for each model, but has different costs. Our analysis show that Mobile-COP has a very high starting consumption cost due to node displacement. Since CoMNet incorporates the moving cost in its objective function, the initial cost remains low for CoMNet variants. Note that when moving is much more costly than sending (Figure 5(e)), ORouting on the move consumes the least energy, followed by M ove (r) . This is due to the inherent goals of each variant: ORouting on the move has a bias towards the move cost so it tries to minimize the move distance while M ove (DSr) 's priority is to minimize the sending cost. M ove (r) which aims at minimizing both energy costs has an average performance compared to the other variants. When sending cost is higher (Figure 5(c)) or close to moving cost (Figure 5(a)), the performance of all schemes are comparable. In this case, MobileCOP and M ove (r) have the best performance.

For all models, the cost difference between CoMNet and MobileCOP is even bigger for low node densities since nodes in MobileCOP move longer distances as candidate nodes are on average located farther away from the direct line between the source and the destination. This phenomena is counter-balanced in the other schemes by the fact that the lower the network density, the higher the proportion of dominating nodes. Thus, dominating nodes are more likely to be selected as next hop candidates, as a result routing will not incur any move cost.

Effect of network density on protocol performance. We select a sourcedestination pair (S and D) and allow 10 consecutive route computations between S and D. Figures 5(b), 5(d) and 5(f) display the cumulative energy consumed after all consecutive routing tasks using each CoMNet variant and MobileCOP. The analysis is conducted for various network densities and for different mobility models. Note that MobileCOP runs independently of the mobility cost. The energy consumed smoothly decreases as the number of nodes increases. This is due to the fact that for high network density MobileCOP can select a relay node at the right location saving the energy to move it. To the contrary, CoM-Net variants behave differently. When moving cost is equivalent to sending cost, CoMNet variants behave similar to MobileCOP. When moving is less costly than sending, CoMNet variants try to favor sending rather than moving. Nevertheless, for low densities, by construction of the CDS, almost every node belongs to the CDS and thus can not move. The energy consumed is thus mainly the energy consumed to send messages. Since sending is more expensive than moving, CoM-Net consumes more energy than MobileCOP. As the network density increases, the proportion of dominant nodes decreases and thus, more nodes can move to the proper positions and hence, CoMNet variants outperform MobileCOP. For these settings, ORouting variant is the best performing technique (since it tries to minimizes the moving distance), followed by M ove (r) and M ove (DSr) . The latter tries to favor sending over mobility energy consumption. When sending is more expensive than moving, CoMNet adapts to the network density (unlike MobileCOP where the moving cost is not considered in the optimized cost). For various network density, CoMNet outperforms MobileCOP. In addition, in all cases, CoMNet ensures the network connectivity and is memoryless, which is not the case for MobileCOP. For these settings, the best CoMNet variant is M ove (DSr) which tries to minimize the energy consume to send messages.

Conclusion and Future work

We introduce a novel protocol, CoMNet, that takes advantage of node mobility for efficient routing while ensuring network connectivity. The robustness of CoMNet compared to existing methods is due to the fact that it incorporates all costs (transmission as well as moving costs) in its routing decision. Note that the cost model does not have to be restricted to the transmission and moving costs it can be generalized to include receiving energy and other costs. Through extensive simulations, we show that CoMNet outperforms existing methods in terms of energy consumption and memory overhead. CoMNet has three variants that have different objective functions and mobility models. Each variant has its specific applications. Our future work will focus on the behavior of CoMNet in the presence of node conflict-when a node is solicited by more than one flow. Preliminary results show that M ove (r) is the most appropriate variant since it does not try to align nodes. Another interesting extension to this work would consider non-connected networks and explore node mobility to achieve network connectivity. This work is efficient for a single flow transmission, extending it to multiple flows may be an interesting problem; as flows share the same nodes and create bottlenecks, node conflict may complicate the routing schemes.

Fig. 1 .

 1 Fig. 1. ORouting on the move. Red arrows show possible displacement of nodes with associated moving costs. Solid links from S to Ai are associated to sending costs.

Fig. 2 .

 2 Fig. 2. M ove (DSr) . Red arrows show possible displacement of nodes with associated moving costs. Dashed links are associated to the sending cost.

Fig. 3 .

 3 Fig. 3. M ove (r) . Red arrows show possible displacement of nodes with associated moving costs. Dashed links are associated to the sending cost.

Figure 4 (

 4 Figure 4(b) shows the percentage of routing success with respect to network node density. Results show that CoMNet outperforms MobileCOP for low densities. For higher densities CoMNet and MobileCOP have similar performance. As expected, ORouting on the move, which routing behavior is the closest to MobileCOP, performs very close to MobileCOP. The best performing technique is M ove (r) variant.

  MobileCOP: Proportion of times the network gets disconnected wrt number of routes computed.

  Routing success rate: comparative analysis of CoMNet variants and MobileCOP routing.

Fig. 4 .

 4 Fig. 4. Disconnection and success rate

Fig. 5 .

 5 Fig. 5. Cumulative energy consumption for each algorithm.

This work was partially supported by CPER Nord-Pas-de-Calais/FEDER Campus Intelligence Ambiante and the ANR BinThatThinks project.