Keywords: Problème de Robin, problème inverse à la frontière, corrosion

Stabilité Lipschitzienne pour le problème inverse de Robin

Résumé : On considère un problème inverse lié a la détection de la corrosion. On prouve un résultat de stabilité Lipschitz pour un coefficient de Robin scalaire sur une partie inaccessible de la frontière depuis des mesures électrostatiques disponibles sur la partie accessible.

Introduction

In the following we shall consider the inverse problem of recovering a scalar Robin coefficient. The direct problem associated with the inverse one, is the following

             ∆u = 0 , in Ω , ∂u ∂ν = g , on Γ 2 , ∂u ∂ν = -Lu , on Γ 1 , u = 0 , on Γ D , (1.1)
where Γ 1 and Γ 2 are two open, disjoint portions of ∂Ω such that Γ D = ∂Ω \(Γ 1 ∪Γ 2 ).

According to this model, Ω represents the electrostatic conductor, u is the harmonic potential, g is the prescribed current density on the accessible part of the boundary Γ 2 , whereas on Γ 1 , the portion which is out of reach, we prescribe a Robin boundary condition with a scalar Robin coefficient L.

The aim of the inverse problem that we shall study is to recover the scalar Robin coefficient L when a fixed choice of the current density g and the measurement of the corresponding boundary voltage u| Γ2 on Γ 2 are available.

In the following we shall treat the stability issue for this inverse problem. Such an issue has been already discussed by Chaabane and Jaoua in [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF]. The authors provided an identifiability result as well as a Lipschitz stability estimate à la Bellout and Friedman for a Robin coefficient which depends on a scalar parameter only. The new feature of the present work consists in a quantitative evaluation of such a Lipschitz rate of stability, which is no longer a directional stability as in [START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF] and which is obtained with a constructive approach. Moreover, Alessandrini, Del Piero, Rondi [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF] and Chaabane, Fellah, Jaoua, Leblond [START_REF] Chaabane | Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse oroblems[END_REF] have established a logarithmic stability result for a non necessarily constant Robin coefficient, namely of the type

L = L(x) , x ∈ Γ 1 . (1.2)
In [START_REF] Chaabane | Parameter identification for the Laplace equation and approximation in Hardy classes[END_REF], Chaabane, Jaoua and Leblond have provided a constructive procedure in order to solve the Robin problem. We refer also to Fasino and Inglese [START_REF] Fasino | An inverse Robin problem for Laplace s equation: theoretical results and numerical methods[END_REF][START_REF] Fasino | Discrete methods in the study of an inverse problem for Laplace's equation[END_REF][START_REF] Fasino | Stability of the solutions of an inverse problem for Laplace's equation in a thin strip[END_REF], who have introduced numerical methods relied on the thin-plate approximation and the Galerkin method.

In the corrosion literature we can find another type of boundary condition which postulates a nonlinear relationship between the potential and the normal current density and it is associated to the name of Butler and Volmer. Such type of problem has been studied by Brian, Kavian, Vogelius and Xu [START_REF] Bryan | Singular solutions to a nonlinear elliptic boundary value problem originating from corrosion modeling[END_REF][START_REF] Kavian | On the existence and "blow-up" of solutions of a two dimensional nonlinear boundary value problem arising in corrosion modeling[END_REF][START_REF] Vogelius | A nonlinear elliptic boundary value problem related to corrosion modeling[END_REF]. The authors have dealt with the uniqueness and the existence issues for this problem with a special choice of the nonlinearity, that is of the type

∂u ∂ν = λ(exp(αu) -exp(-(1 -α)u)) . (1.3) RR n°6103 E. Sincich
Moreover, in [START_REF] Alessandrini | Detecting nonlinear corrosion by electrostatic measurements[END_REF][START_REF] Alessandrini | Solving elliptic Cauchy problems and the identification of nonlinear corrosion[END_REF] Alessandrini and the author have discussed the same problem but with a more general choice of the nonlinear profile of the form

∂u ∂ν = f (u) , (1.4)
establishing a logarithmic stability estimate as well as a reconstruction result. The present work fits in the framework of the study of the stability for coefficients which depend on a finite number of parameters only. Such an issue has been developed by many authors, among them we cite Alessandrini, Bacchelli, Beretta, Chaabane, Jaoua, Vessella [START_REF] Alessandrini | Determining linear cracks by boundary measurements: Lipschitz stability[END_REF][START_REF] Alessandrini | Lipschitz Stability for the Inverse Conductivity Problem[END_REF][START_REF] Bacchelli | Lipschitz stability for a stationary 2-D inverse problem with unknown poligonal boundary[END_REF][START_REF] Chaabane | Identification of Robin coefficients by the means of boundary measurements[END_REF]. Indeed, for what concerns the present problem, in [START_REF] Alessandrini | Stable determination of corrosion by a single electrostatic boundary measurement[END_REF] it has been shown in a 2-dimensional setting the stability for a functional Robin coefficient is at best logarithmic. Unfortunately this kind of stability is inconvenient for the application. For this reason is usefull to introduce some reasonable simplifications which are phisically meaningfull (as remarked in [START_REF] Alessandrini | Lipschitz Stability for the Inverse Conductivity Problem[END_REF]) and that lead to a better stability, namely of Lipschitz type.

Main assumptions and results

A priori information on the domain

We shall assume throughout that Ω is a bounded, connected domain in R n , n 2 such that diam(Ω) D and with Lipschitz boundary ∂Ω with constants r 0 , M . Moreover, we assume that the portions of the boundary Γ i are contained respectively into surfaces S i , i = 1, 2 which are C 1,α smooth with constants r 0 , M . We also suppose that the boundary of Γ i , within S i , is of C 1,α class with constants r 0 , M . We introduce some notation that we shall use in the sequel, for every ρ > 0 and i = 1, 2, we set

U i ρ = {x ∈ Ω : dist(x, ∂Ω \ Γ i ) > ρ} , (2.1)
Γ i,ρ = U 1 ρ ∩ Γ i . (2.2)
In some places, it will be necessary to isolate one privileged coordinate direction, to this purpose, we shall use the following notation for a point

x ∈ R n , x = (x , x n ), with x ∈ R n-1 , x n ∈ R.

A priori information on the boundary data

The current flux g is a prescribed function such that

g C 0,α (Γ2) G , (2.3)
g L ∞ (Γ2,2r 0 ) m > 0 .
(2.4)

INRIA

A priori information on the scalar Robin coefficient

Given Λ > 0, we assume that the scalar Robin coefficient L satisfies

0 < L Λ . (2.5)
From now on we shall refer to the a priori data as to the set of quantities r 0 , M, α, Λ, G, D, m.

In the sequel we shall denote by η(t) and ω(t), two positive increasing functions defined on (0, +∞), that satisfy

η(t) exp - t c -γ
, for every

0 < t G , (2.6
)

ω(t) C |log(t)| -θ , for every 0 < t < 1 , (2.7)
where c > 0, C > 0, γ > 1, 0 < θ < 1 are constants depending on the a priori data only.

The statement of the main result is the following.

Theorem 2.1. There exists a constant C > 0, depending on the a priori data only, such that

|L 0 -L 1 | C u 0 -u 1 L 2 (Γ2) , (2.8)
where u i , i = 0, 1 is the solution to (1.1) with L = L i .

Lipschitz stability

Given L 0 , L 1 > 0 and t ∈ [0, 1] , we shall denote by L t the following

L t = -[L 0 + t(L 1 -L 0 )] . (3.1)
Moreover, let u t ∈ H 1 (Ω) be the weak solution to

             ∆u t = 0 , in Ω , ∂u t ∂ν = g , on Γ 2 , ∂u t ∂ν = -L t u t , on Γ 1 , u t = 0 , on Γ D . (3.2)
As a weak solution to the problem (3.2) we shall mean a function u t ∈ H 1 (Ω), such that u t = 0 which satisfies

Ω ∇u t ∇ρ = Γ2 gρ - Γ1 L t u t ρ (3.3) for every ρ ∈ H 1 (Ω) such that ρ = 0 on Γ D .
The existence and the uniqueness of the solution to (3.2) is a consequence of standard theory on boundary value problem.

Proposition 3.1. There exists a constant C > 0 depending on the a priori data only such that

Ω |∇u t | 2 C (3.4)
for each t ∈ [0, 1] .

Proof.

Let us choose ρ = u t , then by (3.3) we have that

Ω |∇u t | 2 = Γ2 gu t - Γ1 L t u 2 t . (3.5)
From which follows

Ω |∇u t | 2 Γ2 gu t . (3.6)
Let ε > 0, by the Schwartz inequality we have that

Ω |∇u t | 2 1 ε Γ2 g 2 + ε Γ2 |u t | 2 . (3.7)
By a standard trace inequality (see for instance [START_REF] Adams | Sobolev Spaces[END_REF]) we have that there exists a constant C > 0 depending on the a priori data only such that

Ω |∇u t | 2 1 ε Γ2 g 2 + Cε Γ2 |∇u t | 2 . (3.8)
Choosing ε = 1 2C we have that

Ω |∇u t | 2 1 C Γ2 g 2 . (3.9)
Hence by (2.3) the thesis follows.

Theorem 3.2. Let t ∈ [0, 1] and let u t ∈ H 1 (Ω) be the solution to (3.2), there exists a constant C > 0 depending on the a priori data only such that

u t C 0,α (Γ1) C. (3.10)
Moreover, for any ρ ∈ (0, r 0 ), u t ∈ C 1,α (U 1 ρ ) and there exists a constant C ρ > 0 depending on the a priori data and on ρ only, such that the following estimate holds

u t C 1,α (U 1 ρ ) C ρ E , (3.11)
where 0 < α < 1 is a constant depending on r 0 , M, n only.

INRIA

Proof.

For any z 0 ∈ Γ 1 and for any ρ > 0, we shall denote

Γ ρ (z 0 ) = Ω ∩ B ρ (z 0 ) , (3.12
)

∆ ρ (z 0 ) = Γ ρ (z 0 ) ∩ ∂Ω. (3.13) Let 0 < ρ 1 < ρ 2 r 0 and let us consider a test function ϕ ∈ C 1 (Ω) such that i) 0 ϕ 1; ii) ϕ = 1 in Γ ρ1 (z 0 ) and ϕ = 0 in Ω \ Γ ρ2 (z 0 ); iii) |∇ϕ| 2 ρ 2 -ρ 1 .
For a sake of brevity, in the following we shall replace u t with u and L t with L. For any integer s 2, let us define the function ψ = |u| s-2 uϕ 2 . Hence, choosing ψ as test function in the weak formulation of the problem (3.3) we have that

Γρ 2 (z0) (s -1)|∇u| 2 |u| s-2 ϕ 2 + Γρ 2 (z0) 2∇u • ∇ϕ|u| s-2 uϕ = ∆ρ 2 (z0) L|u| s ϕ 2 . (3.14)
Hence,

Γρ 2 (z0) (s -1)|∇u| 2 |u| s-2 ϕ 2 Γρ 2 (z0) 2∇u • ∇ϕ|u| s-2 uϕ + (3.15)
+ ∆ρ 2 (z0) L|u| s ϕ 2 .
By applying the Hölder inequality to the first term on the right hand side of (3.15), we obtain

Γρ 2 (z0) 2∇u • ∇ϕ|u| s-2 uϕ 4 ρ 2 -ρ 1 Γρ 2 (z0) |∇u| 2 |u| s-2 ϕ 2 1 2 Γρ 2 (z0) |u| s 1 2
By the Schwartz inequality, it then follows that for every ε > 0

Γρ 2 (z0) 2∇u • ∇ϕ|u| s-2 uϕ (3.16) ε Γρ 2 (z0) |∇u| 2 |u| s-2 ϕ 2 + 16 (ρ 2 -ρ 1 ) 2 ε Γρ 2 (z0) |u| s

RR n°6103

Let us now consider the second term on the right hand side of (3.15). Furthermore by a trace inequality, see for instance [START_REF] Adams | Sobolev Spaces[END_REF]Theorem 5.22], we infer that

∆ρ 2 (z0) L|u| s ϕ 2 CL Γρ 2 (z0) |∇(|u| s ϕ 2 )| (3.17)
where C > 0 is a constant depending on the a priori data only. Hence by the Schwartz inequality, it then follows that for every ε > 0

∆ρ 2 (z0) L|u| s ϕ 2 (3.18) ε Γρ 2 (z0) |u| s-2 |∇u| 2 ϕ 2 + s 2 C 2 L 2 ε Γρ 2 (z0) |u| s + 4CL ρ 2 -ρ 1 Γρ 2 (z0) |u| s
Inserting (3.16) and (3.18) in (3.15), we obtain

(1 -2ε) Γρ 2 (z0) |u| s-2 |∇u| 2 ϕ 2 16 (ρ 2 -ρ 1 ) 2 ε + L 2 s 2 C 2 ε + 4CL ρ 2 -ρ 1 Γρ 2 (z0) |u| s Choosing ε = 1
4 in the above inequality we have that

Γρ 1 (z0) |u| s-2 |∇u| 2 32 (ρ 2 -ρ 1 ) 2 ε + 2L 2 s 2 C 2 ε + 2CL ρ 2 -ρ 1 Γρ 2 (z0) |u| s
By the Sobolev inequality, see for instance [1, Chap. 5], we have that

Γρ 1 (z0) |u| ns n-2 n-2 ns C(1 + s) ρ 2 -ρ 1 2 s Γρ 2 (z0) |u| s 1 s
, where n = n for n > 2, 2 > 2 and C > 0 is a constant depending on the a priori data only. Now, dealing as in [START_REF] Gilbarg | Elliptic Partial Differential Equations od Second Order[END_REF]Chap. 8], we observe that the above inequality can be iterated.

Indeed, setting s = s m = 2 n n-2 m and ρ m = r0 4 + 2 -m r0 4 , m = 0, 1, . . . , by (3.19) it follows u L sm Γ r 0 4 (z0) C n n -2 4m( n n-2 ) -m u L 2 Γ r 0 2 (z0) . (3.19) Letting m tends to ∞ in (3.19), we can infer that u L ∞ Γ r 0 4 (z0) C u L 2 Γ r 0 2 (z0) , (3.20)
∈ C 1 (Ω) such that i) 0 η 1; ii) η = 1 in Γ r1 (z 0 ) and η = 0 in Ω \ Γ r2 (z 0 ); iii) |∇η| 2 r 2 -r 1 .
By (3.21), we have that

M 2 = sup x∈Γr 2 (z0) u(x) < +∞. (3.22)
Let us define the following non-negative function

v(x) = M 2 -u(x), for every x ∈ Γ r2 (z 0 ). (3.23)
Let us introduce the following quantities.

For every ρ ∈ (0

, r0 4 ), let i) b = 2LC, ii) h = bM 2 ; iii) k = k(ρ) = ρ δ h, iv) b = b 2 + k -2 h 2 ; v) v = v + k.
where C > 0 is the constant appearing in the inequality (3.17) and δ is such that 0 < δ < 1. Let us define, for β ∈ R \ {0} the function χ = η 2 vβ . Hence choosing χ as test function in the weak formulation (3.3), it follows that Γr 2 (z0)

|∇v| 2 vβ-1 η 2 + 2 β Γr 2 (z0) ∇v • ∇ηηv β = - 1 β ∆r 2 (z0) f (M 2 -v)η 2 vβ . ( 3
|∇v| 2 vβ-1 η 2 + 2 β Γr 2 (z0) ∇v • ∇ηηv β 1 |β| ∆r 2 (z0) L|M 2 -v|η 2 vβ . (3.25) RR n°6103
Furthermore by the trace inequality used in (3.17), we have that

Γr 2 |∇v| 2 vβ-1 η 2 + 2 β Γr 2 ∇v • ∇ηηv β LC |β| Γr 2 (z0) |∇[(M 2 -v)η 2 vβ ]|.
After straightforward calculations, we have that

Γr 2 (z0) |∇v| 2 vβ-1 η 2 -LC Γr 2 (z0) |M 2 -v||∇v|v β-1 η 2 2 |β| Γr 2 (z0) |∇v||∇η|ηv β + 2LC |β| Γr 2 (z0) |M 2 -v||∇η|ηv β + + LC |β| Γr 2 (z0) |∇v|η 2 vβ . (3.26)
By the Schwartz inequality it follows that for every ε > 0

LC Γr 2 (z0) |M 2 -v||∇v|v β-1 η 2 ε Γr 2 (z0) |∇v| 2 vβ-1 η 2 + (3.27) + L 2 C 2 ε Γr 2 (z0) |M 2 -v| 2 vβ-1 η 2 .
Hence choosing ε = 1 2 in (3.27), we obtain

Γr 2 (z0) |∇v| 2 vβ-1 η 2 -LC Γr 2 (z0) |M 2 -v||∇v|v β-1 η 2 (3.28) 1 2 Γr 2 (z0) |∇v| 2 vβ-1 η 2 -2L 2 C 2 Γr 2 (z0) |M 2 -v| 2 vβ-1 η 2 1 2 Γr 2 (z0) |∇v| 2 vβ-1 η 2 -b 2 Γr 2 (z0) v 2 vβ-1 η 2 -h 2 Γr 2 (z0) vβ-1 η 2 .
Moreover, observing that b 2 v 2 + h 2 bv 2 , by (3.28) we can infer that Γr 2 (z0)

|∇v| 2 vβ-1 η 2 -LC Γr 2 (z0) |M 2 -v||∇v|v β-1 η 2 (3.29) 1 2
Γr 2 (z0)

|∇v| 2 vβ-1 η 2 -2 b Γr 2 (z0) vβ+1 η 2
On the other hand we have also that

Γr 2 (z0) |∇v||∇η|ηv β + LC Γr 2 (z0) |M 2 -v||∇η|ηv β = Γr 2 (z0) 1 2 (2|∇v| + bv + h) |∇η|ηv β . ( 3
|∇v| 2 vβ-1 η 2 -2 b Γr 2 (z0) vβ+1 η 2 (3.33) ε |β| Γr 2 (z0) |∇v| 2 η 2 vβ-1 + 1 ε|β| Γr 2 (z0) |∇η| 2 vβ+1 + + 1 2|β| Γr 2 (z0) |∇η| 2 vβ+1 + b 2|β| Γr 2 (z0) η 2 vβ+1 + + ε Γr 2 (z0) |∇v| 2 vβ-1 η 2 + L 2 C 2 ε|β| 2 Γr 2 (z0) η 2 vβ+1 .
From the above inequality it follows that

1 2 - ε |β| -ε Γr 2 (z0) |∇v| 2 vβ-1 η 2 Γr 2 (z0) 2 b + b 2|β| + L 2 C 2 ε|β| η 2 vβ+1 + Γr 2 (z0) 1 2|β| + 1 ε|β| |∇η| 2 vβ+1 . ( 3
|∇v| 2 vβ-1 η 2 Ĉ Γr 2 (z0) η 2 + |∇η| 2 vβ+1 , (3.35)
where Ĉ is a positive constant depending on |β|, L, C, M 2 , ρ, δ.

RR n°6103

Let w be a function defined as follows

w = v β+1 2 , if β = -1, log v, if β = -1.
Hence we can reformulate (3.35) as follows

Γr 2 (z0) |η∇w| 2        (β + 1) 2 Ĉ Γr 2 (z0) η 2 + |∇η| 2 w 2 , if β = -1 , Ĉ Γr 2 (z0) η 2 + |∇η| 2 , if β = -1.
(3.36)

By the Sobolev inequality, see for instance [1, Chap. 5], we have that

ηw 2 L 2n n-2 (Γr 2 (z0)) C Γr 2 (z0) |η∇w| 2 + |w∇η| 2 (3.37)
where n = n for n > 2, 2 > 2 and C > 0 is a constant depending on the a priori data only. Combining (3.36) and (3.37) we obtain

ηw 2 L 2n n-2 (Γr 2 (z0)) c(β + 1) 2 Γr 2 (z0) (η 2 + |∇η| 2 )w 2 , (3.38)
where c > 0, depending on the a priori data, on ρ, on |β| and on δ only, is bounded when |β| is bounded away from zero. Hence from (3.38) we obtain

w 2 L 2n n-2 (Γr 1 (z0)) c (|β + 1| + 1) r 2 -r 1 w L 2 (Γr 2 (z0)) (3.39)
where c > 0, depending on the a priori data, on ρ and on δ only. At this stage arguing as in [START_REF] Gilbarg | Elliptic Partial Differential Equations od Second Order[END_REF]Theorem 8.18], we obtain the following weak Harnack inequality for the function v. For every 0 < ρ < r0 16 , we have that

ρ -n v L 1 (Γ2ρ(z0)) C inf Γρ(z0) v + ρ δ |M 2 | , (3.40)
where C > 0 is a constant only depending on the a priori data . On the other hand by (3.21) we have also that,

m 2 = inf x∈Γr 2 (z0) u(x) < +∞. (3.41)
Then, we define the following non-negative function

z(x) = u(x) -m 2 for every x ∈ Γ r2 (z 0 ). (3.42) INRIA
Hence, by analogous arguments to those developed for the function v, we find also the following weak Harnack inequality for the function z. For every 0 < ρ < r0 16 , we have that

ρ -n z L 1 (Γ2ρ(z0)) C inf Γρ(z0) z + ρ δ |m 2 | , (3.43)
where C > 0 is a constant only depending on the a priori data .

For every ρ ∈ (0, r0 16 ), let us denote

M (ρ) = sup Γρ(z0) u , (3.44)
m(ρ) = inf Γρ(z0)
u.

(3.45) By (3.21),(3.40) and (3.43), we have that there exists a constant K > 0 depending on the a priori data only, such that

ρ -n Γ2ρ(z0) (M 2 -u) K M 2 -M + ρ δ , (3.46) ρ -n Γ2ρ(z0) (u -m 2 ) K m -m 2 + ρ δ . (3.47)
Moreover, let us observe that being the boundary ∂Ω of Lipschitz class, we have that there exists a constant c 1 > 0, depending on r 0 , M only, such that for every ρ ∈ (0, r0 16 ) where c 2 = 2K and γ = 1 -c1 K . By the arguments in [START_REF] Gilbarg | Elliptic Partial Differential Equations od Second Order[END_REF]Lemma 8.23], it follows that for any µ ∈ (0, 1) and any 0 < ρ ρ 0

ρ -n |Γ 2ρ (z 0 )| c 1 . ( 3
r0 16 ω(ρ) C ρ ρ 0 α ω(ρ 0 ) + c 2 ρ µδ ρ 0 (1-µ)δ , (3.51)
where C is a constant depending on the a priori data only, whereas α is such that α = (1µ)( log(γ) log( 14 ) ). Hence choosing µ such that (1µ) log(γ) log( 14 ) < µδ, we have that (3.51) leads to

ω(ρ) ρ α c ρ 0 -α ω(ρ 0 ) + ρ β , (3.52)
where c is a constant depending on the a priori data only and β is such that β = µ(δ -1)α + 1 > 0. Furthermore, we have that the above inequality and (3.21) lead to

ω(ρ) ρ α c ρ 0 -α 2CE + ρ β , (3.53)
where C is a constant depending on the a priori data only.

Hence we can infer that for any

z 0 ∈ Γ 1 u C 0,α Γ r 0 16 (z0) CE. (3.54)
where C > 0, 0 < α < 1 are constants depending on the a priori data only. Since, we know that u ∈ C 0,α (Γ 1 ), we have that

∂u ∂ν (x) = -Lu(x) ∈ C 0,α (Γ 1 )
.

By well-known regularity bounds for the Neumann problem (see for instance [2, p.667]) it follows that u ∈ C 1,α (U 1 ρ ) and the following estimate holds

u C 1,α (U 1 ρ ) C u C 0,α (Γ ρ 2 1 ) + ∂u ∂ν C 0,α (Γ ρ 2 1 ) + ∇u L 2 (Ω) C ∂u ∂ν C 0,α (Γ ρ 2 1 ) + E (3.55)
where C > 0 depends on the a priori data and on ρ only. Moreover, we can estimate the

C 0,α norm of ∂u ∂ν in terms of E, in fact ∂u ∂ν C 0,α (Γ ρ 2 1 ) = sup x∈Γ ρ 2 1 ∂u(x) ∂ν + ρ 2 α sup x,y∈Γ ρ 2 1 ∂u(x) ∂ν -∂u(y) ∂ν |x -y| α = = sup x∈Γ ρ 2 1 |Lu(x)| + L ρ 2 α sup x,y∈Γ ρ 2 1 |u(x) -u(y)| |x -y| α . INRIA Hence we have ∂u ∂ν C 0,α (Γ ρ 2 1 )
L sup

x∈Γ ρ 2 1 |u(x)| + L ρ 2 α sup x,y∈Γ ρ 2 1 |u(x) -u(y)| |x -y| α L u C 0,α (Γ1) CE . (3.56)
So inserting this estimate in (3.55) we obtain the thesis.

Lemma 3.3. The mapping

S t : [0, 1] → H 1 (Ω) (3.57) t → u t (3.58)
is differentiable and its derivative u t is Lipschitz continuous with respect to t. Moreover, there exists a constant C > 0, depending on the a priori data only, such that

u t H 1 (Ω) C|L 1 -L 0 |, ∀ t ∈ [0, 1] , (3.59) u t H 1 (Ω) C|L 1 -L 0 | 2 , ∀ t ∈ [0, 1] . (3.60)
Proof.

First we show that the map S t is well defined. Indeed, we observe that since L t > 0, the well-posedness of the problem (3.2) follows as consequence of the Lax-Milgram theorem. We show that the mapping S t is Lipschitz continuous. Let us pick t 1 , t 2 ∈ [0, 1], by the weak formulation of the problem (3.2) we have that

Ω |∇(u t1 -u t2 )| 2 = - Γ1 [L 0 + t 1 (L 1 -L 0 )]u t1 (u t1 -u t2 ) (3.61) + Γ1 [L 0 + t 2 (L 1 -L 0 )]u t2 (u t1 -u t2 ) . (3.62) Hence, Ω |∇(u t1 -u t2 )| 2 = Γ1 (t 2 -t 1 )(L 1 -L 0 )u t1 (u t1 -u t2 ) (3.63) + Γ1 -[L 0 + t 2 (L 1 -L 0 )](u t1 -u t2 ) 2 . (3.64)
Observing that the second integral on the right hand side is negative, we obtain that

Ω |∇(u t1 -u t2 )| 2 Γ1 |t 2 -t 1 ||L 1 -L 0 ||u t1 ||u t1 -u t2 | . (3.65) E. Sincich
Let ε > 0, then by the Schwartz inequality and well-known trace inequality (see for instance [START_REF] Adams | Sobolev Spaces[END_REF]) we have that

Ω |∇(u t1 -u t2 )| 2 1 ε Γ1 |t 2 -t 1 | 2 |L 1 -L 0 | 2 |u t1 | 2 + Cε Ω |∇(u t1 -u t2 )| 2 ,
where C > 0 is a constant depending on the a priori data only. Hence choosing ε = 1 2C we obtain that

Ω |∇(u t1 -u t2 )| 2 C Γ1 |t 2 -t 1 | 2 |L 1 -L 0 | 2 |u t1 | 2 , (3.66)
where C > 0 is a constant depending on the a priori data only. Hence by (3.10)

Ω |∇(u t1 -u t2 )| 2 C Γ1 |t 2 -t 1 | 2 |L 1 -L 0 | 2 , (3.67)
where C > 0 is a constant depending on the a priori data only. By (3.67) we deduce that the map S t is Lipschitz continuous and therefore, for almost every t, the derivative u t exists and satisfies

             ∆u t = 0 , in Ω , ∂u t ∂ν = 0 , on Γ 2 , ∂u t ∂ν + L t u t = -(L 1 -L 0 )u t , on Γ 1 , u t = 0 , on Γ D . (3.68)
Moreover by (3.67), (3.59) follows for almost every t.

By analogous arguments we can prove that u t is Lipschitz continuous. Therefore, (3.59) holds for every t and, for almost every t, the second derivative u t satisfies

             ∆u t = 0 , in Ω , ∂u t ∂ν = 0 , on Γ 2 , ∂u t ∂ν + L t u t = -2(L 1 -L 0 )u t , on Γ 1 , u t = 0 , on Γ D . (3.69)
We have that (3.60) follows for almost every t ∈ [0, 1] dealing with similar argument to those used to prove (3.59). Moreover, arguing as before, we observe that the map t → u t is Lipschitz continuous and we can conclude that (3.60) holds for every t ∈ [0, 1].

Theorem 3.4. There exists a constant C > 0, depending on the a priori data only, such that

u t L 2 (Γ2) C|L 1 -L 0 |, ∀ t ∈ [0, 1] , (3.70)
u t L 2 (Γ2) C|L 1 -L 0 | 2 , ∀ t ∈ [0, 1] . (3.71) INRIA Proof.
By a standard trace inequality (see for instance [START_REF] Adams | Sobolev Spaces[END_REF]) and by the previous Lemma, we deduce that

u t L 2 (Γ2) C|L 1 -L 0 |, ∀ t ∈ [0, 1] , (3.72)
u t L 2 (Γ2) C|L 1 -L 0 | 2 , ∀ t ∈ [0, 1] . (3.73)
By the Taylor formula we have that

u 1 -u 0 = u 0 + R on Γ 2 (3.74) where R = 1 0 (t -1)u t dt satisfies R L 2 (Γ2) C|L 0 -L 1 | 2 .
(3.75) Lemma 3.5. There exists a constant K > 0 depending on the a priori data only such that

u 0 L ∞ (Γ1) K . (3.76)
Proof.

Dealing as in [7, Theorem 2.1] we obtain that

u 0 L ∞ (Γ1) η( g L ∞ (Γ2,2r 0 ) ) 0 (3.77)
where η is the function defined in the formula (2.15) in [START_REF] Alessandrini | Detecting nonlinear corrosion by electrostatic measurements[END_REF]. Hence the thesis follows.

Lemma 3.6. Let u 0 be the solution to (3.68) with t = 0, then for any ρ ∈ (0, r 0 ) u 0 ∈ C 1,α (U 1 ρ ) and there exists a constant C ρ > 0, depending on the a priori data and on ρ only, such that the following estimate holds

u 0 C 1,α (U 1 ρ ) C ρ , (3.78)
where ρ ∈ (0, r 0 ) and 0 < α < 1 is a constant depending on r 0 , M, n only.

Proof.

First we shall prove an upper bound for the energy of u 0 . Indeed, by the weak formulation of (3.68) we have that

Ω |∇u 0 | 2 + Γ1 L 0 |u 0 | 2 = Γ1 (L 0 -L 1 )u 0 u 0 . (3.79)
Let ε > 0, applying the Scwhartz inequality the right hand side we have

Ω |∇u 0 | 2 + Γ1 L 0 |u 0 | 2 ε Γ1 |u 0 | 2 + 1 ε Γ1 |L 0 -L 1 |u 2 0 . (3.80) E. Sincich
Moreover, observing that the second term on the left hand side is positive we have that

Ω |∇u 0 | 2 ε Γ1 |u 0 | 2 + 2 ε ΛE. (3.81)
Choosing ε > 0 sufficiently small and by a trace inequality we have that, there exists a constant C > 0 depending on the a priori data only, such that

Ω |∇u 0 | 2 C. (3.82)
By Theorem 3.2 we know that u 0 ∈ C 0,α (Γ 1 ). Hence by well-known regularity results for the Neumann problem and using the same tecniques used in Theorem (3.2) as well as in [START_REF] Sincich | Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements[END_REF]Chap. 3] we deduce the thesis.

Proposition 3.7. There exists a constant C > 0 depending on the a priori data only, such that

u 0 L 2 (Γ2) c|L 0 -L 1 | . (3.83)
Proof.

Let us define the function

ũ0 = u 0 L 0 -L 1 (3.84)
where u 0 is the solution to the problem (3.69) with t = 0. It follows that ũ0 is the solution to the following problem

             ∆ ũ0 = 0 , in Ω , ∂ ũ0 ∂ν = 0 , on Γ 2 , ∂ ũ0 ∂ν + L 0 ũ0 = u 0 , on Γ 1 , ũ0 = 0 , on Γ D . (3.85)
Elaborating a classical estimate of unique continuation from the boundary [START_REF] Trytten | Pointwise bound for solution of the Cauchy problem for elliptic equations[END_REF], see also [START_REF] Sincich | Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements[END_REF][Chap.2] we have that

ũ0 L 2 (Bρ(P0)∩U 2 2r 0 ) c ũ0 L 2 (Γ2) + ∇ ũ0 H 1 (Ω) 1-δ ũ0 δ L 2 (Γ2) , (3.86)
where ρ ∈

M 4 √ 1+M 2 r0 , 3M 4 √ 1+M 2 r0 , P 0 = P 1 + M 4 √
1+M 2 r0 ν, ν is outer unit normal to Ω at P 1 and c > 0, 0 < δ < 1 are constants depending on the a priori data and on ρ only. By (3.59) we have that ũ0 H 1 (Ω) C .

(3.87)

INRIA

Hence we have that

ũ0 L 2 (Bρ(P0)∩U 2 2r 0 ) c ũ0 L 2 (Γ2) + C 1-δ ũ0 δ L 2 (Γ2) (3.88)
where C > 0 is the constant introduced in (3.59).

By the Lipschitz regularity of the boundary we have that if Q is a point of ∂Ω, then there exists a rigid transformation of coordinates under which we have Q = 0. We consider the finite cone

C = x : |x| < r 0 , x • ξ |x| > cos θ
with axis in the direction ξ and width 2θ, where θ = arctan 1 M , we have that C ⊂ Ω. Let us consider now a point Q ∈ Γ 2,r0 and let Q 0 be a point lying on the axis ξ of the cone with vertex in Q = 0 such that d 0 = dist(Q 0 , 0) < r0 2 . Following Lieberman [START_REF] Lieberman | Regularized distance and its applications[END_REF], we introduce a regularized distance d from the boundary of Ω. We have that there exists d such that d ∈ C 2 (Ω) ∩ C 0,1 ( Ω), satisfying the following properties

• γ 0 dist(x, ∂Ω) d(x) γ 1 , • |∇ d(x)| c 1 , for every x such that dist(x, ∂Ω) br 0 , • d C 1,α c 2 r 0 ,
where γ 0 , γ 1 , c 1 , c 2 , b are positive constants depending on M, α only, (see also [START_REF] Alessandrini | Optimal Stability for Inverse Elliptic Boundary Value Problems with Unknown Boundaries[END_REF]Lemma 5.2]). Let us define for every ρ > 0 Ωρ = {x ∈ Ω : d(x) > ρ} .

We have that, there exists a, 0 < a 1, only depending on M, α such that for every ρ, 0 < ρ ar 0 , Ωρ is connected with boundary of class C 1 and c1 ρ dist(x, ∂Ω) c2 ρ for every x ∈ ∂ Ωρ ∩ Ω (3.89) where c1 , c2 are positive constants depending on M, α only. By (3.89) it follows that

Ω c2ρ ⊂ Ωρ ⊂ Ω c1ρ .
We define the point P = P 0 -

1 4 √ 1+M 2 r 0 • ν and ρ 0 = min{ 1 32M √ 1+M 2 r 0 , r0 4
sin θ}. Moreover, let γ be a path in Ω ρ 0 c1 joining P to Q 0 and let us define {y i }, i = 0, . . . , s as follows y 0 = Q 0 ,

y i+1 = γ(t i ), where t i = max{t s.t. |γ(t) -y i | = 2ρ 0 } if |P -y i | > 2ρ 0 otherwise let i = s
and stop the process.

E. Sincich

Now, we will use the three spheres inequality for harmonic functions (see for instance [START_REF] Korevaar | Logarithmic convexity for supremum norms of harmonic functions[END_REF] or [6, Appendix E]) that is

B3ρ 0 (y0) ũ0 2 Bρ 0 (y0) ũ0 2 τ • B4ρ 0 (y0) ũ0 2 1-τ
where 0 < τ < 1 is an absolute constant. Now since B ρ0 (y 0 ) ⊂ B 3ρ0 (y 1 ) and since by (3.87) ũ0 H 1 (Ω) C, then we have

Bρ 0 (y0) ũ0 2 B3ρ 0 (y1) ũ0 2 τ • C 1-τ .
An iterated application of the three spheres inequality leads to

Bρ 0 (y0) ũ0 2 Bρ 0 (ys) ũ0 2 τ s • C 1-τ s .
Finally, since we have

B ρ0 (y s ) ⊂ B 3M 4 √ 1+M 2 r0 (P 0 ) ∩ U 2 2r0
, then by the Proposition (3.88) it follows

Bρ 0 (y0) ũ0 2 C (ε + C) 1-δ • ( ũ0 L 2 (Γ2) ) δ τ s .
We shall construct a chain of balls B ρ k (Q k ) centered on the axis of the cone, pairwise tangent to each other and all contained in the cone

C = x : |x| < r 0 , x • ξ |x| > cos θ
where θ = arcsin ρ0 d0 . Let B ρ0 (Q 0 ) be the first of them, the following are defined by induction in such a way

Q k+1 = Q k -(1 + µ)ρ k ξ , ρ k+1 = µρ k , d k+1 = µd k , with µ = 1 -sin θ 1 + sin θ .
Hence, with this choice, we have ρ k = µ k ρ 0 and B ρ k+1

(Q k+1 ) ⊂ B 3ρ k (Q k ).
Let us now consider the following estimate obtained by a repeated application of the three

INRIA spheres inequality ũ0 L 2 (Bρ k (Q k )) ũ0 L 2 (B3ρ k-1 (Q k-1 )) ũ0 τ L 2 (Bρ k-1 (Q k-1 )) ũ0 1-τ L 2 (B4ρ l-1 (Q k-1 ))
C ũ0 (3.99)

Proof.

For the proof we refer to [7, Theorem 2.2], see also [START_REF] Sincich | Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements[END_REF].

L ∞ Γ r 0 4 ( 2 r0 4

 424 INRIAwhere C > 0 is a constant depending on the a priori data only. Hence combining (3.4) and (3.20) the inequality u us now prove the inequality (3.10). Let 0 < r 1 < r and let us consider a test function η

2 -

 2 m 2 ) + 2Kρ δ . (3.49) Denoting by ω(ρ) = osc Γρ(z0) u, we have that by (3.49) it follows ω(ρ) γω(4ρ) + c 2 ρ δ , (3.50)

τ k L 2 ( 2 (.Lemma 3 . 8 .

 2238 Bρ 0 (Q0))C ( ũ0 L 2 (Γ2) + C) 1-δ • ( ũ0 L 2 (Γ2) ) δ τ s τ k . (3.90)For every r, 0 < r < d 0 , let k(r) be the smallest positive integer such that d k r, then since d k = µ k d 0 , it follows by (3.90), we haveũ0 L 2 (B ρ k (r) (Q k (r))) C ( ũ0 L 2 (Γ2) + C) 1-δ • ( ũ0 L 2 (Γ2) ) δ τ s τ k(r) . (3.92) Let x ∈ Γ 1,2r0 , x ∈ B ρ k(r)-1 Q k(r)-1 ), since ũ0 ∈ C 1,α (U 1 2r0 ) we have ∂ ũ0 (x) ∂ν ∂ ũ0 (x) ∂ν + C|x -x| α .Hence by (3.85) we have that|u 0 (x)| |L 0 ũ0 (x)| + ∂ ũ0 (x) ∂ν + C|x -x| α . (3.93)Moreover, by the Hölder continuity of ũ0 we have that|u 0 (x)| |L 0 ũ0 (x)| + ∂ ũ0 (x) ∂ν + C(1 + L 0 )|x -x| α . (3.94)Hence we have|u 0 (x)| |L 0 ũ0 (x)| + ∂ ũ0 (x)0 is a constant depending on the a priori data only. Moreover, by the estimate (3.76) we have thatK |L 0 ũ0 (x)| + ∂ ũ0 (x)function on the right hand side, with respect to r, with r ∈ (0, r0 4 ), we deduceK C log 1 σ( ũ0 L 2 (Γ2) ) -2α ν+2 .Hence, solving for ũ0 L 2 (Γ2) , we can computeũ0 L 2 (Γ2) C exp -K -ν+2 2αBy the definition of ũ0 we obtain the thesis. There exists a continuous, increasing function ω : [0, +∞) → [0, +∞), with ω(0) = 0, such that|L 0 -L 1 | ω( u 0u 1 L 2 (Γ2) ) .

Proof. Theorem 2 . 1 .

 21 If |L 0 -L 1 | c 2 where c > 0 is the constant in (3.83), then from (3.74)-(3.83), we deduceu 1u 0 L 2 (Γ2) |L 1 -L 0 |(c -C|L 0 -L 1 |) c 2 |L 0 -L 1 |(3.100)Then (2.8) follows.On the other hand if |L 0 -L 1 | c 2 , we have that by|L 1 -L 0 | 2Λ ω -1 ( c 2C ) u 0u 1 L 2 (Γ2) ,(3.101)thus (2.8) follows as well.

  .30)

	Noticing that bv + h 2	√ bv, we have that (3.30) yields
		Γr 2 (z0)	|∇v||∇η|ηv β + LC	Γr 2 (z0)	|M 2 -v||∇η|ηv β
		Γr 2 (z0)	|∇v| + bv |∇η|ηv β .	(3.31)
	Hence inserting (3.29) and (3.31) in (3.26) we obtain
		1 2	Γr 2 (z0)	|∇v| 2 vβ-1 η 2 -2 b	Γr 2 (z0)	vβ+1 η 2	(3.32)
	+	1 |β| Γr 2 (z0) LC |β| Γr 2 (z0) ηv β |∇η||∇v| + |∇v|η 2 vβ .	1 |β| Γr 2 (z0)	ηv β+1 b|∇η| +
	Moreover, by the Schwartz inequality and by (3.32) we obtain that for every ε > 0
		1
		2	Γr 2 (z0)
							INRIA

RR n°6103

E. Sincich

Integrating over B ρ k(r)-1 2 (Q k(r)-1 ), we deduce that

Applying the Caccioppoli inequality, we have

and since k is the smallest integer such that d k r, then d k-1 > r, it follows

From (3.92), we deduce

thus the previous inequality becomes

where ν = -log 1 µ log τ . We have

INRIA


Nancy Rodriguez

Development of a Wheelchair Simulator for Children with Multiple Disabilities

Keywords: virtual reality, simulator, disability, multiple disabilities, wheelchair, learning, augmented and alternative communication, interaction devices I.3.1 [Computer Graphics]: Hardware Architecture -Input devices, I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism -Virtual reality, H.5.1 [Information Interfaces And Presentation]: Multimedia Information Systems -Artificial, augmented, and virtual realities H.5.2 [Information Interfaces And Presentation]: User Interfaces -Input devices and strategies

Virtual reality allows to create situations which can be experimented under the control of the user, without risks, in a very flexible way. This allows to develop skills and to have confidence to work in real conditions with real equipment. VR is then widely used as a training and learning tool. More recently, VR has also showed its potential in rehabilitation and therapy fields because it provides users with the ability of repeat their actions several times and to progress at their own pace. In this communication, we present our work in the development of a wheelchair simulator designed to allow children with multiple disabilities to familiarize themselves with the wheelchair.

INTRODUCTION

Training is an application area perfectly suited to virtual reality. VR allows the development of specific skills and to evolve independently VR simulation create a safe learning environment, user can test the consequences of his actions without risk in the real world. With a simulation, it is possible to expand the range of possible situations and to adapt them to different users, capabilities and knowledge [START_REF] Mellet-D'huart | Les environnements Informatiques pour l'Apprentissage Humain[END_REF] [START_REF] Gall | Applications des tech-niques de réalité virtuelle à la neuropsychologie clinique[END_REF]. In addition, the virtual environment allows to easy and quantitatively assessing the progress of the user and his weaknesses [START_REF] Sahnoun | Conception et Simulation d'une Commande à Retour d'Effort pour Fauteuil Roulant Electrique[END_REF].

The objective of the current work we present in this article is the realization of an electric wheelchair simulator for children with multiple disabilities, aged between 13 and 16 years. This simulator will prepare children to wheelchair driving, taking into account motor and cognitive aspects. The first stage of this work concerns the development of a demonstrator to assess children's interest in the IT tool, which will then be integrated into their learning project.

"Multiples disabilities" is a serious handicap associating motor impairment and severe mental retardation, resulting in a extreme restriction of autonomy and possibilities of perception, expression and relationship [START_REF] Arpaillance | Évaluer la communication de la personne polyhandicapée[END_REF]. In fact, drive autonomously a wheelchair provides better mental and motor development by initiating new movements and by increasing socialization. But learning to drive a wheelchair in real conditions can be dangerous and tiresome sometimes even scary for patients, which can lead to slow this crucial stage in their development. The use of a virtual environment can help in this learning process and also allow them to use an electric wheelchair earlier, once children habituate and control the emotions provoked by driving. In addition, the handling of the vehicle is not intuitive and requires practice, especially for patients with severe motor dysfunction preventing them from using conventional devices such as the joystick. A simulator may help find which devices are suitable for the individual and teach him how to use it without taking any risk. This project is carried out in collaboration between the Institute for Children and Adolescents with multiple disabilities Coste Rousse located in Prades-le-Lez (Hérault, France) and the Laboratory of Informatics, Robotics and Microelectronics of Montpellier (LIRMM) in Montpellier. We work closely with the multidisciplinary team (educators, orthoptist, physiotherapist) of the center to make the simulator accessible from this conception.

RELATED WORK

Several wheelchair simulators are available; some designed to learn to drive or to evaluate the wheelchair and others to understand the difficulties of using a wheelchair and evaluate constructions accessibility. But there is not, to our knowledge, a simulator designed or adaptable for children with multiple disabilities. Marchal-Crespo presents in [START_REF] Marchal-Crespo | A simulator and assist-as-needed control strategy for learning to drive a power wheelchair[END_REF] a wheelchair prototype to facilitate the learning to drive, especially for the mentally handicapped. The system provides a chair with force feedback joystick and sensors to detect collisions. The prototype is able to avoid the walls and other obstacles to help movement.

In the work of Steyn [START_REF] Steyn | Virtual Reality Platform Modelling and Design for Versatile Electric Wheelchair Simulation in an Enabled Environment[END_REF], the simulator uses a immersive virtual reality interface displayed on 4 screens and a physical platform with a force feedback chair. ACCESSIM [START_REF] Schmid | Accessim[END_REF] uses a very realistic representation of urban areas to educate architects and decision makers on the difficulties faced by wheelchair users. The simulator VIEW [START_REF] Morère | VIEW : Virtual electric wheelchair[END_REF]has a virtual reality interface and virtual chair is directed using a conventional joystick electric wheelchair (Figure 1). It is possible to visualize the virtual environment using a conventional screen or a helmet with track head movements. Wheelsim [START_REF] Lifetool | [END_REF] is a simulator constructed on the basis of a conventional racing game. Several courses (levels) are available. The duration of each course is recorded and the time spent outside the planned route. Note that a very important work was carried out on the sound environment in order to amplify immersion. All these systems have realistic sets with many details and animations. This type of scene is not suitable for children * nancy.rodriguez@lirmm.fr suffering from cognitive disorders. Indeed, the details used to increase the realism and immersion of the user will cause difficulties in perception and understanding of the environment.

DESIGN

The simulator is developed under Unity3D, a game engine allowing to create and manage interactive virtual environments. The application provides many pre-programmed features and an ergonomic and intuitive interface. Unity offers several functionalities such as a physics engine, a tool for build animations and the possibility to integrate many input and output devices. Concerning the design and development of our prototype, all choices regarding look and sound, the different levels, gamification (positive reinforcement) and tests were carried out in consultation with the Coste Rousse team in a collaborative and iterative approach. Central to the simulation is the virtual chair. It is the only object manipulated by the user and the goal is that its behaviour is fairly close to the behaviour of a real chair. For rapid prototyping, we use a predefined model of a car wheels simulation. This simulation tool has four independent wheels, allowing simulating the behaviour of the wheels of various models of real wheelchairs. Of course, we also tested the behaviour of a wheelchair by using it, in order to better understand the feelings caused by its use. We then set the correspondence between the different movements of the wheels and the values of the joystick in order to better reproduce in the simulator the real behaviour of the wheelchair.

Visual aspects

Given the cognitive and perceptive troubles of children, it is essential to build a tool visually adapted. It is therefore necessary to identify the elements and codes that will allow conveying important information of the simulation. As we have stated above, the environments are usually very realistic with rich details for amplifying the immersion of the user. In our case, this approach does not make sense. Indeed, the accumulation of elements in a scene will tend to interfere with the user and prevent it from properly analyse the significant elements of the scene. We have therefore chosen to work on very refined environments by keeping only the necessary items. The colours have also been adapted to be sufficiently contrasted and then highlight the important objects and increase their visibility [START_REF] Arditi | Designing for people with partial sight and color deficiencies[END_REF]. As a part of positive reinforcement, we decided to reward the player when he is on the right track with a discreet symbol (the Coste Rousse smiley) and colourful fireworks at the end of each level. Even if this effect is more intrusive, once the level is completed, the child's distraction is no longer a concern (see Figure 2). At present, concerning sound feedback, there is a sound when the wheelchair starts to move and a sound (applause) associated with the fireworks at the end of the level. There is no yet haptic feedback in our system.

Interaction devices

A long-term goal of our project is the development of a software component to quickly integrate various interaction devices. This is a challenging task, since there is no standard for the input/output of of-the-shelf interaction devices, forcing to work on a case-bycase basis. We focused on the integration of a joystick, the default device on electric wheelchairs. After trying several joystick models, we decided to "build" our own joystick combining an unused joystick of a wheelchair (Figure 3) and a circuit board of a game joystick for translating data from analogic to digital format. This "hybrid" solution (a joystick from real wheelchair with a USB connexion) has enabled children to successfully control the virtual chair without being distracted by the interaction device. We have evaluated the prototype with four children of the Coste Rousse center, who know already drive a wheelchair, and have some computer skills. During various test sessions we observed a lot of curiosity about the simulator, and different ways of approaching it. Two children were focused on the screen and joystick while others tend to be more easily distracted. We did manage to get feedback on the recognition of objects in the virtual environment when the vehicle is stationary. However, recognition is more difficult in motion. The children managed to run the simulator for a while (2-3 minutes). But after a few minutes, we found difficulties in the use of devices and users wanted to stop. Indeed, the use of the joystick in a long time is difficult because of their mobility impairment and therefore it is necessary in the future to work with alternative devices.

In this project, the target users have extreme difficulties in communication; the level of verbal expression and even body language is very low. But studies and experiences in educational centers like Coste Rousse, show that children are able to express themselves and learn when you provide the right tools. Working with the team Coste Rousse and a small group of children, we managed to get ways to improve our work and offer a tool used by children with multiple disabilities. It is important to note that our system can only be tested with children in a advanced state of development. Indeed, application bugs of an early prototype usually provoke frustration and even rejection to use the simulator. For these reasons the phases of development and evaluation of our system are distant.

CONCLUSION

We have developed an electric wheelchair simulator suitable for children with multiple disabilities. We were able to experiment with a group of children and observed that it is an interest to add virtual reality in their learning project.

This ongoing work has allowed us to highlight the potential of virtual reality and challenges of such an approach in the field of health, to enable children with disabilities to grow in better conditions, develop new skills and express themselves. The keyword of our future work is "formalization". We now need to study various models of user centered design, interaction and evaluation in a context where users cannot communicate their needs and expectations. This will allow us, in collaboration with the multidisciplinary team in charge of children, to define an experiment enabling to evaluate the benefits of virtual reality for learning when users are experiencing multiple disabilities.

Figure 1 :

 1 Figure 1: VIEW, a simulator for learning to drive

Figure 2 :

 2 Figure 2: Visual feedback

Figure 3 :

 3 Figure 3: Joystick build for our simulator