Isabeau Birindelli

Françoise Demengel

Keywords: 35J70, 35J75 Isabeau Birindelli, Françoise Demengel, Fabiana 2010 Mathematical Subject Classification : 35J70, 35J75

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Dirichlet problems for fully nonlinear equations with "subquadratic" HamiltoniansMathematical

Dirichlet problems for fully nonlinear equations with "subquadratic" Hamiltonians

Contents of the paper

In this paper we prove some existence, uniqueness and Lipschitz regularity results for solutions of the Dirichlet problems associated with a class of fully nonlinear equations, whose principal part is given by second order operators which are singular or degenerate elliptic at the points where the gradient of the solution vanishes. Precisely, let us consider the problem

   -F (∇u, D 2 u) + b(x)|∇u| β + λ|u| α u = f in Ω u = ϕ on ∂Ω (1.1)
where Ω ⊂ R N is a bounded, C 2 open set and the operator F satisfies the structural assumptions (H1) F : R N \ {0} × S N → R is a continuous function, S N being the set of N × N symmetric matrices;

(H2) F (p, M ) is homogeneous of degree α > -1 wrt p, positively homogeneous of degree 1 wrt M , and satisfies, for some constants A ≥ a > 0,

a|p| α tr(N ) ≤ F (p, M + N ) -F (p, M ) ≤ A|p| α tr(N ) (1.2)
for any M, N ∈ S N , with N ≥ 0, and, for some c > 0,

|F (p, M ) -F (q, M )| ≤ c|M | ||p| α -|q| α | (1.3)
for any p, q ∈ R N \ {0}, and M ∈ S N .

We further assume that the first order coefficient b is Lipschitz continuous, the forcing term f is bounded and continuous, and the boundary datum ϕ is Lipschitz continuous. Moreover, we will consider the cases λ > 0 and β ∈ (0, α + 2]. In the case α = 0, the conditions on β reduce to 0 < β ≤ 2, hence the terminology "subquadratic". At least in the case α < 0, the definition of viscosity solution we adopt must be clarified. We use the definition firstly introduced in [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF], which is equivalent to the usual one in the case α ≥ 0, and allows not to test points where the gradient of the test function is zero, except in the locally constant case, when α < 0. A solution is a continuous function which is both a supersolution and a subsolution.

The results of the present paper have been announced in [START_REF] Birindelli | Ergodic pairs for singular or degenerates fully nonlinear operators[END_REF], where we consider solutions u = u λ of problem (1.1) with either ϕ = 0 or ϕ = +∞ and the behavior of u λ as λ → 0 is studied. Clearly, the first step to perform this analysis is a detailed description of the existence, uniqueness and regularity properties of the solutions of problems (1.1) in the case λ > 0, which is precisely the object of this paper. Our results can be summarized in the following main theorem. Theorem 1.2. Under the above assumptions, problem (1.1) has a unique viscosity solution, which is Lipschitz continuous up to the boundary.

We recall that the restrictions on the exponent β for having existence of solutions already appear in the non singular nor degenerate case α = 0, when generalized solutions satisfying the boundary conditions in the viscosity sense can be constructed, but loss of boundary conditions may occur, see [START_REF] Barles | On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF][START_REF] Capuzzo Dolcetta | Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian[END_REF].

Theorem 1.2 is obtained as a consequence of the classical Perron's method, which in turn relies on the comparison principle given by Theorem 3.3. Observe that when α < 0 and the gradient of the involved test functions is 0, the information on the solution is recovered through the result in Lemma 3.2, which is analogous to one used in [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF] in the "sublinear" case i.e. β ≤ α + 1. Moreover when b is not constant it is necessary to check that the gradient of the test functions be uniformly bounded. This comes from a priori interior Lipschitz estimates, proved in Theorem 2.1, which are of independent interest.

We prove Lipschitz estimates up to the boundary for solutions of the Dirichlet problem (1.1). Our proof follows by the Ishii-Lions technique, see [START_REF] Ishii | Lions Viscosity solutions of fully-nonlinear second order elliptic partial differential equations[END_REF], which has to be adapted to the present singular/degenerate case. We borrow ideas from [START_REF] Barles | Hölder continuity of solutions of secondorder non-linear elliptic integro-differential equations[END_REF][START_REF] Birindelli | C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations[END_REF], and we first prove Hölder estimates and then push the argument up to the Lipschitz result. Regularity results for degenerate elliptic equations have been obtained also in [START_REF] Imbert | C 1,α regularity of solutions of some degenerate fully nonlinear elliptic equations[END_REF], where equations having only a principal term of the form F (∇u, D 2 u) have been considered. However, we observe that any scaling argument relying on the homogeneity of the operator is no applicable in our case when β = α + 1, due to the different homogeneities of the terms in the equation. We also emphasize that we make no assumptions on the sign of the first order coefficient b, meaning that the estimates we obtain rely only on the ellipticity properties of the second order term, despite its singularity or degeneracy. The estimates we obtain here are radically different from the local Lipschitz regularity result proved in [START_REF] Birindelli | Ergodic pairs for singular or degenerates fully nonlinear operators[END_REF], where only positive hamiltonians with "superlinear" exponent β > α + 1 are considered. In that case, the obtained Lipschitz estimates, which do not depend on the L ∞ norm of the solution, are consequence of the coercivity of the first order term and require that the forcing term f is Lipschitz continuous.

The Lipschitz estimates are proved in Section 2. In Section 3, after the construction of sub and supersolutions vanishing on the boundary, we prove the comparison principle and obtain the proof of Theorem 1.2 in the case ϕ = 0. A Strong Maximum Principle and Hopf Principle are also included. Finally, the changes to be done in order to prove Theorem 1.2 for any boundary datum ϕ are detailed in Section 4.

Lipschitz estimates.

The main result in this section is the following Lipschitz type estimate, where we denote by B 1 the unit ball in R N . Theorem 2.1. Suppose that u is a bounded viscosity subsolution of

-F (∇u, D 2 u) + b(x)|∇u| β ≤ g in B 1
and v is a bounded viscosity supersolution of

-F (∇v, D 2 v) + b(x)|∇v| β ≥ f in B 1 ,
with f and g bounded and b Lipschitz continuous. Then, for all r < 1, there exists c r such that for all (x, y)

∈ B 2 r u(x) -v(y) ≤ sup B 1 (u -v) + c r |x -y|.
In order to prove Theorem 2.1 we first obtain the following Hölder estimate: Lemma 2.2. Under the hypothesis of Theorem 2.1, for any γ ∈ (0, 1), there exists c r,γ > 0 such that for all (x, y)

∈ B 2 r u(x) -v(y) ≤ sup B 1 (u -v) + c r,γ |x -y| γ . (2.1)
Proof of Lemma 2.2. We borrow ideas from [START_REF] Ishii | Lions Viscosity solutions of fully-nonlinear second order elliptic partial differential equations[END_REF], [START_REF] Barles | Hölder continuity of solutions of secondorder non-linear elliptic integro-differential equations[END_REF], [START_REF] Birindelli | C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations[END_REF]. Fix x o ∈ B r , and define

φ(x, y) = u(x) -v(y) -sup B 1 (u -v) -M |x -y| γ -L(|x -x o | 2 + |y -x o | 2 )
with L = 4(|u|∞+|v|∞)

(1-r) 2 and M = (|u|∞+|v|∞)+1 δ γ , δ will be chosen later small enough depending only on the data and on universal constants. We want to prove that φ(x, y) ≤ 0 in B 1 which will imply the result, taking first x = x o and making x o vary.

We argue by contradiction and suppose that sup B 1 φ(x, y) > 0. By the previous assumptions on M and L the supremum is achieved on (x, ȳ) which belongs to B 2 and it is such that 0 < |x -ȳ| ≤ δ. By Ishii's Lemma [START_REF] Ishii | Viscosity solutions of nonlinear partial differential equations[END_REF], for all > 0 there exist X and Y in S such that (q x , X ) ∈ J 2,+ u(x), (q y , -Y ) ∈ J 2,-v(ȳ) with

q x = γM |x -ȳ| γ-2 (x -ȳ) + 2L(x -x o ) q y = γM |x -ȳ| γ-2 (x -ȳ) -2L(ȳ -x o ). Hence -F (q x , X ) + b(x)|q x | β ≤ g(x), -F (q y , -Y ) + b(ȳ)|q y | β ≥ f (ȳ) (2.2)
Furthermore,

-( 1 + |B|) I 0 0 I ≤ X 0 0 Y -2L I 0 0 I (2.3) ≤ B + 2 B 2 -B -2 B 2 -B -2 B 2 B + 2 B 2 with B = M γ|x -ȳ| γ-2 I -(2 -γ) (x-ȳ)⊗(x-ȳ) |x-ȳ| 2
. It is immediate to see that, as soon as δ is small enough, with our choice of M and L, there exists c 1 and c 2 , such that

c 1 δ -γ |x -ȳ| γ-1 ≤ |q x |, |q y | ≤ c 2 δ -γ |x -ȳ| γ-1 .
We take

= 1 8|B|+1 = 1 8M γ|x-ȳ| γ-2 (N -γ)+1
and drop the index for X and Y . By standard considerations on the eigenvalues of B, in particular note that B + B 2 has a negative eigenvalue less than -3γ(1-γ) 4 M |x -ȳ| γ-2 , and, using (2.3), the following holds

X + Y ≤ (2L + )I and inf λ i (X + Y ) ≤ 4 inf λ i (B + B 2 ) ≤ -3γ(1 -γ)M |x -ȳ| γ-2 .
Hence, as soon as δ is small enough, for some constant c depending only on a, A, γ and N ,

F (q x , X) -F (q x , -Y ) ≤ |q x | α A λ i >0 λ i (X + Y ) + a λ i <0 λ i (X + Y ) ≤ |q x | α (2AN (2L + 1) -3aM γ(1 -γ)) |x -ȳ| γ-2 ≤ -cδ -γ(α+1) |x -ȳ| γ(α+1)-(α+2) .
(2.4) And the following standard inequalities hold:

If 0 ≤ α < 1, ||q y | α -|q x | α | ≤ |q x -q y | α ≤ cL α If α ≥ 1, ||q y | α -|q x | α | ≤ |α||q x -q y |(|q x | + |q y |) α-1 ≤ c(δ -γ |x -ȳ| (γ-1) ) (α-1) If -1 < α < 0, ||q y | α -|q x | α | ≤ |α||q x -q y | inf(|q x |, |q y |) α-1 ≤ c(δ -γ |x -ȳ| (γ-1) ) (α-1)
This implies that, for any α > -1,

|F (q x , X) -F (q y , X)| ≤ c max(δ -γ |x -ȳ| γ-2 , δ -αγ |x -ȳ| αγ-α-1 ). (2.5) Next, we need to evaluate |b(x)|q x | β -b(ȳ)|q y | β |.
One easily has, for some constant which depends only on β and universal constants,

|b(x)|q x | β -b(ȳ)|q y | β | ≤ Lip b cδ -γβ |x -ȳ| γβ-β+1 (2.6) + c|b| ∞ max(1, δ -γ(β-1) |x -ȳ| γ(β-1)-β+1 ).
Observe that choosing δ small, the terms in (2.5) and in (2.6) are of lower order with respect to (2.4). We then have, for some constant c, subtracting the inequalities in (2.2),

-g(x) ≤ F (q x , X) -b(x)|q x | β ≤ F (q y , Y ) -cM 1+α |x -ȳ| γ-2+(γ-1)α -b(ȳ)|q y | β ≤ -f (ȳ) -cδ -γ(1+α) |x -ȳ| γ(α+1)-(2+α)
This is a contradiction with the fact that f and g are bounded, as soon as δ is small enough.

We are now in a position to prove the Lipschitz estimate. The proof proceeds analogously to the Hölder estimate above, but with a modification in the term depending on |x -y| in the function φ(x, y).

Proof of Theorem 2.1. For fixed τ ∈ (0, 1 2 ) for α ≤ 0 and τ ∈ (0, inf(1,α)

2

)) for α > 0, and

s o = (1 + τ ) 1 τ we define for s ∈ (0, s o ) with ω(s) = s - s 1+τ 2(1 + τ ) , (2.7)
which we extend continuously after s o by a constant. Note that ω(s) is C 2 on s > 0, s < s o and satisfies ω > 0, ω < 0 on ]0, 1[, and s > ω(s) ≥ s 2 . As before in the Hölder case, with L = 4(|u|∞+|v|∞)+1

(1-r) 2 and M = (|u|∞+|v|∞)+1 δ we define

φ(x, y) = u(x) -v(y) -sup B 1 (u -v) -M ω(|x -y|) -L(|x -x o | 2 + |y -x o | 2 ).
Classically, as before, we suppose that there exists a maximum point (x, ȳ) such that φ(x, ȳ) > 0, then by the assumptions on M , and L, x, ȳ belong to B(x o , 1-r 2 ), hence they are interior points. This implies, using (2.1) in Lemma 2.2 with γ < 1 such that

γ 2 > τ inf(1,α) for α > 0 and γ 2 > τ when α ≤ 0 that, for some constant c r , L|x -x o | 2 ≤ c r |x -ȳ| γ . (2.8)
and then one has |x -

x o | ≤ cr L 1 2 |x -ȳ| γ 2 .
Furthermore, for all > 0, there exist X and Y in S such that (q x , X ) ∈ J 2,+ u(x), (q y , -Y ) ∈ J 2,-v(ȳ) with

q x = M ω (|x -ȳ|) x - ȳ |x -ȳ| + L(x -x o ), q y = M ω (|x -ȳ|) x - ȳ |x -ȳ| -L(ȳ -x o ).
While X and Y satisfy (2.3) with

B = M ω (|x -ȳ|) |x -ȳ| (I - x -ȳ ⊗ x - ȳ |x -ȳ| 2 ) + ω (|x -ȳ|) x -ȳ ⊗ x - ȳ |x -ȳ| 2 .
Note that as soon as δ is small enough M

2 ≤ |q x |, |q y | ≤ 3M 2 . Also, M ω (|x -ȳ|) = -M τ 2 |x -ȳ| τ -1
is an eigenvalue of B which is large negative as soon as δ is small enough.

Taking = 1 8|B|+1 and dropping the in the notations arguing as in the above proof, we get that there exists some constant c such that

F (q x , X) -F (q x , -Y ) ≤ -cδ -(1+α) |x -ȳ| τ -1 .
(2.9)

Note that by (2.3) using the explicit value of B one has

|X| + |Y | ≤ c(M |x -ȳ| -1 + 4(L + 1)N ) ≤ cδ -1 |x -ȳ| -1 (2.10)
as soon δ is small enough. On the other hand, using the mean value theorem, that for some universal con-

stant if α ≥ 1, or α < 0, ||q x | α -|q y | α | ≤ cδ -α+1 |x -ȳ| γ 2 , while if 0 < α ≤ 1, ||q x | α -|q y | α | ≤ c|x -ȳ| γα 2 .
In each of these cases one easily obtains, using ( 2.10)

||q x | α -|q y | α | |X| ≤ cδ -1-α |x -ȳ| -1+ γ 2 if α ≤ 0, or α > 1, ||q x | α -|q y | α | |X| ≤ cδ -1 |x -ȳ| -1+ γα 2 if α ∈]0, 1[
We have obtained, with the above choice of γ that , as soon as δ is small enough, |F (q x , X) -F (q y , X)| is small with respect to cδ -1-α |x -ȳ| τ -1 . We now treat the terms involving b with similar considerations. If

β ≥ 1, |b(x)|||q x | β -|q y | β | ≤ |b| ∞ |q x -q y |M β-1 ≤ cδ -β+1 |x -ȳ| γ 2 , while if β ≤ 1, |b(x)|||q x | β -|q y | β | ≤ |b| ∞ c|x -ȳ| γβ 2 .
Observe also that

|b(x) -b(ȳ)||q x | β ≤ c lip b|x -ȳ|δ -β .
Finally,

|b(x) -b(ȳ)||q x | β ≤ c|x -ȳ| τ -1 δ -β+2-τ = c|x -ȳ| τ -1 δ -1-α δ -β+3-τ +α Since 3 + α -β -τ > 0 , this term is also small wrt to δ -1-α |x -ȳ| τ -1 .
Putting all the estimates together we get

-g(x) ≤ F (q x , X) -b(x)|q x | β ≤ F (q y , -Y ) -b(ȳ)|q y | β -cδ -1-α |x -ȳ| -1+τ ≤ -f (ȳ) -cδ -1-α |x -ȳ| -1+τ .
This is clearly a contradiction as soon as δ is small enough since f and g are bounded.

3 Existence and uniqueness results for homogenous Dirichlet conditions.

The existence of solutions for (1.1) with the boundary condition ϕ = 0 will be classically obtained as a consequence of the existence of sub-and supersolutions, of some comparison result, and the Perron's method.

We start with a result on the existence of sub and supersolutions of equations involving the Pucci's operators

M + (M ) = A λ i >0 λ i + a λ i <0 λ i M -(M ) = a λ i >0 λ i + A λ i <0 λ i defined for any M ∈ S N , M ∼ diag(λ 1 , . . . , λ N ).
Proposition 3.1. Let λ > 0 and b > 0 be given. For all M > 0, there exists a continuous function ϕ ≥ 0 supersolution of

-|∇ϕ| α M + (D 2 ϕ) -b|∇ϕ| β + λϕ 1+α ≥ M in Ω ϕ = 0 on ∂Ω (3.1)
and symmetrically, for all M > 0, there exists a continuous function ϕ ≤ 0 subsolution

-|∇ϕ| α M -(D 2 ϕ) + b|∇ϕ| β + λ|ϕ| α ϕ ≤ -M in Ω ϕ = 0 on ∂Ω.
Proof of Proposition 3.1. We shall construct explicitly a positive supersolution, the subsolution is just the negative of it.

Since Ω is a smooth domain, d(x), the function distance from the boundary, is C 2 in the neighborhood {d(x) < δ 0 } for some δ 0 > 0, hence we will suppose to extend it to a C 2 function in Ω, which is greater than δ 0 outside of a δ o neighborhood of the boundary.

Let us fix a positive constant κ satisfying λ log(1 + κ) 1+α > M and, for C > 2κ δ 0 , consider the function ϕ(x) = log(1 + Cd(x)). Observe that |∇d| = 1 in {d(x) < δ 0 }.

Suppose that ϕ is showed to be a supersolution in the set {Cd(x) < 2κ}. Then, the function

φ(x) := log(1 + Cd(x)) in {Cd(x) < κ} log(1 + κ) in {Cd(x) ≥ κ}
is the required supersolution, being the minimum of two supersolutions. We now prove that in {Cd(x) < 2κ}, ϕ is a supersolution. Easily, one gets

∇ϕ = C∇d (1 + Cd) , D 2 ϕ = CD 2 d 1 + Cd - C 2 ∇d ⊗ ∇d (1 + Cd) 2 .
We suppose first that

β < α + 2. Let C 1 be such that D 2 d ≤ C 1 Id. For C satisfying furthermore aC 2(1 + 2κ) ≥ AN C 1 , a 2
C 1 + 2κ 2+α-β > 2b, aC 2+α 4(1 + 2κ) 2+α > M , one gets that |∇ϕ| α M + (D 2 ϕ) + b|∇ϕ| β ≤ -M ,
from which the conclusion follows.

We now suppose that β = α + 2. We begin to observe that the calculations above can be extended to β = α + 2 as soon as b < a 4 . So suppose that = a 4b , that ϕ is some barrier for the equation

-|∇ϕ| α M + (D 2 ϕ) + a 4 |∇ϕ| β ≥ M 1+α Then ψ = ϕ satisfies -|∇ψ| α M + (D 2 ψ) + b|∇ψ| β ≥ M.
Recall that, for α < 0, the equation we are considering are singular. Hence, we need to treat differently the solutions when the test function has a vanishing gradient. That is the object of the following lemma, which is proved on the model of [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF] where we treat the case β = α + 1. We give the detail of the proof for the convenience of the reader. Lemma 3.2. Let γ and b be continuous functions. Assume that v is a supersolution of

-|∇v| α M -(D 2 v) + b(x)|∇v| β + γ(v) ≥ f in Ω such that, for some C > 0 and q ≥ α+2 α+1 , x ∈ Ω is a strict local minimum of v(x) + C|x -x| q . Then f (x) ≤ γ(v(x)).
Proof. Without loss of generality we can suppose that x = 0. If v is locally constant around 0 the conclusion is the definition of viscosity supersolutions. If v is not locally constant, since q > 1, for any δ > 0 sufficiently small, there exist (z δ , t δ ) ∈ B 2 δ such that

v(t δ ) > v(z δ ) + C|z δ -t δ | q . (3.2)
The idea of the proof is to construct a test function near 0 whose gradient is not zero. Since 0 is a strict minimum point of v(x) + C|x| q , there exist R > 0 and, for any

0 < η < R, (η) > 0 satisfying min η≤|x|≤R (v(x) + C|x| q ) ≥ v(0) + (η).
Let η > 0 be fixed. We choose δ = δ(η) > 0 such that Cδ q ≤ (η) 4 . Note that δ → 0 as η → 0. With this choice of δ, we have

min |x|≤R (v(x) + C|x -t δ | q ) ≤ v(0) + C|t δ | q ≤ v(0) + (η) 4 .
On the other hand, restricting further δ such that qδC(R + 1) q-1 < (η) 4 , we get

min η≤|x|≤R (v(x)+C|x-t δ | q ) ≥ min η≤|x|≤R (v(x)+C|x| q )-qC|t δ |(R +|t δ |) q-1 ≥ v(0)+ 3 (η) 4 .
This implies that min |x|≤R (v(x) + C|x -t δ | q ) is achieved in B η . Furthermore, it cannot be achieved in t δ by (3.2). Hence, there exists

y δ ∈ B η , y δ = t δ , such that v(y δ ) + C|y δ -t δ | q = min |x|≤R (v(x) + C|x -t δ | q ).
Let us now consider the test function

ϕ(z) = v(y δ ) + C|y δ -t δ | q -C|z -t δ | q ,
that touches v from below at y δ . Since v is a supersolution, we obtain

N A(q -1)q α+2 C α+1 |y δ -t δ | q(α+1)-(α+2) + C β |b(y δ )||y δ -t δ | (q-1)β + γ(v(y δ )) ≥ f (y δ ). (3.3)
On the other hand, we observe that

v(y δ ) ≤ v(y δ ) + C|y δ -t δ | q ≤ v(0) + C|t δ | q ≤ v(0) + Cδ q .
By the lower semicontinuity of v, this implies that v(y δ ) → v(0) as η → 0 .

Thus, letting η → 0 in (3.3), by the continuity of γ and f it follows that γ(v(0)) ≥ f (0) .

We are now in a position to prove the following comparison principle that will be essential to the proof of the existence of the solution.

Theorem 3.3. Suppose that Ω is a bounded domain and that γ is a non decreasing function. Assume that, in Ω, u is an upper semicontinuous bounded from above viscosity subsolution of

-F (∇u, D 2 u) + b(x)|∇u| β + γ(u) ≤ g
and that v is a lower semicontinuous bounded from below viscosity supersolution of

-F (∇v, D 2 v) + b(x)|∇v| β + γ(v) ≥ f ,
with f and g bounded and b Lipschitz continuous. Suppose furthermore that -either g ≤ f and γ is increasing , -or g < f . Then

u ≤ v on ∂Ω =⇒ u ≤ v in Ω.
Proof. The case α ≥ 0. We use classically the doubling of variables. So we define for all j ∈ N , ψ j (x, y) = u(x) -v(y) -j 2 |x -y| 2 . Suppose by contradiction that u > v somewhere, then the supremum of u -v is strictly positive and achieved inside Ω.

Then one also has sup ψ j > 0 for j large enough and it is achieved on (x j , y j ) ∈ Ω 2 . Using Ishii's lemma , [START_REF] Ishii | Viscosity solutions of nonlinear partial differential equations[END_REF], there exist X j and Y j in S such that (j(x j -y j ), X j ) ∈ J 2,+ u(x j ), (j(x j -y j ), -Y j ) ∈ J 2,-v(y j ). It is clear that Theorem 2.1 in section 2 can be extended to the case where Ω replaces B(0, 1) and Ω ⊂⊂ Ω replaces B(0, r). Since (x j , y j ) converges to (x, x), both of them belong, for j large enough, to some Ω . We use Theorem 2.1 to obtain that j|x j -y j | is bounded.

Indeed u(x j ) -v(y j ) -j 2 |x j -y j | 2 ≥ sup(u -v), hence j 2 |x j -y j | 2 ≤ u(x j ) -v(y j ) -sup(u -v) ≤ sup(u -v) + c|x j -y j | -sup(u -v).
We obtain

g(x j ) -γ(u(x j )) ≥ -F (j(x j -y j ), X j ) + b(x j )|j(x j -y j )| β ≥ -F (j(x j -y j ), -Y j ) + b(y j )|j(x j -y j )| β + ω(j|x j -y j | 2 ) -o(x j -y j )(j|x j -y j |) β ≥ f (y j ) + o(j|x j -y j | 2 )|(j|x j -y j |) α -γ(v(y j )).
By passing to the limit, one gets on the point x limit of a subsequence of

x j g(x) -γ(u(x)) ≥ f (x) -γ(v(x))
and in both cases we obtain a contradiction. The case α < 0. We recall that in the case α < 0 one must use a convenient definition of viscosity solutions, see [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF].

We suppose by contradiction that sup(u -v) > 0 then it is achieved inside Ω, and taking q > α+2 α+1 the function

ψ j (x, y) = u(x) -v(y) - j q |x -y| q
has also a local maximum on (x j , y j ), with x j = y j . Then, there are X j , Y j ∈ S N such that (j(|x j -y j | q-2 (x j -y j ), X j ) ∈ J 2,+ u(x j ) (j(|x j -y j | q-2 (x j -y j ), -Y j ) ∈ J 2,-v(y j ) and

-4jk j I 0 0 I ≤ X j 0 0 Y j ≤ 3jk j I -I -I I
where k j = 2 q-3 q(q -1)|x j -y j | q-2 .

Note that : (i) from the boundedness of u and v one deduces that |x j -y j | → 0 as j → ∞. Thus up to subsequence (x j , y j ) → (x, x).

(ii) One has lim inf ψ j (x j , y j ) ≥ sup(u -v);

(iii) lim sup ψ j (x j , y j ) ≤ lim sup u(x j ) -v(y j ) = u(x) -v(x) (iv) Thus j|x j -y j | q → 0 as j → +∞ and x is a maximum point for u -v.

Furthermore by the Lipschitz estimates in Theorem 2.1 j|x j -y j | q ≤ u(x j )v(y j ) -sup(u -v) ≤ c|x j -y j | since (x j , y j ) belong to a compact set inside Ω. This implies that j|x j -y j | q-1 is bounded.

Claim: For j large enough, there exist x j and y j such that (x j , y j ) is a maximum pair for ψ j and x j = y j . Indeed suppose that x j = y j . Then one would have

ψ j (x j , x j ) = u(x j ) -v(x j ) ≥ u(x j ) -v(y) - j q |x j -y| q ; ψ j (x j , x j ) = u(x j ) -v(x j ) ≥ u(x) -v(x j ) - j q |x -x j | q ;
and then x j would be a local minimum for Φ := v(y) + j q |x j -y| q , and similarly a local maximum for Ψ := u(x) -j q |x j -x| q . We first exclude that x j is both a strict local maximum and a strict local minimum. Indeed in that case, by Lemma 3.2 γ(v(x j )) ≥ f (x j ), and γ(u(x j )) ≤ g(x j ). This is a contradiction with the assumptions on γ and f and g, once we pass to the limit when j goes to ∞ since we get

γ(v(x)) ≥ f (x) ≥ g(x) ≥ γ(u(x)).
Hence x j cannot be both a strict minimum for Φ and a strict maximum for Ψ. Suppose that this is the case for Φ, then there exist δ > 0 and R > δ such that B(x j , R) ⊂ Ω and

v(x j ) = inf δ≤|x-x j |≤R {v(x) + j q |x -x j | q }.
Then if y j is a point on which the minimum above is achieved, one has

v(x j ) = v(y j ) + j q |x j -y j | q ,
and (x j , y j ) is still a maximum point for ψ j .

We can now conclude. By Ishii's Lemma there exist X j and Y j such that j|x j -y j | q-2 (x j -y j ), X j ∈ J 2,+ u(x j ) and j|x j -y j | q-2 (x j -y j ), -Y j ∈ J 2,-v(y j ).

We can use the fact that u and v are respectively sub and super solution to obtain, (recalling that j|x j -y j | q-1 is bounded) :

g(x j ) ≥ F (j|x j -y j | q-2 (x j -y j ), X j ) + b(x j )|j(x j -y j | q-1 | β + γ(u(x j )) ≥ F (j|x j -y j | q-2 (x j -y j ), -Y j ) + b(y j )|j(x j -y j | q-1 | β -lipb|x j -y j ||j|x j -y j | q-1 | β + γ(v(y j )) + (-γ(v(y j )) + γ(u(x j ))) ≥ f (y j ) -γ(v(y j )) + γ(u(x j )) + O(x j -y j )
Passing to the limit one obtains

g(x) ≥ f (x) -γ(v(x)) + γ(u(x))
We have obtained a contradiction in each of the cases "f > g and γ is non decreasing", or "f ≥ g and γ is increasing".

We derive from the construction of barriers and the comparison theorem the following existence result for Dirichlet homogeneous boundary condition Theorem 3.4. Let Ω be a bounded C 2 domain, λ > 0, f ∈ C(Ω) and b ∈ W 1,∞ (Ω). Under the assumptions (H1) and (H2), there exists a unique u which satisfies

-F (∇u, D 2 u) + b(x)|∇u| β + λ|u| α u = f in Ω u = 0 on ∂Ω (3.4)
Furthermore, u is Lipschitz continuous up to the boundary.

Proof. The existence result is an easy consequence of the comparison principle and Perron's method adapted to our framework, as it is done in [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF]. In order to prove that u is Lipschitz continuous up to the boundary, observe that, by construction, there exist C > c > 0 such that cd ≤ u ≤ Cd, where d is the distance function from ∂Ω. Then, consider the function

φ(x, y) = u(x) -u(y) -M ω(|x -y|) ,
where ω has been defined in (2.7), and suppose that M > 2C. Arguing as in the proof of Theorem 2.1, we assume by contradiction that φ is positive somewhere, say on (x, ȳ). Then, neither x nor ȳ belong to the boundary, due to the inequality, for y ∈ ∂Ω,

u(x) ≤ Cd(x) ≤ M 2 d(x) ≤ M ω(|x -y|).
A similar reasoning proves that x cannot belong to the boundary. The rest of the proof runs as in Theorem (2.1), with even simpler computations, since we do not need the additional term |x -x o | 2 + |y -y o | 2 in the auxiliary function φ.

We end this section with some Strong maximum principle and Hopf principle.

Theorem 3.5. Suppose that u is a non negative solution in Ω of

-|∇u| α M -(D 2 u) + b(x)|∇u| β ≥ 0
Then either u > 0 inside Ω, or u ≡ 0. Moreover if x is in ∂Ω so that an interior sphere condition holds and u(x) = 0, then "∂ n u(x)" < 0

Proof. Suppose that there exists an interior point x 1 such that u(x 1 ) = 0. We can choose x 1 such that there exists

x o satisfying |x o -x 1 | = R, the ball B(x o , 2R) ⊂ Ω and u > 0 in B(x o , R). Since u is lower semi-continuous , let δ < inf(1, inf B(xo, R 2
) u), and define in the crown B(x o , 2R) \ B(x o , R

2 ) v(r) = δ(e -cr -e -cR ) where c > 2(N -1)A Ra , and a 2 c 2+α-β > |b| ∞ , if β < α + 2, ( which implies that δ 1+α a 2 c 2+α-β > δ β |b| ∞ ), if β = α + 2, note that we can take δ so that δ 1+α-β a 2 > |b| ∞ . Then one has v ≤ u on the boundary of the crown, and

-|v | α (av + A N -1 r v ) + |b| ∞ |v | β < 0
Using the comparison principle one gets that u ≥ v. Then v is a C 2 function which achieves u on below on x 1 and this is a contradiction with the fact that u is a super-solution. The last statement is proved . Suppose that x is on the boundary and consider an interior sphere B(x o , R) ⊂ Ω with |x o -x 1 | = R, and the function v as above. We still have by construction that v ≥ u. Then taking forh > 0 small,

x h = hx o + (1 -h)x 1 one has |x h -xo| = (1 -h)R and u(x h ) -u(x 1 ) x h -x 1 ≥ v(x h ) -v(x 1 ) x h -x 1 ≥ cRe -cR > 0
which implies the desired Hopf's principle.

Non homogeneous boundary conditions

In order to obtain solutions for the non homogeneous boundary condition, we need the construction of barriers, and a Lipschitz estimate near the boundary, as follows Then, for all r < 1 and for all γ < 1, there exists δ o depending on f ∞ , a, A, b ∞ , Ω, r and Lipϕ, |u| ∞ such that for all δ < δ o , if u be a USC bounded by above sub-solution of

-F (∇u, D 2 u) + b|∇u| β ≤ f in B ∩ {x N > η(x )} u ≤ ϕ on B ∩ {x N = η(x )} (4.1) such that u ≤ 1 then it satisfies u(x , x N ) ≤ ϕ(x ) + log(1 + 2 δ d) in B r (0) ∩ {x N = η(x )}.
We have a symmetric result for supersolutions bounded by below.

Proof. First case. We suppose that β < α + 2. We also write the details of the proof for ϕ = 0. The changes to bring in the case ϕ = 0 will be given at the end of the proof, the detailed calculation being left to the reader. It is sufficient to consider the set where d(x) < δ since the assumption u ∞ ≤ 1 implies the result elsewhere.

We begin by choosing δ < δ 1 , such that on d(x) < δ 1 the distance is C 2 and satisfies |D 2 d| ≤ C 1 . We shall also later choose δ smaller depending of (a, A, f ∞ , b ∞ , N ).

In order to use the comparison principle we want to construct w a super solution of

-|∇w| α M + (D 2 w) -b|∇w| β ≥ f ∞ , in B ∩ {x N > η(x ), d(x) < δ} (4.2) such that w ≥ u on ∂(B ∩ {x N > η(x ), d(x) < δ}).
We then suppose that δ < 1-r 9 , define C = 2 δ and

w(x) = log(1 + Cd) for |y| < r log(1 + Cd) + 1 (1-r) 3 (|x| -r) 3 for 1 ≥ |x| ≥ r.
In order to prove the boundary condition, let us observe that, on {d(x) = δ}, w ≥ log 3 ≥ 1 ≥ u, on {|x| = 1} ∩ {d(x) < δ}, w ≥ 1 (1-r) 3 (1 -r) 3 ≥ u and finally on B ∩ {x N = η(x )}, w ≥ 0 = u. We need to check that w is a super solution. For that aim, we compute

∇w = C 1+Cd ∇d when |x| < r C 1+Cd ∇d + x |x| 3 (1-r) 3 (|x| -r) 2 if |x| > r. Note that |∇w| ≥ C 2(1+Cd) ≥ 1 3δ since δ ≤ 1-r 9 . One also has |∇w| ≤ 3C 2(1+Cd) . By construction w is C 2 and D 2 w = CD 2 d 1 + Cd - C 2 ∇d ⊗ ∇d (1 + Cd) 2 + H(x)
where

|H(x)| ≤ 6 (1-r) 2 + 3N r(1-r) . In particular -M + (D 2 w) ≥ a C 2 (1 + Cd) 2 -A C|D 2 d| ∞ 1 + Cd -A 6 (1 -r) 2 + 3N r(1 -r) ≥ a C 2 (1 + Cd) 2 -AC|D 2 d| ∞ -A 6 (1 -r) 2 + 3N r(1 -r) ≥ a C 2 4(1 + Cd) 2
as soon as δ is small enough, depending only on r, A,a.

Hence ( we do the computations for α > 0 and leave to the reader the case α < 0, which can easily be deduced easily since Furthermore the desired lower bound on u is easily deduced by considering -w in place of w in the previous computations and restricting to B r ∩ {x N > η(x )}. This ends the case ϕ ≡ 0.

To treat the case where ϕ is non zero, let ψ be a solution of M + (ψ) = 0, ψ = ϕ on ∂Ω It is known that ψ ∈ C 2 (Ω), ψ is Lipschitz, |∇ψ| ≤ K|∇ϕ| ∞ . Then in the previous calculation it is sufficient to define as soon as δ is small enough in order that Then, for all r < 1, there exists c r > 0 depending on r, a, A, Lip(b) and N such that, for all x and y in B r ∩ {x N ≥ η(x )}), u(x) -v(y) ≤ sup

B 1 ∩{y N ≥η(y )}) (u -v) + c r |f | 1 1+α ∞ + |u| ∞ + |ψ| W 1,∞ (T ) |x -y|
In particular, when u is a solution we have a Lipschitz local estimate up to the boundary.

Definition 1 . 1 .

 11 Let Ω be an open set in R N , let f : Ω × R → R be a continuous function. An upper (lower) semicontinuous function u a is a subsolution (supersolution) of -F (∇u, D 2 u) + b(x)|∇u| β = f (x, u) in Ω if for any x 0 ∈ Ω, either u is locally constant around x 0 and f (x o , u(x 0 )) ≥ 0(≤), or for any test function φ of class C 2 around x 0 such that u -φ has a local maximum (minimum) point in x 0 and ∇φ(x 0 ) = 0, then -F (∇φ(x 0 ), D 2 φ(x 0 )) + b(x 0 )|∇φ(x 0 )| β ≤ (≥)f (x 0 , u(x o )) .

Lemma 4 . 1 .

 41 Let B be the unit ball in R N -1 , and let ϕ ∈ W 1,∞ (B ). Let η ∈ C 2 (B ) such that η(0) = 0 and ∇η(0) = 0. Let d be the distance to the hypersurface {x N = η(x )}.

C 2 (2

 2 1+Cd) ≤ |∇w| ≤ 3C 2(1+Cd) ). -|∇w| α M + (D 2 w) -b|∇w| β ≥ a C 2+α 2 2α+2 (1 + Cd) 2+α -b C 2(1 + Cd) β ≥ a C 2+α 2 4+α (1 + Cd) 2+α ≥ |f | ∞ as soon as δ is small enough in order that b < a2 β-2α-4 C 3 4+α (1+Cd) 2+α > |f | ∞ . By the comparison principle, Theorem 3.3, u ≤ w in B ∩ {x N > η(x )} ∩ {d(x) < δ}.

1 +

 1 Cd) + ψ(x) for |x| < r log(1 + Cd) +1 (1-r) 3 (|x| -r)3 + ψ(x) for |x| ≥ r.And we choose C = 2 δ large enough in order that C 2 > 2(|∇ψ| ∞ + 1 1-r ) and also large enough in order that -|∇w| α M + (D 2 w) -b|∇w| β > |f | ∞ which can be easily done by the same argument as before, using-|∇w| α M + (D 2 w) -b|∇w| β ≥ -2 -|α| |∇(log(1 + Cd)| α M + (log(1 + Cd)) -b2 -β |∇(log(1 + Cd)| β .Second case Suppose now that β = α + 2. It is sufficient to construct a convenient super-solution w.Recall that the previous proof used (4.3), which reduces when α + 2 = β, to the condition b < a2 β-2α-4 . Let then = a2 β-2α-4 b , and let w which equals ϕ on the boundary, and satisfies-|∇w| α M + (D 2 w) -b|∇w| β ≥ 1+α |f | ∞Then we obtain by the comparison principle that u ≤ w and then u ≤ w. This enables us to prove the following Lipschitz estimate up to the boundary. Proposition 4.2. Let ϕ be a Lipschitz continuous function on the part T = B(0, 1)∩ {x N = η(x )}, that f ∈ C(Ω) and b is Lipschitz continuous. Suppose that u and v are respectively sub and supersolution which satisfy (4.1) in B(0, 1) ∩ {x N > η(x )}, with u = ψ = v on T .

Acknowledgment. Part of this work has been done while the first and third authors were visiting the University of Cergy-Pontoise and the second one was visiting Sapienza University of Rome, supported by INDAM-GNAMPA and Laboratoire AGM Research Center in Mathematics of the University of Cergy-Pontoise.

Proof. We must prove Hölder's case and deduce from it the Lipschitz one as in the proof of Theorem 2.1 . We do not give the details, in each of the Hölder's and Lipschitz case, we define in B(0, 1)

where ω is as in the proof of Theorem 2.1. We want to prove that φ ≤ 0, which will classically imply the result. We argue by contradiction and need to prove first that if (x, ȳ) is a maximum point for φ, then neither x nor ȳ belongs to x N = η(x ). We notice that, if ȳ ∈ {x N = η(x )}, then

which contradicts Lemma 4.1 for M large enough. The case when x ∈ {x N = η(x )} is excluded in the same manner, and the rest of the proof follows the lines of Theorem 2.1

Remark 4.3. When the boundary condition is prescribed on the whole boundary, we have a simpler proof, as we noticed in the homogeneous case, taking φ(x, y) = u(x) -v(y) -sup

In this case the localizing terms are not needed, since it is immediate to exclude that the maximum point (x, ȳ) is on the boundary.


S Savoye

N Macé

S Lefèvre

J C Robinet

Cement-based Materials for Nuclear Wastes Diffusive behavior of tritiated water and 36 Cl through partiallysaturated cement-based materials

Keywords: Diffusion, desaturation, 36 Cl, HTO, osmosis, saline saturation method, cement

Diffusion into cement-based materials has been extensively investigating since this transport phenomenon is a key parameter for modelling processes related to contaminant transport in cement-based barriers, assessment of rebar corrosion in civil concrete structures and long-term behavior of nuclear waste disposal systems based on cement-based engineered barriers. Moreover, there are many situations where cement-based materials can be partially water-saturated, like in deep geological facility for the disposal of long-lived intermediate and high level nuclear waste. In this case, the saturation state of cement-based materials will be controlled during the exploitation period by the surrounding relative humidity imposed by ventilation of the underground drifts and shafts and during the post-closure period by gas generation (radiolysis, anoxic corrosion…). However, few data are available in literature on radionuclide diffusion behavior in cement-based materials according to saturation. Since determining the solute diffusion coefficients through cement-based material remains a challenging work for fullysaturated conditions, acquiring them for controlled partial saturation conditions constitutes thus a knotty task.

In the current study, we present three innovative and complementary techniques allowing an accurate determination of diffusive parameters for tritiated water (HTO) and 36 Cl through hardened cement pastes (HCP) undergone water-saturation ranging from 20% to 100%. The first technique is based on an adapted through-diffusion experiment with a control of the suction (up to 9 MPa) using osmosis process. The two other techniques perform diffusion experiments in humidity chambers controlled by salt solutions. One allows 36 Cl to diffuse from a spiked fresh cement paste into a cementitious material having a given saturation. After a controlled diffusion time, the 36 Cl activity profiles within the samples are acquired by means of the abrasive peeling method. The last technique is based upon a vapor exchange method where HTO is added in the salt solution and penetrates the studied material under its gas form. Using this principle, steady-state through-diffusion experiments can be performed for HTO allowing the determination of its effective diffusion coefficient and retention properties for several saturation states. The comparison of the data acquired by means of the three techniques shows a good consistency (Figure 1). The evolution of the effective diffusion coefficients with saturation are discussed as a function of species type using empirical models such as the Archie's second law.

Figure 1 :

 1 Figure 1: Effective diffusion coefficient (De) values normalized by the values obtained at full-saturation for HTO and 36 Cl as a function of the water saturation of the cement pastes.


Daniele Astolfi

Pauline Bernard

Romain Postoyan

Lorenzo Marconi

Redesign of discrete-time nonlinear observers with state estimate constraints guaranteed in prescribed convex set

We propose a technique to modify a given discrete-time (nonlinear) observer so that the state estimate remains in a given convex set, without altering the observer performances in terms of convergence and robustness to external disturbances. The proposed approach can be used to remove the peaking phenomenon or to attenuate the effect of impulsive outliers in the measures. it assumes that it is possible to execute a certain number of computations between any two sampling times in order to refine the current estimate and bring it back into the prescribed set. The proposed technique can be applied to any class of nonlinear observers for which a quadratic Lyapunov function is used to prove stability.

INTRODUCTION

In the context of observer design, it is well known that a wrong initialization of the estimate may induce very large estimation errors during transient behaviors. This phenomenon, also named peaking, see [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF], can be observed in both continuous and discrete time observers. For instance, peaking may be induced by large gains selected to guarantee fast convergence of the estimation error, or by dynamics possessing large overshoots caused by unstable poles on the open loop system, or by underdamping oscillatory behaviors, see, e.g., (Seron et al., 2012, §1, §8, §9). Moreover, large estimation errors can also occur in presence of large measurement disturbances of impulsive nature, denoted also as outliers, see [START_REF] Alessandri | Stubborn state observers for linear time-invariant systems[END_REF]. This phenomena may make the observer impractical or unsafe to use during transient if not properly handled, see, e.g. [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF], and renders implementation tricky from a numerical perspective, in particular if the computational power is limited as in embedded systems, see, e.g., [START_REF] Henriksson | Resource-constrained embedded control and computing systems[END_REF], [START_REF] Kopetz | Real-time systems: design principles for distributed embedded applications[END_REF]. A simple solution could be to keep the observer dynamics intact but saturate its estimate. However, this approach does not solve the numerical issues and does not attenuate the effect of impulsive disturbances inducing undesired peaking in steady-state behavior. Another natural idea would be to saturate the observer dynamics and force the observer to remain in the prescribed set, but this may destroy the observer convergence properties as we show on an example.

For continuous-time systems, some solutions have been proposed, such as reducing the peaking in high-gain observers in [START_REF] Andrieu | A hybrid scheme for reducing peaking in high-gain observers for a class of nonlinear systems[END_REF], [START_REF] Astolfi | Low-power peaking-free high-gain observers[END_REF], or ensuring the invariance of a convex set for the highgain observers in [START_REF] Astolfi | Output feedback stabilization for siso nonlinear systems with an observer in the original coordinates[END_REF]; [START_REF] Maggiore | A separation principle for a class of non-uco systems[END_REF], and for linear Kalman filters with linear constraints in [START_REF] Berkane | Hybrid constrained estimation for linear time-varying systems[END_REF] and references therein. For discrete-time systems, many techniques have been developed to address the problem of outliers, as in [START_REF] Alessandri | Moving-horizon estimation with guaranteed robustness for discrete-time linear systems and measurements subject to outliers[END_REF], [START_REF] Alessandri | Stubborn state observers for linear time-invariant systems[END_REF], De Palma and Indiveri (2017), [START_REF] Gandhi | Robust kalman filter based on a generalized maximum-likelihood-type estimator[END_REF]. Yet, these approaches do not guarantee the state of the observer to remain in some given prescribed set.

We propose a technique to modify a given discrete-time observer that guarantees the estimate to remain in a prescribed convex compact set, while preserving the observer convergence properties. This technique is inspired from the convex design method first proposed in [START_REF] Astolfi | Output feedback stabilization for siso nonlinear systems with an observer in the original coordinates[END_REF] for continuous-time observers and can be applied to any class of discrete-time nonlinear observer provided that a quadratic Lyapunov function for the error dynamics is known, see, for instance, [START_REF] Califano | On the observer design in discrete-time[END_REF], [START_REF] Ciccarella | Observers for discrete-time nonlinear systems[END_REF], [START_REF] Ibrir | Circle-criterion approach to discrete-time nonlinear observer design[END_REF], [START_REF] Kazantzis | Discrete-time nonlinear observer design using functional equations[END_REF], [START_REF] Xie | Robust kalman filtering for uncertain discrete-time systems[END_REF], [START_REF] Zemouche | Comments on a note on observers for discrete-time lipschitz nonlinear systems[END_REF]. We assume for this purpose that each discrete time corresponds to a sampling time of the plant and that some computations can be carried out between each consecutive sampling times. More precisely, if the estimate -following the initially given observer's dynamics -exits the prescribed set at a sample time, we propose a refining strategy that brings it back into that viable set before the next sampling event. We guarantee that the Lyapunov function does not increase at these intermediate computations. Hence, the convergence properties of the observer are not altered.

The rest of the paper is organized as follows. The problem formulation is stated in Section 2 and the redesign approach presented in Section 3. Then, our main result is given in Section 4 and a numerical simulation is shown in Section 5. Finally, conclusions and future perspectives are discussed in Section 6.

Notation:

We denote by R the set of real numbers, R ≥0 is the set of non-negative real numbers, N is the set of nonnegative integers, and N >0 is the set of positive integers. We define

R n ≥0 := R ≥0 × • • • × R ≥0 with n ∈ N >0 .
Given a set A ⊂ R n , int(A) stands for the interior of A, namely the set of all points in A that do not belong to its boundary. Given a real number x ∈ R ≥0 , we denote with ceil(x) the ceiling function, namely ceil(x) = min{y ∈ N : y ≥ x}. We define | • | as the standard Euclidean norm. Given a matrix A ∈ R n×n , we denote by |A| its induced norm, and eig(A) the set of its eigenvalues. Given x ∈ R, we define sat r (x) := max(min(x, r), -r). A continuous function α :

≥0 → R ≥0 is of class K if it is strictly increasing and α(0) = 0.

PROBLEM STATEMENT

Consider the discrete-time system

x k+1 = f (x k , u k , d k ), y k = h(x k , d k ) (1) where x k ∈ R nx is the state, u k ∈ R nu a known input, y k ∈ R ny is the output, and d k ∈ R n d is
a perturbation acting on the dynamics and/or the measurement, at time k ∈ N. The following assumption is stated. Assumption 1. We are given compact subsets

X 0 ⊂ R nx , U ⊂ R nu , D ⊂ R n d ,
and X ⊂ R nx , such that any trajectory of (1) initialized in X 0 , with input in U and perturbation in D, remains in X for all k ∈ N.

We suppose to know an observer of the form xk+1 = F (x k , u k , y k )

(2) where xk ∈ R nx is the state estimate at time k ∈ N. By denoting the estimation error e := x -x and the observer error dynamics as

e k+1 = ∆(x k , e k , u k , d k ) := F (e k + x k , u k , h(x k , d k )) -f (x k , u k , d k ) , (3
) the performances of observer (2) are characterized in the following assumption. Assumption 2. There exist a matrix P = P > 0 and a map β : R ≥0 × R n d → R ≥0 , such that, by defining V (e) := e P e (4) on R nx , the following inequality holds for all (x, e, u, d) ∈

X × R nx × U × D V (∆(x, e, u, d)) ≤ β V (e), d .
(5) Remark 1. At this point, no particular assumption is made on the map β which characterizes the performances of the observer. For an exponentially stable observer which is input-to-state stable (ISS) with respect to the perturbation, the function β is typically of the form

β(v, d) = ρ v + α(|d|) (6) with ρ ∈ [0, 1) and α ∈ K.
In the following, λ and λ are the minimum and maximum eigenvalue of P , respectively, so that for all e in R nx ,

λ|e| 2 ≤ V (e) ≤ λ|e| 2 . ( 7
)
Although the plant state x k is known to remain in X , the estimate xk may leave X , especially during transient behaviors. As a consequence, our goal is to modify the observer dynamics (2) to ensure the estimate xk remains in a given compact set X ⊃ X , while preserving the original performances stated in Assumption 2.

In the following, we suppose that the discrete-time instances k correspond to sampling times t k where a measurement y k is available, and that between any two consecutive sampling events t k and t k+1 , we are able to carry out θ ∈ N -to be defined -computations in order to refine the estimates given by ( 2). This assumption is realistic when system (1) is the exact or approximate discrete-time model obtained by sampling1 a continuous-time system at instants t k , t k+1 , . . ., if the digital controller has a computational frequency which is higher than the sampling frequency. In other words, we run the following algorithm depicted in Figure 1:

( We thus address the following problem.

1) at instant t k , we compute xk,1 = F (x k , u k , y k ) . ( 8
) (2) if xk,1 ∈ X , we select xk+1 = xk,1 . Otherwise, if xk,1 ∈ X ,
Problem 1. For a given convex compact set X ⊃ X , design a refining strategy, i.e. a map G : R nx → R nx and a number θ ∈ N so that, by combining (8)-( 9)

with xk,i+1 = G(x k,i ) , i = 1 . . . θ , k ∈ N, (10) the resulting new observer dynamics xk+1 = F m (x k , u k , y k ) (11) with F m : R nx × R nu × R ny → R nx , verify (1) the Lyapunov function (4) still satisfies (5) with ∆ replaced by ∆ m (x k , e k , u k , d k ) := F m (e k + x k , u k , h(x k , d k )) -f (x k , u k , d k ) , (12)
i.e. the performances of the observer are preserved, (2) for any initial conditions pair (x 0 , x0 ) ∈ X 0 × X , any input in U and perturbation in D, the corresponding solution to (11) lies in X for all k ∈ N.

REDESIGN APPROACH

Convexity assumption

The method we propose requires convexity properties of the set X in which we want to keep the estimate trajectories. For this, let us define v max as

v max := max (x,x,d)∈X × X ×D β V (x -x), d , (13)
with β given by Assumption 2, and the compact sets Ω max and X max as

Ω max := {e ∈ R nx : V (e) ≤ v max }, (14)
X max := {x ∈ R nx : ∃x ∈ X , x -x ∈ Ω max }.
(15) According to Assumption 1, and in view of (5), v max defines the size of the largest Lyapunov level sets of V after a jump from x k ∈ X , xk ∈ X , with u k ∈ U and d k ∈ D. In other words, xk+1 is in X max and e k+1 is in 2). We make the following assumption. Assumption 3. There exist n c ∈ N, a C 1 function c : R nx → R nc ≥0 and real numbers ε max > ε min > 0 and δ > 0 such that, by letting c i the i-th component of c and

Ω max when (x k , xk , u k , d k ) is in X × X × U × D along the dynamics (
C 0 := {x ∈ R nx : c(x) = 0}, C min := {x ∈ R nx : |c(x)| ≤ ε min }, C max := {x ∈ R nx : |c(x)| ≤ ε max }, (16) the following properties hold. (1) X ⊂ C 0 ⊂ C min ⊂ X ⊂ C max . (2) X max ⊆ C max . (3) Each function c i : R nx → R ≥0 is convex on C max , i.e. c i (x) ≤ c i (x) + dc i dx (x)(x -x) ∀ x, x ∈ C max . (17) (4) sup x∈Cmax dc dx (x) ≤ δ.
If the set X can be arbitrarily chosen, we have the following result.

Lemma 1. Consider Q = Q > 0 and r > 0, such that X ⊂ {x ∈ R nx : x Qx ≤ r}. Then, Assumption 3 is verified with X = {x ∈ R nx : x Qx ≤ r}, c(x) := max x Qx r -1, 0 2 , ( 18
)
for some r > r > r and ε max > ε min > 0.

If X , X are imposed but a compact convex intersection of half-planes and quadrics fits between X and X , c can be taken multi-dimensional with each c i built as in (18) from linear or quadratic maps, as showed in the next lemma. Lemma 2. Suppose there exist real numbers r i and

r i i = 1 . . . n x such that X ⊂ nx i=1 [r i , r i ] ⊂ X . Then, Assumption 3 is verified with c(x) = (c 1 (x), c 1 (x), . . . , c i (x), c i (x), . . . , c nx (x), c nx (x)), c i (x) = max {r i -x i , 0} 2 , c i (x) = max {x i -r i , 0} 2 , ( 19
) for some ε max > ε min > 0.
The proofs of Lemma 1 and 2 are omitted for space reasons.

Refining strategy

We introduce the map M : C max → R nx defined as

M (x) = -γP -1 dc dx (x) c(x) , (20)
with a positive scalar γ, and P coming from Assumption 2. As proved in Lemma 3 of Appendix A, if γ is sufficiently small, modifying the estimate x as x + M (x) makes V (xx) decrease no matter where the plant state x actually is, with V the quadratic Lyapunov function defined in (4). Besides, V decreases strictly (by γε 2 min ) when x is outside int(C min ), with C min defined in (16).

Since C min ⊆ X , a possible strategy is therefore to repetitively use M to bring the estimate back in X at each iteration k. For that, consider v min > 0 as

v min := max{v ∈ R ≥0 : ∀ x ∈ X , {x ∈ R nx : V (x -x) ≤ v} ⊆ C min } .
In other words, v min is the largest positive number such that the following property holds

x ∈ X and V (x -x) ≤ v min =⇒ x ∈ C min ⊂ X . (21)
Since the true state x is known to be in X , a way to bring x back into X is thus to bring V below the threshold v min . This can be done by applying M at most θ ∈ N times, with θ defined as

θ := ceil max{v max -v min , 0} γε 2 min . ( 22
)
Remark 2. By definition of X max , it may occur that X max ⊂ C min and that v max < v min . This means that after every jump of the original observer (2), the estimate remains in the desired set X . In such a case, no modification is needed, namely θ = 0 and all the analysis is trivially satisfied since no refining is needed.

The redesign we propose consists in correcting the state estimate with the map M as long as the estimate is outside C min , namely the function ( 10) is selected as

G(x k,i ) = xk,i + M (x k,i ) if xk,i ∈ C max \ C min xk,i otherwise (23)
for i = 1 . . . θ. As it can be noted by the expression of M , the correction (23) uses the gradient of the convext map c, namely -dc/dx, to bring xk,i+1 closer to X along level sets of V . The recursive algorithm stops when we cross C min . This strategy is depicted in Figure 2. Note that this could not be achieved in one iteration because γ needs to be sufficiently small to ensure that V decreases, as shown in Lemma 3 in the Appendix, which justifies the θ steps. Remark 3. This redesign generates new observer dynamics (11) with a function F m recursively defined through (23).

If an explicit expression of F m is available known, as in [START_REF] Berkane | Hybrid constrained estimation for linear time-varying systems[END_REF] for the particular case of linear dy-namics with linear constraints, then the θ computational steps of the refining algorithm are not necessary.

MAIN RESULT

With the refining strategy proposed in the previous section, we obtain the next result, addressing Problem 1. Theorem 1. Suppose Assumptions 2 and 3 hold and take

0 < γ ≤ λ δ 2 . ( 24
)
Then, the functions V , F m , ∆ m , defined by (4), ( 11), ( 12) with the algorithm (23), satisfy

V (∆ m (x, e, u, d)) ≤ β V (e), d (25a)
F m (x, u, h(x, d)) ⊂ X , (25b)
for any (x, x, u, d) ∈ X × X ×U ×D such that f (x, u, d) ∈ X , and with e = x -x.

Proof. The proof of the theorem is omitted for space reasons. It can be deduced by direct application of Lemma 3 given in appendix. In particular, property (25a) can be proved by using the fact that at each step i ∈ {1, . . . , θ}, the decrease of the Lyapunov function is preserved in view of inequality (A.1a). Property (25b) can be similarly proved by applying inequality (A.1b) θ times and by using item 1) of Assumption 3. 2

In Theorem 1, (25a) states that inequality ( 5) is preserved when using the modified dynamics ( 11). ( 25b) implies the invariance of X for the dynamics (11) at times k ∈ N, namely that xk+1 ∈ X if xk+1 is computed according to the modified dynamics along (23), for any initial condition (x k , xk ) ∈ X × X , input in U and perturbation in D. We immediately deduce the next corollary.

Corollary 1. Consider system (1) and suppose Assumptions 1, 2 and 3 hold with β of the type (6). There exist ν > 0 and a class K function α 0 such that for any γ verifying (24) and for any solution x k to (1) initialized in X 0 , any solution xk to (11) initialized in X , with input

(u, d) in U × D and y k = h(x k , d k ), satisfies xk ∈ X and |x k -x k | ≤ ν|x 0 -x 0 |ρ k/2 + α 0 max i≤k |d i | (26)
for all k ∈ N.

The same ISS gain that we would have obtained without modifying the observer is ensured in ( 26), but we have gained the fact that x remains in the prescribed set X .

In order to use the algorithm in ( 23) we need to be able to make (at most) θ + 1 recursive computations between t k and t k+1 . As a consequence, depending on the sampling period t k+1 -t k and the computational power of the system, this algorithm may have some limitations in practical applications. Note however that the Lyapunov function decreases even if we perform less than θ + 1 steps. Therefore, in case of limited computational power, it is still interesting to do as many θ m steps, , with θ m ∈ {0, . . . , θ}, as possible, as long as the estimate is outside of C min : the Lyapunov function will decrease by θ m γε 2 min which will make the algorithm converge faster and make the estimate stay closer to X .

The number θ defined in ( 22) gives the largest number of refining steps needed to bring x in C min . However, this number may be significantly smaller, as the analysis is made on the conservative assumption that at each step, the decrease rate of V is γε 2 min . But according to Lemma 3 in Appendix A, the true decrease is γ|c(x)| 2 , which is larger than γε 2 min when x is far from C min . In fact, the definition of the refining strategy (23) provides, from a computational point of view, a fast way to interrupt the algorithm once x is in C min .

An important observation is that although xk stays in X , the intermediary estimates xk,i can leave this set. However, according to the proof, they remain in X max defined in (15). Therefore, X max gives the magnitude of the numbers to be computed throughout the algorithm. This information can be useful to optimize the number encoding and increase numerical precision in case of limited memory, for instance in embedded systems. Actually, this set can be made more precise by considering the evolution of V given by ( 5) and the maximal initial error e 0 = x0 -x 0 . For instance, in the case of Corollary 1, V remains smaller than vmax := V (e 0,max

) + 1 1 -ρ α(d max )
where e 0,max and d max are bounds for the initial error and the disturbance respectively. Therefore, xk,i actually remain in

X max := x ∈ R nx : ∃x ∈ X , V (x-x) ≤ min{v max , ṽmax }
contained in X max . However, if the bounds e 0,max and d max are small, it can happen that ṽmax ≤ v max , in which case xk,i are not guaranteed to be in a smaller set than if we had not modified the observer. Nevertheless, those bounds are extremely conservative, and it is very likely in practice that xk,i remain closer to X .

All the aforementioned considerations are illustrated in Section 5 on an example. Remark 4. The robustness property of the modified observer with respect to the perturbation d is characterized by (25a), and is therefore the same as with the initial observer. The added computational steps are not affected by perturbations, they are carried out independently from the plant and the observer. Therefore, the only robustness property we could be concerned about is the robustness of Theorem 1 with respect to numerical errors in the refining steps. The discrete dynamics ( 23) are not outer semicontinuous on the boundary of C min , so that sequential compactness of solutions is not guaranteed. However, it turns out that the same result holds when allowing to use the correction M on the boundary, and we can show using the framework of discrete inclusion that the result of Theorem 1 is robust to computational errors.

SIMULATIONS

As a simple illustration, we consider the linear system The corresponding solution to ( 27) is constant, namely x k = (-1, -1) for all k ∈ N, and lives therefore in the set

x k+1 = Ax k , y k = Cx k + d k (27) with x k ∈ R 2 , initial condition x 0 = (-1, -1), d k ∈ R a measurement disturbance and A = 0 1 1 0 , C = (1 0) .
X = [-1, 1] × [-1, 1]. Consider the observer xk+1 = Ax k + L(y k -C xk ) , (28
d k = 0, 0 ≤ k ≤ 150 or k ≥ 154, 3, 150 < k < 154
is given in Figure 3a). Two large peaking phenomena are present, caused by the wrong initial conditions and by the effect of the disturbance d.

We would like to preserve the convergence of observer (28) while keeping x in the set X = [-r, r] × [-r, r], for some scalar r > 1. Note that we cannot implement the saturated observer xk+1 = sat r (F (x k , u k , y k )) , (29) where the function sat is taken component-wise, because the convergence is not preserved in absence of disturbances if r is taken too small. For instance, by selecting x0 = (1, 1) and r = 2, solutions to (29) remain stuck in xk = (2, -2) for all k > 0.

We thus apply the method presented in this paper. We know that for all λ > max | eig(A -LC)|, there exists 2 P solution to (A -LC) P (A -LC) ≤ λ 2 P , P > 0 . We choose for instance λ = 0.955, P = 10 5 • 1.259 1.319 1.319 1.381 , P -1 = 0.039 -0.037 -0.037 0.036 .

Then, Assumption 2 holds along the lines of Remark 1, so that we can follow Section 4 to modify the observer (28). By following Lemma 2, we choose the function c satisfying Assumption 3 as in ( 19) with r 1 = -1.1, r1 = 1.1, r 2 = -1.1, r2 = 1.1, and ε min = 0.01 which defines the set C min that will be made invariant for the observer. As for ε max and C max , notice that they are not used in the design apart from providing an upper bound on γ in (24) through δ defined in Assumption 3. Here we take γ = 17. Instead of computing the theoretical number of steps θ by using ( 22), we suppose that, due to computation limit constraints, we cannot carry out more than θ m computations in-between 2 Because 1 λ (A -LC) is Schur. sample times to apply the refining algorithm (23). We performed two different scenarios: θ m = 3 and θ m = 10. Figure 3 shows the behavior of the estimation error, the trajectory of the refining steps, and the number of refining steps required at each k for θ m = 10 and for the same initial condition x0 = (1, 1). The benefits of the refining algorithm are obvious: the convergence rate is fastened and the estimate is constrained in a set close to X which is far smaller than the set where solutions to (28) naturally evolve. As a comparison, the results of the same simulation for θ m = 3 are presented on Figure 4: the solution goes further from X during the transients, but the benefits of the modification algorithm are still clearly visible. Table 1 shows the maximum values of each component in the three cases. Moreover, we see from Figures 3 and4 how the modification algorithm enables to reject more efficiently the impulsive measurement disturbance d k .

CONCLUSION

We proposed a technique that guarantees the estimate to remain in a prescribed convex set for a given observer's dynamics, without modifying its convergences properties.

For this, we need to know a quadratic Lyapunov function for the error dynamics and we assume that, between any two samples, we are able to make a certain number of computations in order to refine the estimate of the observer.

The proposed methodology allows one to attenuate the peaking phenomenon caused by wrong initial conditions and/or by measurement outliers, as illustrated by the numerical example. Future works aim at extending the proposed technique to observers with a state-dimension different from the plant's, as in e.g. [START_REF] Astolfi | Low-power peaking-free high-gain observers[END_REF], observers with time-varying dynamics to handle Kalmanlike designs as in [START_REF] Besançon | Further results on high gain observers for nonlinear systems[END_REF] or [START_REF] Boutayeb | Convergence analysis of the extended kalman filter used as an observer for nonlinear deterministic discrete-time systems[END_REF], and finally observers whose convergence is proved in other coordinates [START_REF] Astolfi | Output feedback stabilization for siso nonlinear systems with an observer in the original coordinates[END_REF], [START_REF] Bernard | Hybrid implementation of observers in plant's coordinates with a finite number of approximate inversions and global convergence[END_REF]Marconi (2018) Ciccarella et al. (1993)).

Figure 2 .

 2 Figure 2. Refining strategy in (23). Red ellipsoides: C min , C max . Green polygon: X . Dotted blue ellipsoides: V (e) ≤ v min and V (e) ≤ v max . Dashed black line: path of xk,i+1 , i = 0, . . . , θ.

Figure 3

 3 Figure 3. a) Error e k , for k ∈ {0, . . . , 300}, provided by observer (28) without modification (blue line) and with modification (23) (red line) with θ m = 10. b) Number of refining steps of observer (28) with modification (23) at each k ∈ {0, . . . , 300}, with θ m = 10. c) Trajectory of xk,i , for k ∈ {0, . . . , 150} and i ∈ {0, . . . , 10}, of observer (28) with modification (23).

  ) with L chosen such that the matrix A -LC is Schur, namely its eigenvalues lie in the open unit disk. The result of a simulation of observer (28) with x0 = (1, 1), L = [-1.87, 1874] (eig(A -LC) = {0.95, 0.92}), and a disturbance d k selected as

Figure 4 .

 4 Figure 4. Error components e k , k ∈ {0, . . . , 300}, provided by observer (28) without modification (blue line) and with modification (23) (red line) with θ m = 3.

See for instance[START_REF] Nešić | Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations[END_REF] and references therein.

Appendix A. PROPERTIES OF THE MAP M Lemma 3. Suppose Assumptions 1 to 3 hold. If γ verifies (24), the functions V and M defined in (4) and (20) satisfy

where e = x -x.

Proof. Take x in X and x in C max . In view of (20),

. By using the convexity properties in ( 17), we compute

3) where we used the fact that c i takes nonnegative values and c(x) = 0 for all x ∈ X by Assumption 3. By combining (A.2), (A.3), and by using (20), we obtain

in which we used |P -1 | ≤ λ -1 . Hence (A.1a) holds. Moreover, when x ∈ C max \int(C min ), we have |c(x)| ≥ ε min and hence (A.1b) holds.

2