Naila Hayek
email: naila.hayek@u-paris2.fr

Hasan Yilmaz

INFINITE-DIMENSIONAL MULTIOBJECTIVE OPTIMAL CONTROL IN CONTINUOUS TIME

Keywords: Pontryagin maximum principle, Pareto optimality, multiobjective optimization, piecewise continuous functions June, 29, 2022

This paper studies multiobjective optimal control problems in the continuous-time framework when the space of states and the space of controls are infinite-dimensional and with lighter smoothness assumptions than the usual ones. The paper generalizes to the multiobjective case existing results for single-objective optimal control problems in that framework. The dynamics are governed by differential equations and a finite number of terminal equality and inequality constraints are present. Necessary conditions of Pareto optimality are provided namely Pontryagin maximum principles in the strong form. Sufficient conditions are also provided.

Introduction

In this paper we study multiobjective optimal control problems, with open loop information structure, in the continuous-time framework, when the space of states and the space of controls are infinite-dimensional. We derive necessary conditions and sufficient conditions of Pareto optimality. We rely on lighter smoothness assumptions than the usual ones. The paper extends to the multiobjective case, results obtained for single-objective optimal control problems in infinite dimension.

In the continuous-time framework, some results of multiobjective optimal control problems can be found in Bellaassali and Jourani [START_REF] Bellaassali | Necessary optimality conditions in multiobjective dynamic optimization[END_REF], in Zhu [START_REF] Zhu | Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints[END_REF], in Bonnel and Kaya [START_REF] Bonnel | Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems[END_REF], in Gramatovici [START_REF] Gramatovici | Optimality conditions in multiobjective control problems with generalized invexity[END_REF], in de Oliveira and Nunes Silva [START_REF] De Oliveira | On sufficient optimality conditions for multiobjective control problems[END_REF] and in references therein. Differential games are widely used in economic theory, see [START_REF] Leitmann | Cooperative and Noncooperative Many Player Differential Games[END_REF], [START_REF] Dockner | Differential Games in Economics and Management Science[END_REF], [START_REF] Reddy | Necessary and sufficient conditions for Pareto optimality in infinite horizon cooperative differential games[END_REF] , [START_REF] Engwerda | Necessary and sufficient conditions for Pareto optimal solutions of cooperative differential games[END_REF] and [START_REF] Stalford | Criteria for Pareto optimality in cooperative differential games[END_REF] and Pareto optimality plays a central role in analyzing these problems. In the discrete-time framework, results on infinite-horizon multiobjective optimal control problems can be found in Hayek [START_REF] Hayek | Infinite horizon multiobjective optimal control problem in the discrete time case[END_REF] and [START_REF] Hayek | A generalization of mixed problems with an application to multiobjective optimal control[END_REF], [START_REF] Hayek | Infinite-horizon multiobjective optimal control problems for bounded processes[END_REF], in Ngo-Hayek [START_REF] Ngo | Necessary Conditions of Pareto Optimality for Multiobjective Optimal Control Problems under Constraints[END_REF]. Bachir and Blot [START_REF] Bachir | Infinite Dimensional Infinite-horizon Pontryagin Principles for Discrete-time Problems[END_REF], [START_REF] Bachir | Infinite Dimensional Multipliers and Pontryagin Principles for Discrete-time Problems[END_REF] extended infinite-horizon single-objective optimal control problems in the discrete-time framework, to the case of infinite-dimensional spaces of states and controls and Hayek [START_REF] Hayek | Infinite-dimensional Infinite-horizon Multiobjective Optimal Control in Discrete Time[END_REF] extended these results to multiobjective optimal control problems.

In this paper we rely on the results of Blot and Yilmaz in [START_REF] Blot | A generalization of Michel's result on the Pontryagin Maximum Principle[END_REF] and [START_REF] Blot | Pontryagin Principle and Envelope theorem[END_REF] to study multiobjective optimal control problems in an infinite-dimensional setting and in continuous time. We obtain necessary conditions of Pareto optimality under the form of Pontryagin Principles and we provide sufficient conditions of Pareto optimality.

We start by providing necessary conditions of optimality for Mayer multiobjective optimal control problems and we deduce necessary conditions for Bolza problems with lighter smoothness assumptions. The Hadamard differential of a mapping between Banach spaces, which is stronger than the Gâteaux differential but weaker than the Fréchet differential, has been applied many times in the literature. In finite dimension, the Hadamard differential coincides with the Fréchet differential , but for infinite-dimensional spaces the Fréchet differential is much stronger, even for Lipschitz functions.

We provide different results relying on different constraint qualifications namely to obtain non trivial multipliers associated to the objective functions. For the sufficient conditions we follow Mangasarian [START_REF] Mangasarian | Nonlinear programming[END_REF] and Seierstadt-Sydsaeter [START_REF] Seierstad | Sufficient Conditions in Optimal Control Theory[END_REF] and we rely on weaker assumptions than the usual ones namely the concavity at a point and the quasi-concavity at a point.

The plan of this paper is as follows. Section 2 is devoted to definitions and assumptions. In section 3 the problems are presented: multiobjective optimal control problems governed by a differential equation when the space of states and the space of controls are infinite-dimensional, in the continuous-time framework. The notions of Pareto optimality and weak Pareto optimality are defined. In section 4 the theorems on necessary conditions of Pareto optimality are stated namely Pontryagin maximum principles in the strong form for a Mayer's problem and for a Bolza's problem. In section 5 we give sufficient conditions. The proofs of the necessary conditions theorems are provided in section 6 and those of the sufficient ones in section 7.

Definitions and assumptions

We set N the set of positive integers and N * = N \ {0}. R denotes the set of real numbers and R + the set of non-negative real numbers. When X and Y are Hausdorff space, C 0 (X, Y ) denotes the space of continuous mappings from X into Y . When Y be a Hausdorff space and T ∈ R * + =]0, +∞[. As in [START_REF] Blot | A generalization of Michel's result on the Pontryagin Maximum Principle[END_REF], a function u : [0, T ] → Y is called piecewise continuous when there exists a subdivision 0 = τ 0 < τ

1 < ... < τ k < τ k+1 = T such that • For all i ∈ {0, ..., k}, u is continuous on ]τ i , τ i+1 [.
• For all i ∈ {0, ..., k}, the right-hand limit u(τ i +) exists in Y .

• For all i ∈ {1, ..., k + 1}, the left-hand limit u(τ i -) exists in Y . The space of piecewise continuous mappings from [0, T ] to Y is denoted by

P C 0 ([0, T ], Y ). A function u ∈ P C 0 ([0, T ], Y
) is called a normalized piecewise continuous function when moreover u is right continuous on [0, T [ and when u(T -) = u(T ) cf. [START_REF] Blot | A generalization of Michel's result on the Pontryagin Maximum Principle[END_REF]. We denote by N P C 0 ([0, T ], Y ) the space of such functions. As in [START_REF] Blot | A generalization of Michel's result on the Pontryagin Maximum Principle[END_REF], when Y is a real Banach space, a function x : [0, T ] → Y is called piecewise continuously differentiable when x ∈ C 0 ([0, T ], Y ) and there exists a subdivision (τ i ) 0≤i≤k+1 of [0, T ] such that the following conditions are fulfilled.

• For all i ∈ {0, ..., k}, x is continuously differentiable on ]τ

i , τ i+1 [ • For all i ∈ {0, ..., k}, x ′ (τ i +) exists in Y • For all i ∈ {1, ..., k + 1}, x ′ (τ i -) exists in Y
The (τ i ) 1≤i≤k+1 are the corners of the function x.

We denote by P C 1 ([0, T ], Y ) the space of such functions.

When G is an open subset of Y , P C 1 ([0, T ], G) is the set of functions x ∈ P C 1 ([0, T ], Y ) such that x([0, T ]) ⊂ G. When x ∈ P C 1 ([0, T ], Y
) and (τ i ) 0≤i≤k+1 are the corners of the function x, we define the function dx : [0, T ] → Y , called the extended derivative of x, by setting

dx(t) :=    x ′ (t) if t ∈ [0, T ] \ {τ i : i ∈ {0, ..., k + 1}} x ′ (τ i +) if t = τ i , i ∈ {0, ..., k} x ′ (T -) if t = T. (2.1)
Notice that, contrary to the usual derivative of x, the extended derivative of x is defined on [0, T ] all over. Note that dx ∈ N P C 0 ([0, T ], Y ) and we have the following relation between x, dx and the Riemann integral:

for all a < t in [0, T ], x(t) -x(a) = t a dx(s)ds, Besides, d is a bounded linear operator from P C 1 ([0, T ], Y ) into N P C 0 ([0, T ], Y ).
All these properties motivated the authors of [START_REF] Blot | A generalization of Michel's result on the Pontryagin Maximum Principle[END_REF] to introduce the notion of extended derivative for piecewise continuously differentiable functions. When X and Y are real normed vector spaces, L(X, Y ) denotes the space of the bounded linear mappings from X into Y and X * denotes the topological dual of X. We denote by • L the usual norm of L(X, Y ). Let G be a non-empty open subset of X, let f : G → Y be a mapping and let x ∈ G. The mapping f is called Gâteaux differentiable at x when there exists D G f(x) ∈ L(X, Y ) such that for all h ∈ X, lim t→0+

f(x+th)-f(x) t = D G f(x) • h. Moreover, D G f(x) is called the Gâteaux differential of f at x.
We say that f is Hadamard differentiable at x when there exists D H f(x) ∈ L(X, Y ) such that for each K compact in X, lim t→0+ sup h∈K

f(x+th)-f(x) t -D H f(x)•h = 0. Moreover, D H f(x) is called the Hadamard differential of f at x. When f is Hadamard differentiable at x, f is also Gâteaux differentiable at x and D H f(x) = D G f(x).
But the converse is false in general when the dimension of X is greater than 2. Notice that Hadamard differentiability and Gâteaux differentiability always coincide for locally Lipschitz functions in any normed vector space. When it exists,

D F f(x) denotes the Fréchet differential of f at x. When f is Fréchet differentiable at x, f is Hadamard differentiable at x and D F f(x) = D H f(x). But the converse is false in general when the dimension of X is infinite. When X is a finite product of n real normed spaces, X = 1≤i≤n X i , if k ∈ {1, ..., n}, D F,k f(x) (respectively D H,k f(x), respectively D G,k f(x)
) denotes the partial Fréchet (respectively Hadamard, respectively Gâteaux) differential of f at x with respect to the k-th vector variable. More information on these notions of differentials can be found in [START_REF] Tm | Differential analysis[END_REF]. Next, we introduce definitions of notions of concavity at a point in infinite dimension cf. Mangasarian [START_REF] Mangasarian | Nonlinear programming[END_REF] for the finite dimension. This concepts will be used for sufficient conditions. Let g : G → R be a mapping. The mapping g is said to be concave at x when for all y ∈ G, for all t ∈ [0, 1] s.t. (1t)x + ty ∈ G, g((1t)x + ty) ≥ (1t)g(x) + tg(y). When g is Gâteaux differentiable at x, the function g is said to be pseudo-concave

at x when for all y ∈ G, [D G g(x) • (y -x) ≤ 0 ⇒ g(y) ≤ g(x)].
The mapping g is said to be quasi-concave at x when for all y ∈ G, for all t ∈ [0, 1]

s.t. (1 -t)x + ty ∈ G,[g(x) ≤ g(y) ⇒ g(x) ≤ g((1 -t)x + ty)].
When g is Gâteaux differentiable at x and g is quasi-concave at x, we have, for all

y ∈ G, [g(y) ≥ g(x) ⇒ D G g(x) • (y -x) ≥ 0].

The multiobjective optimal control problems

Let T ∈]0, +∞[, E is a real Banach space, Ω is a non-empty subset of E, U is a Hausdorff topological space and ξ 0 ∈ Ω. We consider the functions f :

[0, T ] × Ω × U → E, f 0 i : [0, T ] × Ω × U → R when i ∈ {1, .
.., l}, g 0 i : Ω → R when i ∈ {1, ..., l}, g α : Ω → R when α ∈ {1, ..., m} and h β : Ω → R when β ∈ {1, ..., q}, when (l, m, q) ∈ (N * ) 3 . For all i ∈ {1, ..., l} we consider also the function J i :

P C 1 ([0, T ], Ω) × N P C 0 ([0, T ], U ) → R defined by, for all (x, u) ∈ P C 1 ([0, T ], Ω) × N P C 0 ([0, T ], U ), J i (x, u) := g 0 i (x(T )) + T 0 f 0 i (t, x(t), u(t))dt.
With these elements, we can build the following multiobjective Bolza problem (B)

           Maximize (J 1 (x, u), ..., J l (x, u)) subject to x ∈ P C 1 ([0, T ], Ω), u ∈ N P C 0 ([0, T ], U ) ∀t ∈ [0, T ], dx(t) = f (t, x(t), u(t)), x(0) = ξ 0 ∀α ∈ {1, ..., m}, g α (x(T )) ≥ 0 ∀β ∈ {1, ..., q}, h β (x(T )) = 0.
Our problem is a reformulation of the multiobjective classical Bolza problem where the controlled dynamical system is formulated as follows : x ′ (t) = f (t, x(t), u(t)) when x ′ (t) exists, and the control function u ∈ P C 0 ([0, T ], U ). In [START_REF] Blot | A generalization of Michel's result on the Pontryagin Maximum Principle[END_REF], we explain that the present formulation is equivalent to the classical one, for the singleobjective Bolza problem. By using the same reasoning, we remark that this formulation is also equivalent for the multiobjective Bolza problem. When for all i ∈ {1, ..., l}, f 0 i = 0, (B) is called a multiobjective Mayer problem and it is denoted by (M). We denote by Adm(B) (respectively Adm(M)) the set of the admissible processes of (B) (respectively (M)). It is clear that Adm(B) = Adm(M). When (x, u) is an admissible process for (B) or (M), we consider the following constraint qualifications, when the functions defining the terminal constraints and the terminal parts of the criterion are Hadamard differentiable at x(T ).

(QC 0 )            If (b i ) 1≤i≤l ∈ R l + , (c α ) 1≤α≤m ∈ R m + , (d β ) 1≤β≤q ∈ R q satisfy (∀α ∈ {1, ..., m}, c α g α (x(T )) = 0), and l i=1 b i D H g 0 i (x(T )) + m α=1 c α D H g α (x(T )) + q β=1 d β D H h β (x(T )) = 0, then (∀i ∈ {1, ..., l}, b i = 0), (∀α ∈ {1, ..., m}, c α = 0) and (∀β ∈ {1, ..., q}, d β = 0). and (QC 1 )        If (c α ) 1≤α≤m ∈ R m + , (d β ) 1≤β≤q ∈ R q satisfy (∀α ∈ {1, ..., m}, c α g α (x(T )) = 0), and m α=1 c α D H g α (x(T )) + q β=1 d β D H h β (x(T )) = 0,
then (∀α ∈ {1, ..., m}, c α = 0) and (∀β ∈ {1, ..., q}, d β = 0). Definition 3.1. An admissible process (x, u) for (B) is a Pareto optimal solution for (B) when there does not exist an admissible process (x, u) for (B) such that for all i ∈ {1, ..., l}, J i (x, u) ≥ J i (x, u) and for some i 0 ∈ {1, ..., l}, J i0 (x, u) > J i0 (x, u). Definition 3.2. An admissible process (x, u) for (B) is a weak Pareto optimal solution for (B) when there does not exist an admissible process (x, u) for (B) such that for all i ∈ {1, ..., l}, J i (x, u) > J i (x, u). Now, we formulate a list of conditions which will become the assumptions of our theorems. Let (x 0 , u 0 ) be an admissible process for (B) or (M).

Conditions on the vector field.

(Av1) f ∈ C 0 ([0, T ] × Ω × U, E), for all (t, ξ, ζ) ∈ [0, T ] × Ω × U , D G,2 f (t, ξ, ζ) exists, for all (t, ζ) ∈ [0, T ] × U , D F,2 f (t, x 0 (t), ζ) exists and [(t, ζ) → D F,2 f (t, x 0 (t), ζ)] ∈ C 0 ([0, T ] × U, L(E, E)). (Av2) For all non-empty compact K ⊂ Ω, for all non-empty compact M ⊂ U , sup (t,ξ,ζ)∈[0,T ]×K×M D G,2 f (t, ξ, ζ) L < +∞.
Conditions on the integrands of the criterion. (Ai1) For all i ∈ {1, ..., l}, f

0 i ∈ C 0 ([0, T ]×Ω×U, R), for all (t, ξ, ζ) ∈ [0, T ]×Ω×U , D G,2 f 0 i (t, ξ, ζ) exists, for all (t, ζ) ∈ [0, T ] × U , D F,2 f 0 i (t, x 0 (t), ζ) exists and [(t, ζ) → D F,2 f 0 i (t, x 0 (t), ζ)] ∈ C 0 ([0, T ] × U, E * ). (Ai2) For all i ∈ {1, ..., l}, for all non-empty compact K ⊂ Ω, for all non-empty compact M ⊂ U , sup (t,ξ,ζ)∈[0,T ]×K×M D G,2 f 0 i (t, ξ, ζ) L < +∞.
Conditions on the functions defining the terminal constraints and terminal parts of the criterion (At1) For all i ∈ {1, ..., l}, g 0 i is Hadamard differentiable at x 0 (T ). (At2) For all α ∈ {1, ..., m}, g α is Hadamard differentiable at x 0 (T ). (At3) For all β ∈ {1, ..., q}, h β is continuous on a neighborhood of x 0 (T ) and Hadamard differentiable at x 0 (T ).

M : [0, T ]×Ω×U ×E * → R defined by, for all (t, x, u, p) ∈ [0, T ] × Ω × U × E * , H M (t, x, u, p) := p • f (t, x, u).
Theorem 4.2. (Pontryagin Principle for the Mayer problem) When (x 0 , u 0 ) is a Pareto optimal solution of (M), under (Av1), (Av2), (At1), (At2) and (At3), there exists

(θ i ) 1≤i≤l ∈ R l , (λ α ) 1≤α≤m ∈ R m , (µ β ) 1≤β≤q ∈ R q
and an adjoint function p ∈ P C 1 ([0, T ], E * ) which satisfy the following conditions.

(NN) ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 (Si) For all i ∈ {1, ..., l}, θ i ≥ 0 and for all α ∈ {1, ..., m}, λ α ≥ 0.

(Sℓ) For all α ∈ {1, ..., m}, λ α g α (x 0 (T )) = 0. (TC)

l i=1 θ i D H g 0 i (x 0 (T )) + m α=1 λ α D H g α (x 0 (T )) + q β=1 µ β D H h β (x 0 (T )) = p(T ). (AE.M) dp(t) = -D F,2 H M (t, x 0 (t), u 0 (t), p(t)) for all t ∈ [0, T ]. (MP.M) For all t ∈ [0, T ], for all ζ ∈ U , H M (t, x 0 (t), u 0 (t), p(t)) ≥ H M (t, x 0 (t), ζ, p(t)). (CH.M) HM := [t → H M (t, x 0 (t), u 0 (t), p(t))] ∈ C 0 ([0, T ], R).
(NN) is a condition of non nullity, (Si) is a sign condition, (Sℓ) is a slackness condition, (TC) is the transversality condition, (AE.M) is the adjoint equation, (MP.M) is the maximum principle and (CH.M) is a condition of continuity on the Hamiltonian.

Corollary 4.3. In this setting and under the assumptions of Theorem 4.2, if moreover we assume that (QC 1 ) is fulfilled for (x, u) = (x 0 , u 0 ), then, for all t ∈ [0, T ], ((θ i ) 1≤i≤l , p(t)) is never equal to zero.

Corollary 4.4. In this setting and under the assumptions of Theorem 4.2, if moreover we assume that (QC 0 ) is fulfilled for (x, u) = (x 0 , u 0 ), then, for all t ∈ [0, T ], p(t) is never equal to zero.

As in [START_REF] Blot | Pontryagin Principle and Envelope theorem[END_REF], we introduce another condition (Av3) U is a subset of a real normed vector space Y , there exists t ∈ [0, T ] s.t.

U is a neighborhood of u 0 ( t) in Y , D G,3 f ( t, x 0 ( t), u 0 ( t)) exists and it is surjective.

Corollary 4.5. In this setting and under the assumptions of Theorem 4.2, if moreover we assume that (QC 1 ) is fulfilled for (x, u) = (x 0 , u 0 ) and (Av3), then

(θ i ) 1≤i≤l = 0.
We introduce a new condition of linear independence.

(Alib) U is a subset of a real normed vector space Y s.t. U is a neighborhood of u 0 (T ) in Y , D G,3 f (T, x 0 (T ), u 0 (T )) exists and ((D H g α (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))) 1≤α≤m , (D H h β (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))) 1≤β≤q
) are linearly independent.

Corollary 4.6. In this setting and under the assumptions of Theorem 4.2, if moreover we assume (Alib) is fulfilled, then (θ i ) 1≤i≤l = 0.

For each j ∈ {1, ..., l}, we consider the following condition:

(Af) j U is a subset of a real normed vector space Y s.t. U is a neighborhood of u 0 (T ) in Y , D G,3 f (T, x 0 (T ), u 0 (T )) exists and ((D H g 0 i (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))) i =j , (D H g α (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))) 1≤α≤m , (D H h β (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))) 1≤β≤q ) are linearly independent.
Corollary 4.7. In this setting and under the assumptions of Theorem 4.2, if, for each j ∈ {1, ..., l}, we have (Af) j , then θ j = 0 i.e. we can take θ j = 1. Moreover,

((θ i ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) ∈ R l × R m × R q × P C 1 ([0, T ], E * ) with θ j = 1
that verify the conclusions of Theorem 4.2 are unique.

B : [0, T ] × Ω × U × E * × R l → R defined by, for all (t, x, u, p, θ) ∈ [0, T ]×Ω×U ×E * ×R l , H B (t, x, u, p, θ) := l i=1 θ i f 0 i (t, x, u) + p • f (t,
x, u). Theorem 4.9. (Pontryagin Principle for the Bolza problem) When (x 0 , u 0 ) is a Pareto optimal solution of (B), under (Ai1), (Ai2), (Av1), (Av2), (At1), (At2) and (At3), there exists (θ i ) 1≤i≤l ∈ R l , (λ α ) 1≤α≤m ∈ R m , (µ β ) 1≤β≤q ∈ R q and an adjoint function p ∈ P C 1 ([0, T ], E * ) which satisfy the following conditions.

(NN) ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 (Si) For all i ∈ {1, ..., l}, θ i ≥ 0 and for all α ∈ {1, ..., m}, λ α ≥ 0.

(Sℓ) For all α ∈ {1, ..., m}, λ α g α (x 0 (T )) = 0. (TC)

l i=1 θ i D H g 0 i (x 0 (T )) + m α=1 λ α D H g α (x 0 (T )) + q β=1 µ β D H h β (x 0 (T )) = p(T ). (AE.B) dp(t) = -D F,2 H B (t, x 0 (t), u 0 (t), p(t), (θ i ) 1≤i≤l ) for all t ∈ [0, T ]. (MP.B) For all t ∈ [0, T ], for all ζ ∈ U , H B (t, x 0 (t), u 0 (t), p(t), (θ i ) 1≤i≤l ) ≥ H B (t, x 0 (t), ζ, p(t), (θ i ) 1≤i≤l ). (CH.B) HB := [t → H B (t, x 0 (t), u 0 (t), p(t), (θ i ) 1≤i≤l )] ∈ C 0 ([0, T ], R).
Corollary 4.10. In this setting and under the assumptions of Theorem 4.9, if moreover we assume that (QC 1 ) is fulfilled for (x, u) = (x 0 , u 0 ), then, for all t ∈ [0, T ], ((θ i ) 1≤i≤l , p(t)) is never equal to zero.

Corollary 4.11. In this setting and under the assumptions of Theorem 4.9, if moreover we assume that (QC 1 ) is fulfilled for (x, u) = (x 0 , u 0 ) and (Av3), then (θ i ) 1≤i≤l = 0. Corollary 4.12. In the setting and under the assumptions of Theorem 4.9, if moreover we assume (Alib) is fulfilled, then (θ i ) 1≤i≤l = 0.

For each j ∈ {1, ..., l}, we consider the following condition:

(Af) 0 j U is a subset of a real normed vector space Y s.t. U is a neighbor- hood of u 0 (T ) in Y , D G,3 f (T, x 0 (T ), u 0 (T )) exists, ∀i ∈ {1, ..., l}, i = j D G,3 f 0 i (T, x 0 (T ), u 0 (T )) exists and ((D H g 0 i (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) + D G,3 f 0 i (T, x 0 (T ), u 0 (T ))) i =j , (D H g α (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))) 1≤α≤m , (D H h β (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))) 1≤β≤q ) are linearly independent.
Corollary 4.13. In this setting and under the assumptions of Theorem 4.9, if, for each j ∈ {1, ..., l}, we have (Af) 0 j , then θ j = 0 (i.e. we can choose θ j = 1). Moreover, if D G,3 f 0 j (T, x 0 (T ), u 0 (T )) exists, then we have:

((θ i ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) ∈ R l × R m × R q × P C 1 ([0, T ], E * ) with θ j = 1
that verify the conclusions of Theorem 4.9 are unique.

Sufficient conditions of Pareto optimality

Let (x, u) ∈ P C 1 ([0, T ], Ω) × N P C 0 ([0, T ], U ), we consider the following conditions. (St1) For all i ∈ {1, ..., l} g 0 i is concave at x(T ) and Hadamard differentiable at x(T ). (St1-bis) For all i ∈ {1, ..., l} g 0 i is pseudo-concave at x(T ) and Hadamard differentiable at x(T ). (St2) For all α ∈ {1, ..., m}, g α is quasi-concave at x(T ) and Hadamard differentiable at x(T ). (St3) For all β ∈ {1, ..., q}, h β and -h β are quasi-concave at x(T ) and Hadamard differentiable at x(T ). (Si1) For all i ∈ {1, ..., l}, f

0 i ∈ C 0 ([0, T ] × Ω × U, R). (Si2) For all t ∈ [0, T ], for all i ∈ {1, ..., l}, D F,2 f 0 i (t, x(t), u(t)) exists and [t → D F,2 f 0 i (t, x(t), u(t))] ∈ N P C 0 ([0, T ], E * ). (Sv1) f ∈ C 0 ([0, T ] × Ω × U, E).
(Shb1) For each (x, u) ∈ Adm(B), for all t ∈ [0, T ] almost everywhere for the canonical measure of Borel on [0, T ],

H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l )-H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l ) ≥ dp(t)•(x(t)-x(t)),
then we have: if (θ i ) 1≤i≤l = 0, then (x, u) is a weak Pareto optimal solution of (B), if for all i ∈ {1, ..., l}, θ i = 0, then (x, u) is a Pareto optimal solution of (B).

Theorem )] is concave at x(t) and Gâteaux differentiable at x(t), then we have: if (θ i ) 1≤i≤l = 0, then (x, u) is a weak Pareto optimal solution of (B), if for all i ∈ {1, ..., l}, θ i = 0, then (x, u) is a Pareto optimal solution of (B).

Theorem 5.7. When (x, u) ∈ Adm(B), under (St1), (St2), (St3), (Si1), (Si2), (Sv1), (Sv2) if there exists ((θ i ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) belongs to R l+m+q × P C 1 ([0, T ], E * ) verifying all the conclusions of Theorem 4.9 with (x 0 , u 0 ) = (x, u) and if the following condition is satisfied (Shb3) U is a subset of a real normed vector space Y s.t. for all t ∈ [0, T ], U is a neighborhood of u(t), and for all t ∈ [0, T ], [(ξ, ζ) → H B (t, ξ, ζ, p(t), (θ i ) 1≤i≤l ) is Gâteaux differentiable at (x(t), u(t)) and concave at (x(t), u(t)), then we have: if (θ i ) 1≤i≤l = 0, then (x, u) is a weak Pareto optimal solution of (B), if for all i ∈ {1, ..., l}, θ i = 0, then (x, u) is a Pareto optimal solution of (B).

Remark 5.8. By using our constraint qualifications, we can rewrite the conclusion of Theorem 5.6 and Theorem 5.7 as follows. If the condition (Alib) or [(QC 1 ) and (Av3)] is fulfilled for (x 0 , u 0 ) = (x, u) then (x, u) is a weak Pareto optimal solution of (B), if, for each j ∈ {1, ..., l}, (Af) 0 j is fulfilled for (x 0 , u 0 ) = (x, u), then (x, u) is a Pareto optimal solution of (B).

6. Proof of the necessary conditions 6.1. Proof of the Theorem 4.2. Lemma 6.1. For all i ∈ {1, ..., l}, (x 0 , u 0 ) is a solution of the following singleobjective Mayer problem

(M i )                Maximize J i (x, u) := g 0 i (x(T )) subject to x ∈ P C 1 ([0, T ], Ω), u ∈ N P C 0 ([0, T ], U ) ∀t ∈ [0, T ], dx(t) = f (t, x(t), u(t)), x(0) = ξ 0 ∀k ∈ {1, ..., l}, k = i, g 0 k (x(T )) ≥ g 0 k (x 0 (T )) ∀α ∈ {1, ..., m}, g α (x(T )) ≥ 0 ∀β ∈ {1, ..., q}, h β (x(T )) = 0.
Proof. Let i ∈ {1, ..., l}. We proceed by contradiction, we assume that (x 0 , u 0 ) is not a solution of (M i ) i.e. there exists (x, u) an admissible process of (M i ) s.t. g 0 i (x(T )) > g 0 i (x 0 (T )). This can be rewritten (x, u) ∈ Adm(M) s.t. g 0 i (x(T )) > g 0 i (x 0 (T )) and for all k ∈ {1, ..., l}, k = i, g 0 k (x(T )) ≥ g 0 k (x 0 (T )). Therefore, (x 0 , u 0 ) is not a Pareto optimal solution. This is a contradiction.

For each x ∈ Ω, for each i ∈ {2, ..., l}, we set g i (x) = g 0 i (x)g 0 i (x 0 (T )). Thanks to (At1), for each i ∈ {2, ..., l}, g i is Hadamard differentiable at x 0 (T ) and D H g i (x 0 (T )) = D H g 0 i (x 0 (T )). Consequently, by using the Lemma 6.1 and (At2), (At3), (Av1), (Av2), the assumptions of Theorem 2.4 in [START_REF] Blot | Pontryagin Principle and Envelope theorem[END_REF] are fulfilled for (M 1 )

(M 1 )                Maximize g 0 1 (x(T )) subject to x ∈ P C 1 ([0, T ], Ω), u ∈ N P C 0 ([0, T ], U ) ∀t ∈ [0, T ], dx(t) = f (t, x(t), u(t)), x(0) = ξ 0 ∀i ∈ {2, ..., l}, g i (x(T )) ≥ 0 ∀α ∈ {1, ..., m}, g α (x(T )) ≥ 0 ∀β ∈ {1, ..., q}, h β (x(T )) = 0.
Hence, we obtain that there exists

(θ i ) 1≤i≤l ∈ R l , (λ α ) 1≤α≤m ∈ R m , (µ β ) 1≤β≤q ∈ R q and an adjoint function p ∈ P C 1 ([0, T ], E * ) which satisfy the following condi- tions. (NNs) ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0.
(Sis) For all i ∈ {1, ..., l}, θ i ≥ 0 and for all α ∈ {1, ..., m}, λ α ≥ 0. (Sℓs) For all i ∈ {2, ..., l}, θ i g i (x 0 (T )) = 0 and for all α ∈ {1, ..., m}, λ α g α (x 0 (T )) = 0. (TCs)

l i=1 θ i D H g 0 i (x 0 (T )) + m α=1 λ α D H g α (x 0 (T )) + q β=1 µ β D H h β (x 0 (T )) = p(T ). (AE.Ms) dp(t) = -D F,2 H M (t, x 0 (t), u 0 (t), p(t)) for all t ∈ [0, T ]. (MP.Ms) For all t ∈ [0, T ], for all ζ ∈ U , H M (t, x 0 (t), u 0 (t), p(t)) ≥ H M (t, x 0 (t), ζ, p(t)). (CH.Ms) HM := [t → H M (t, x 0 (t), u 0 (t), p(t))] ∈ C 0 ([0, T ], R).
Therefore, since for all i ∈ {2, ..., l}, g i (x 0 (T )) = 0, (NNs), (Sis), (Sℓs), (TCs), (AE.Ms) , (MP.Ms) and (CH.Ms) are equivalent to (NN), (Si), (Sℓ), (TC), (AE.M) , (MP.M) and (CH.M). Therefore, the proof Theorem 4.2 is complete.

6.2. Proof of Corollary 4.3. We proceed by contradiction by assuming that there exists t 1 ∈ [0, T ] such ((θ i ) 1≤i≤l , p(t 1 )) = (0, 0). Since (AE.M) is an homogeneous linear equation, and by using the uniqueness of the Cauchy problem ((AE.M), p(t 1 ) = 0), we obtain that p is equal to zero on [0, T ], in particular we have p(T ) = 0. Hence, by using (TC), (Si), (Sℓ), (QC 1 ), we obtain that (∀α ∈ {1, ..., m}, λ α = 0) and (∀β ∈ {1, ..., q}, µ β = 0). Therefore, since (θ i ) 1≤i≤l = 0, we have ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 which is a contradiction with (NN).

6.3. Proof of Corollary 4.4. We proceed by contradiction by assuming that there exists t 1 ∈ [0, T ] such p(t 1 ) = 0. Since (AE.M) is an homogeneous linear equation, and by using the uniqueness of the Cauchy problem ((AE.M), p(t 1 ) = 0, we obtain that p is equal to zero on [0, T ],

in particular we have p(T ) = 0. Consequently, by using (TC), (Si), (Sℓ), (QC 0 ), we obtain that ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 which is a contradiction with (NN).

6.4. Proof of Corollary 4.5. We proceed by contradiction, we assume that (θ i ) 1≤i≤l = 0. Since D G,3 f ( t, x 0 ( t), u 0 ( t)) exists, D G,3 H M ( t, x 0 ( t), u 0 ( t), p( t)) exists and

D G,3 H M ( t, x 0 ( t), u 0 ( t), p( t)) = p( t) • D G,3 f ( t, x 0 ( t), u 0 ( t)).
Therefore, by using (MP.M), we have

p( t) • D G,3 f ( t, x 0 ( t), u 0 ( t)) = 0. Since D G,3 f ( t, x 0 ( t), u 0 ( t)
) is surjective, we have p( t) = 0. This is a contradiction with the Corollary 4.3, therefore (θ i ) 1≤i≤l = 0.

6.5. Proof the Corolloray 4.6. We proceed by contradiction, we assume that (θ i ) 1≤i≤l = 0. Since D G,3 f (T, x 0 (T ), u 0 (T )) exists, D G,3 H M (T, x 0 (T ), u 0 (T ), p(T )) exists and

D G,3 H M (T, x 0 (T ), u 0 (T ), p(T )) = p(T ) • D G,3 f (T, x 0 (T ), u 0 (T )).
Consequently, by using (MP.M), we have p(T )

• D G,3 f (T, x 0 (T ), u 0 (T )) = 0.
That is why, thanks to (TC) and (θ i ) 1≤i≤l = 0, we obtain that

m α=1 λ α D H g α (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) + q β=1 µ β D H h β (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0.
Hence, thanks to (Alib), we have ((λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0. Consequently, since (θ i ) 1≤i≤l = 0, we have ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 this a contradiction with (NN). 6.6. Proof the Corolloray 4.7. Let j ∈ {1, ..., l}. We assume that (Af) j . We proceed by contradiction, we assume that θ j = 0. Since D G,3 f (T, x 0 (T ), u 0 (T )) exists, D G,3 H M (T, x 0 (T ), u 0 (T ), p(T )) exists and

D G,3 H M (T, x 0 (T ), u 0 (T ), p(T )) = p(T ) • D G,3 f (T, x 0 (T ), u 0 (T )).
Consequently, by using (MP.M), we have p(T ) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0. That is why, thanks to (TC) and θ j = 0, we obtain that

i =j θ i D H g 0 i (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) + m α=1 λ α D H g α (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) + q β=1 µ β D H h β (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0.    Hence, thanks to (Af) j , we have ((θ i ) i =j , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0.
Consequently, since θ j = 0, we have ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 this a contradiction with (NN). We set ∀i ∈ {1, ..., l}, θ ′ i = θi θj , ∀α ∈ {1, ..., m}, λ ′ α := λα θj , ∀β ∈ {1, ..., q}, µ ′ β := µ β θj and p ′ := 1 θj p. Since the set of (

(θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) ∈ R l+m+q × P C 1 ([0, T ], E * ) verifying the conclusions of Theorem 4.2 is a cone, we have ((θ ′ i ) 1≤i≤l , (λ ′ α ) 1≤α≤m , (µ ′ β ) 1≤β≤q , p ′ ) that verifies the conclusions of Theorem 4.2 with θ ′ j = 1. Let ((θ 1 i ) 1≤i≤l , (λ 1 α ) 1≤α≤m , (µ 1 β ) 1≤β≤q , p 1 ) ∈ R l+m+q × P C 1 ([0, T ], E * ) and ((θ 2 i ) 1≤i≤l , (λ 2 α ) 1≤α≤m , (µ 2 β ) 1≤β≤q , p 2 ) ∈ R l+m+q × P C 1 ([0, T ], E * ) s.t.
the conclusions of the Theorem 4.2 are verified with θ 1 j = θ 2 j = 1. Then, we have, for all ℓ ∈ {1, 2}, p ℓ (T ) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0. Therefore, we have (p 1 (T )p 2 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0. By using (TC), we have

i =j (θ 1 i -θ 2 i )D H g 0 i (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) + m α=1 (λ 1 α -λ 2 α )D H g α (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) + q β=1 (µ 1 β -µ 2 β )D H h β (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0.   
Hence, by using (Af) j , ∀(i, α, β) ∈ {1, ..., l}×{1, ..., m}×{1, ..., q}, θ

1 i = θ 2 i , λ 1 α = λ 2 α
and µ 1 β = µ 2 β . Therefore, p 1 (T ) = p 2 (T ); that is why (AE.M), we have : p 1 = p 2 . 6.7. Proof of the Theorem 4.9. In [START_REF] Blot | A generalization of Michel's result on the Pontryagin Maximum Principle[END_REF], by transforming the single-objective Bolza problem into a single-objective Mayer problem, the authors proof the Pontryagin Maximum Principle for the single-objective Bolza problem thanks to the Pontryagin Maximum Principle for the single-objective Mayer problem. For the proof of the Pontryagin Maximum Principle for the multiobjective Bolza problem, we will use the same reasoning. That is why, we introduce the following elements, for all t ∈ [0, T ], for all X = (σ 1 , ..., σ l , x) ∈ R l × Ω, for all u ∈ U , F (t, X, u) := (f 0 1 (t, x, u), ..., f 0 l (t, x, u), f (t, x, u)), G 0 i (X) := σ i + g 0 i (x) for all i ∈ {1, ..., l}, G α (X) := g α (x) for all α ∈ {1, ..., m}, H β (X) := h β (x) for all β ∈ {1, ..., q}. Then, we can introduce the following multiobjective Mayer problem (MB)

           Maximize (G 0 1 (X(T )), ..., G 0 l (X(T ))) subject to X ∈ P C 1 ([0, T ], R l × Ω), u ∈ N P C 0 ([0, T ], U )
dX(t) = F (t, X(t), u(t)), X(0) = (0, ξ 0 ) ∀α ∈ {1, ..., m}, G α (X(T )) ≥ 0 ∀β ∈ {1, ..., q}, H β (X(T )) = 0. Lemma 6.2. For each (x, u) ∈ Adm(B), by setting for all t ∈ [0, T ], for all i ∈ {1, ..., l}, σ i (t) := t 0 f 0 i (s, x(s), u(s))ds, we have ((σ 1 , ..., σ l , x), u) ∈ Adm(MB). Proof. Let (x, u) ∈ Adm(B). Since u ∈ N P C 0 ([0, T ], U ) and x ∈ P C 1 ([0, T ], Ω), by using (Ai1), we have, for each i ∈ {1, ..., l}, [t → f 0 i (t, x(t), u(t))] ∈ N P C 0 d ([0, T ], R). Consequently, for each i ∈ {1, ..., l}, σ i ∈ P C 1 ([0, T ], R) and for all t ∈ [0, T ], dσ i (t) = f 0 i (t, x(t), u(t)). Hence, (σ 1 , ..., σ l , x) ∈ P C 1 ([0, T ], R l × Ω) and for all t ∈ [0, T ],

d(σ 1 , ..., σ l , x)(t) = (dσ 1 (t), ..., dσ l (t), dx(t)) = (f 0 1 (t, x(t), u(t)), ..., f 0 l (t, x(t), u(t)), f (t, x(t), u(t))) = F (t, (σ 1 , ..., σ l , x)(t), u(t))
Moreover, we have, for all α ∈ {1, ..., m}, G α ((σ 1 , ..., σ l , x)(T )) = g α (x(T )) ≥ 0 and ∀β ∈ {1, ..., q}, H β ((σ 1 , ..., σ l , x)(T )) = h β (x(T )) = 0. Therefore, since (σ 1 , ..., σ l , x)(0) = (σ 1 (0), ..., σ l (0), x(0)) = (0, ξ 0 ), we have ((σ 1 , ..., σ l , x), u) ∈ Adm(MB).

Hence, by setting for all i ∈ {1, ..., l}, for all t ∈ [0, T ], σ 0 i (t) := t 0 f 0 i (s, x 0 (s), u 0 (s))ds, by using the Lemma 6.2, we have (X 0 , u 0 ) := ((σ 0 1 , ..., σ 0 l , x 0 ), u 0 ) ∈ Adm(MB).

Lemma 6.3. (X 0 , u 0 ) is a Pareto optimal solution of the multiobjective problem (MB).

Proof. We proceed by contradiction, we assume that (X 0 , u 0 ) is not a Pareto optimal solution for (MB) i.e. there exists (X, u) = ((σ 1 , ..., σ l , x), u) ∈ P C 1 ([0, T ], R l × Ω) × N P C 0 ([0, T ], U ) admissible process for (MB) s.t. for all i ∈ {1, ..., l}, G 0 i (X(T )) ≥ G 0 i (X 0 (T )) and there exists i 0 ∈ {1, ..., l}, G 0 i0 (X(T )) > G 0 i0 (X 0 (T )). Since X ∈ P C 1 ([0, T ], R l × Ω) and ∀t ∈ [0, T ], dX(t) := F (t, X(t), u(t)), we have x ∈ P C 1 ([0, T ], Ω) and for all i ∈ {1, ..., l}, σ i ∈ P C 1 ([0, T ], R) s.t.

∀t ∈ [0, T ], dx(t) = f (t, x(t), u(t)) and dσ i (t) = f 0 i (t, x(t), u(t)). Moreover, we have also for all α ∈ {1, ..., m}, g α (x(T )) ≥ 0 and for all β ∈ {1, ..., q}, h β (x(T )) = 0. Consequently, we have (x, u) ∈ Adm(B). Moreover, for all t ∈ [0, T ], we have σ i (t) = t 0 f 0 i (s, x(s), u(s))ds. Then, for all i ∈ {1, ..., l},

G 0 i (X(T )) = T 0 f 0 i (s, x(s), u(s))ds + g 0 i (x(T )) ≥ G 0 i (X 0 (T )) =
T 0 f 0 i (s, x 0 (s), u 0 (s))ds + g 0 i (x 0 (T )) and there exists i 0 ∈ {1, ..., l},

G 0 i0 (X(T )) = T 0 f 0 i0 (s, x(s), u(s))ds + g 0 i0 (x(T )) > G 0 i0 (X 0 (T )) = T 0 f 0 i0 (s, x 0 (s), u 0 (s))ds + g 0 i0 (x 0 (T )
). This a contradiction with (x 0 , u 0 ) is a Pareto optimal solution. Lemma 6.4. The assumptions of Theorem 4.9 for the multiobjective Mayer problem (MB) with the Pareto optimal solution (X 0 , u 0 ) are verified.

Proof. We consider the linear functions i ∈ {1, ..., l}, w 1 i : R l × E → R defined by, w 1 i (σ 1 , ..., σ l , ξ) = σ i and w 2 : R l × E → E, defined by, w 2 (σ 1 , ..., σ l , ξ) = ξ. For all i ∈ {1, ..., l}, since

G 0 i = w 1 i R l ×Ω + g 0 i • w 2 |R l ×Ω
, by using the property of the chain rule of the Hadamard differentiable function, see [START_REF] Tm | Differential analysis[END_REF] p.267, and (At1), we have

D H G 0 ((σ 0 1 , ..., σ 0 l , x 0 )(T )) = w 1 i + D H g 0 i (x 0 (T )) • w 2 . ( 6
.1) Therefore, (At1) is verified for (MB) with the Pareto optimal solution (X 0 , u 0 ). Next, for all α ∈ {1, ..., m}, since G α = g α • w 2 |R l ×Ω , by using the property of the chain rule of the Hadamard differentiable function, see [START_REF] Tm | Differential analysis[END_REF] p.267, and (At2), we have

D H G α ((σ 0 1 , ..., σ 0 l , x 0 )(T )) = D H g α (x 0 (T )) • w 2 . (6.
2) Hence, (At2) is verified for (MB) with the Pareto optimal solution (X 0 , u 0 ). Moreover, for all β ∈ {1, ..., q}, since H β = h β • w 2 |R l ×Ω , by using the property of chain rule of the Hadamard differentiable function, see [START_REF] Tm | Differential analysis[END_REF] p.267, and (At3), we have

D H H β ((σ 0 1 , ..., σ 0 l , x 0 )(T )) = D H h β (x 0 (T )) • w 2 . (6.3) Since h β is continuous on a neighborhood V β 0 of x 0 (T ) in Ω and w 2 |R l ×Ω ∈ C 0 (R l × Ω, Ω), there exists W β 0 of X 0 (T ) in R l × Ω s.t. w 2|W 0 β ∈ C 0 (W β 0 , V β 0 ). Hence, we have H β |W β 0 ∈ C 0 (W β 0 , R).
Consequently, (At3) is verified for (MB) with the Since the set of ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) ∈ R l+m+q × P C 1 ([0, T ], E * ) verifying the conclusions of Theorem 4.9 is a cone, we have ((θ ′ i ) 1≤i≤l , (λ ′ α ) 1≤α≤m , (µ ′ β ) 1≤β≤q , p ′ ) that verifies the conclusions of Theorem 4.9 with θ ′ j = 1. Now, we assume that D G,3 f 0 j (T, x 0 (T ), u 0 (T )) exists. Let ((θ

1 i ) 1≤i≤l , (λ 1 α ) 1≤α≤m , (µ 1 β ) 1≤β≤q , p 1 ) ∈ R l+m+q × P C 1 ([0, T ], E * ) and ((θ 2 i ) 1≤i≤l , (λ 2 α ) 1≤α≤m , (µ 2 β ) 1≤β≤q , p 2 ) ∈ R l+m+q × P C 1 ([0, T ], E * ) s.t.
the conclusions of the Theorem 4.9 are verified with θ 1 j = θ 2 j = 1. Then, we have, for all ℓ ∈ {1, 2},

p ℓ (T ) • D G,3 f (T, x 0 (T ), u 0 (T )) + D G,3 f 0 j (T, x 0 (T ), u 0 (T ))+ i =j θ ℓ i D G,3 f 0 i (T, x 0 (T ), u 0 (T )) = 0. Therefore, we have (p 1 (T )-p 2 (T ))•D G,3 f (T, x 0 (T ), u 0 (T ))+ i =j (θ 1 i -θ 2 i )D G,3 f 0 i (T, x 0 (T ), u 0 (T )) = 0.
By using (TC), we have

i =j (θ 1 i -θ 2 i )(D H g 0 i (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T ))+ D G,3 f 0 i (T, x 0 (T ), u 0 (T ))) + m α=1 (λ 1 α -λ 2 α )D H g α (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) + q β=1 (µ 1 β -µ 2 β )D H h β (x 0 (T )) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0.       
Hence, by using (Af) 0 j , ∀(i, α, β) ∈ {1, ..., l}×{1, ..., m}×{1, ..., q}, θ

1 i = θ 2 i , λ 1 α = λ 2 α
and µ 1 β = µ 2 β . Therefore, p 1 (T ) = p 2 (T ); that is why, by using (AE.B), we have : p 1 = p 2 . 7. Proof of the sufficient conditions 7.1. Proof of the Theorem 5.1. Let (x, u) ∈ Adm(M). By using (TC), we have

l i=1 θ i D H g 0 i (x(T )) • (x(T ) -x(T )) = p(T ) • (x(T ) -x(T )) - m α=1 λ α D H g α (x(T )) • (x(T ) -x(T )) - q β=1 µ β D H h β (x(T )) • (x(T ) -x(T ))    (7.1)
Moreover, by using (Si) and (Sℓ), we have for each α ∈ {1, ..., m}, λ α g α (x(T )) ≤ λ α g α (x(T )). Consequently, by using (St2), we have for all α ∈ {1, ..., m}, λ α D H g α (x(T )) • (x(T )x(T )) ≥ 0. Moreover, thanks to (St3), we have for all β ∈ {1, ..., q}, µ β D H h β (x(T )) • (x(T )x(T )) = 0. Hence T 0 (H M (t, x(t), u(t), p(t)) -H M (t, x(t), u(t), p(t)))dt+ T 0 (H M (t, x(t), u(t), p(t)) -H M (t, x(t), u(t), p(t)))dt = 0 where we have used (Shm1).

(AE.M) and (MP.M), we have D G,(2,3) H M (t, x(t), u(t), p(t))•(x(t)-x(t), u(t)-u(t)) = -dp(t)•(x(t)-x(t)). Therefore, (Shm1) is verified. Hence, the assumptions of the Theorem 5.1 are verified and the conclusions follow. Furthermore, by using (Si) and (Sℓ), we have for each α ∈ {1, ..., m}, λ α g α (x(T )) ≤ λ α g α (x(T )). Consequently, by using (ST2), we have for all α ∈ {1, ..., m}, λ α D H g α (x(T )) • (x(T )x(T )) ≥ 0. Besides, thanks to (ST3), we have for all β ∈ {1, ..., q}, µ β D H h β (x(T )) • (x(T )x(T )) = 0. Hence, by using Now, we assume that (θ i ) 1≤i≤l = 0. We want to prove that (x, u) is a weak Pareto optimal solution. We proceed by contradiction, we assume that (x, u) is not a weak Pareto optimal solution i.e. there exists (x, u) ∈ Adm(B) such that for all i ∈ {1, ..., l}, J i (x, u) > J i (x, u). Consequently, we have l i=1 θ i J i (x, u) > l i=1 θ i J i (x, u). This is a contradiction with (x, u) is a solution of (P θ ). Next, we assume that for all i ∈ {1, ..., l}, θ i = 0. We want to prove that (x, u) is a Pareto optimal solution. We proceed by contradiction, we assume that (x, u) is not a Pareto optimal solution i.e. there exists

4 . 1 .

 41 Necessary conditions of Pareto optimality 4.Necessary conditions of Pareto optimality for the Mayer problem. Definition 4.1. The Hamiltonian of (M) is the function H

4. 2 .

 2 Necessary conditions of Pareto optimality for the Bolza problem. Definition 4.8. The Hamiltonian of (B) is the function H

l

  i=1 θ i D H g 0 i (x(T )) • (x(T )x(T )) ≤ p(T ) • (x(T )x(T )) = p(0) • (x(0)x(0)) + T 0 d(p(t) • (x(t)x(t)))dt = T 0 dp(t) • (x(t)x(t))dt + T 0 p(t) • d(x(t)x(t))dt ≤

7. 4 .

 4 Proof of the Theorem 5.5. Let (x, u) ∈ Adm(B). By using (ST1), we havel i=1 θ i J i (x, u) = l i=1 θ i g 0 i (x(T )) + T 0 l i=1 θ i f 0 i (t, x(t), u(t))dt ≤ l i=1 θ i g 0 i (x(T )) + l i=1 θ i D H g i (x(T )) • (x(T )x(T ))+ T 0 l i=1 θ i f 0 i (t, x(t), u(t))dt. By using (TC), we havel i=1 θ i D H g 0 i (x(T )) • (x(T )x(T )) = p(T ) • (x(T )x(T )) -m α=1 λ α D H g α (x(T )) • (x(T )x(T )) -q β=1 µ β D H h β (x(T )) • (x(T )x(T ))

l

  i=1 θ i D H g 0 i (x(T )) • (x(T )x(T )) ≤ p(T ) • (x(T )x(T )) = T 0 d(p(t) • (x(t)x(t)))dt = T 0 dp(t) • (x(t)x(t))dt + T 0 p(t) • d(x(t)x(t))dt ≤ T 0 (H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l ) -H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l ))dt+

T0

  (p(t) • f (t, x(t), u(t))p(t) • f (t, x(t), u(t)))dt= T 0 l i=1 θ i f 0 i (t, x(t), u(t))dt -T 0 l i=1 θ i f 0 i (t, x(t), u(t))dt. Therefore, we have l i=1 θ i J i (x, u) ≤ l i=1 θ i g 0 i (x(T )) + T 0 l i=1 θ i f 0 i (t, x(t), u(t))dt -T 0 l i=1 θ i f 0 i (t, x(t), u(t))dt + T 0 l i=1 θ i f 0 i (t, x(t), u(t))dt = l i=1 θ i J i (x, u).. Consequently, (x, u) a solution of the following single optimization problem : (P θ ) Maximize l i=1 θ i J i (x, u) subject to (x, u) ∈ Adm(B).

  ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) belongs to R l+m+q × P C 1 ([0, T ], E * )verifying all the conclusions of Theorem 4.9 with (x 0 , u 0 ) = (x, u) and if the following condition is satisfied (Shb2) for all (t, ξ) ∈ [0, T ] × Ω, H

	5.6. When (x, u) ∈ Adm(B), under (St1), (St2), (St3), (Si1), (Si2),
	(Sv1), (Sv2) if there exists ((θ i

* B (t, ξ, p(t), (θ i ) 1≤i≤l ) = max ζ∈U H B (t, ξ, ζ, p(t), (θ i ) 1≤i≤l ) exists, and for all t ∈ [0, T ], [ξ → H * B (t, ξ, p(t), (θ i ) 1≤i≤l

(Sv2) For all t ∈ [0, T ] D F,2 f (t, x(t), u(t)) exists and [t → D F,2 f (t, x(t), u(t))] ∈ N P C 0 ([0, T ], L(E, E)).

Theorem 5.1. When (x, u) ∈ Adm(M), under (St1-bis), (St2), (St3), (Sv1) if there exists ((θ i ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) ∈ R l+m+q ×P C 1 ([0, T ], E * ) verifying the conclusions (NN), (Si), (Sℓ) and (TC) of Theorem 4.2 with (x 0 , u 0 ) = (x, u) and if the following condition is satisfied (Shm1) For each (x, u) ∈ Adm(M), for all t ∈ [0, T ] almost everywhere for the canonical measure of Borel on [0, T ],

H M (t, x(t), u(t), p(t)) -H M (t, x(t), u(t), p(t)) ≥ dp(t) • (x(t)x(t)), then we have: if (θ i ) 1≤i≤l = 0, then (x, u) is a weak Pareto optimal solution of (M), if for all i ∈ {1, ..., l}, θ i = 0, then (x, u) is a Pareto optimal solution of (M).

Theorem 5.2. When (x, u) ∈ Adm(M), under (St1-bis), (St2), (St3), (Sv1), (Sv2) if there exists ((θ i ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) ∈ R l+m+q ×P C 1 ([0, T ], E * ) verifying all the conclusions of Theorem 4.2 with (x 0 , u 0 ) = (x, u) and if the following condition is satisfied (Shm2) for all (t, ξ) ∈ [0, T ] × Ω, H * M (t, ξ, p(t)) = max ζ∈U H M (t, ξ, ζ, p(t)) exists, and for all t ∈ [0, T ] , [ξ → H * M (t, ξ, p(t))] is concave at x(t) and Gâteaux differentiable at x(t), then we have: if (θ i ) 1≤i≤l = 0, then (x, u) is a weak Pareto optimal solution of (M), if for all i ∈ {1, ..., l}, θ i = 0, then (x, u) is a Pareto optimal solution of (M).

Theorem 5.3. When (x, u) ∈ Adm(M), under (St1-bis), (St2), (St3), (Sv1), (Sv2) if there exists ((θ i ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) ∈ R l+m+q ×P C 1 ([0, T ], E * ) verifying all the conclusions of Theorem 4.2 with (x 0 , u 0 ) = (x, u) and if the following condition is satisfied (Shm3) U is a subset of a real normed vector space Y s.t. for all t ∈ [0, T ], U is a neighborhood of u(t), and for all t

is Gâteaux differentiable at (x(t), u(t)) and concave at (x(t), u(t)), then we have: if (θ i ) 1≤i≤l = 0, then (x, u) is a weak Pareto optimal solution of (M), if for all i ∈ {1, ..., l}, θ i = 0, then (x, u) is a Pareto optimal solution of (M).

Remark 5.4. By using our constraint qualifications, we can rewrite the conclusion of Theorem 5.2 and Theorem 5.3 as follows.

If the condition (Alib) or [(QC 1 ) and (Av3)] is fulfilled for (x 0 , u 0 ) = (x, u) then (x, u) is a weak Pareto optimal solution of (M), if, for each j ∈ {1, ..., l}, (Af) j is fulfilled for (x 0 , u 0 ) = (x, u), then (x, u) is a Pareto optimal solution of (M).

Theorem 5.5. When (x, u) ∈ Adm(B), under (St1), (St2), (St3), (Si1), (Si2) (Sv1), (Sv2) if there exists ((θ i ) 1≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q , p) belongs to R l+m+q × P C 1 ([0, T ], E * ) verifying the conclusions (NN), (Si), (Sℓ) and (TC) of Theorem 4.9 with (x 0 , u 0 ) = (x, u) and if the following condition is satisfied Pareto optimal solution (X 0 , u 0 ). We consider the continuous function χ :

We remark that F := (f 0 1 • χ, ..., f 0 l • χ, f • χ). By using (Ai1) and (Av1), we have, for all i ∈ {1, ..., l}, f 0

By using (Ai1) and (Av1), we have, for all (t, σ, ξ, ζ)

, by using (Ai1) and (Av1), we have D F,2 F (t, X 0 (t), ζ) exists and

). Consequently, by using (Ai1) and (Av1), we have

Therefore, (Av1) is verified for (MB) with the Pareto optimal solution (X 0 , u 0 ). Let K be a non-empty compact s.t. K ⊂ R l × Ω and M be a non-empty compact s.t. M ⊂ U . We consider the linear continuous function ̟ : R l × Ω → Ω, defined by, for all (σ, ξ) ∈ R l × Ω, ̟(σ, ξ) := ξ. Since K is a non-empty compact, K = ̟(K) is a non empty compact s.t. K ⊂ Ω. Consequently, by using (Ai2) and (Av2), we have for all i ∈ {1, ..., l} sup

and sup

Therefore, by using (6.4), we have sup

Hence (Av2) is verified for (MB) with the Pareto optimal solution (X 0 , u 0 ).

By using the Lemma 6.4, by applying the Theorem 4.9, we obtain that, there exists (θ i ) 1≤i≤l ∈ R l , (λ α ) 1≤α≤m ∈ R m , (µ β ) 1≤β≤q ∈ R q and an adjoint function

For all i ∈ {1, ..., l}, θ i ≥ 0 and for all α ∈ {1, ..., m}, λ α ≥ 0.

(iii) For all α ∈ {1, ..., m}, λ α G α (X 0 (T )) = 0.

(iv)

Where the function H MB : [0, T ]×(R l ×Ω)×U ×(R l ×E) * → R is the Hamiltonian of the problem (MB), defined by H MB (t, (σ 1 , ..., σ l , x), u, P ) = P •F (t, (σ 1 , ..., σ l , x), u). We consider the linear continuous function ψ

Therefore, by using (i), (ii) and (iii), we have respectively (NN), (Si) and (Sℓ). By using (iv), we have , for each i ∈ {1, ..., l} P (T ) • (e i , 0) = θ i where (e i ) 1≤i≤l is the canonical basis of R l and ∀ξ ∈ E,

For all i ∈ {1, ..., l}, we consider the linear continuous function

Consequently, by using (vi) and (vii), we have proved (MP.B) and (CH.B). Hence the proof of the Theorem 4.9 is complete. 6.8. Proof of Corollary 4.10. We proceed by contradiction by assuming that there exists t 1 ∈ [0, T ] such ((θ i ) 1≤i≤l , p(t 1 )) = (0, 0). Since (AE.B) becomes an homogeneous linear equation, and by using the uniqueness of the Cauchy problem ((AE.B), p(t 1 ) = 0), we obtain that p is equal to zero on [0, T ], in particular we have p(T ) = 0. Hence, by using (TC), (Si), (Sℓ), (QC 1 ), we obtain that (∀α ∈ {1, ..., m}, λ α = 0) and (∀β ∈ {1, ..., q}, µ β = 0). Therefore, since (θ i ) 1≤i≤l = 0, we have ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 which is a contradiction with (NN). 6.9. Proof of Corollary 4.11. We proceed by contradiction, we assume that (θ i ) 1≤i≤l = 0. Since D G,3 f ( t, x 0 ( t), u 0 ( t)) exists, D G,3 H B ( t, x 0 ( t), u 0 ( t), p( t), 0) exists and

Therefore, by using (MP.B), we have p( t) • D G,3 f ( t, x 0 ( t), u 0 ( t)) = 0. Since D G,3 f ( t, x 0 ( t), u 0 ( t)) is surjective, we have p( t) = 0. Therefore, we have ((θ i ) 1≤i≤l , p( t)) = 0. This is a contradiction with the Corollary 4.10, therefore (θ i ) 1≤i≤l = 0. 6.10. Proof the Corolloray 4.12. We proceed by contradiction, we assume that (θ i ) 1≤i≤l = 0. Since D G,3 f (T, x 0 (T ), u 0 (T )) exists, D G,3 H B (T, x 0 (T ), u 0 (T ), p(T ), 0) exists and

Consequently, by using (MP.B), we have p(T ) • D G,3 f (T, x 0 (T ), u 0 (T )) = 0. That is why, thanks to (TC) and (θ i ) 1≤i≤l = 0, we obtain that

Hence, thanks to (Alib), we have ((λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0. Consequently, since (θ i ) 1≤i≤l = 0, we have ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 this a contradiction with (NN). 6.11. Proof the Corolloray 4.13. Let j ∈ {1, ..., l}. We assume that (Af) j . We proceed by contradiction, we assume that θ j = 0. Since D G,3 f (T, x 0 (T ), u 0 (T )) exists and for all i = j, D G,3 f 0 i (T, x 0 (T ), u 0 (T )) exists, D G,3 H B (T, x 0 (T ), u 0 (T ), p(T ), (θ i ) 1≤i≤l ) exists and

x 0 (T ), u 0 (T )). Consequently, by using (MP.B), we have

That is why, thanks to (TC) and θ j = 0, we obtain that

Hence, thanks to (Af) 0 j , we have ((θ i ) i =j , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0. Consequently, since θ j = 0, we have ((θ i ) 1 ≤i≤l , (λ α ) 1≤α≤m , (µ β ) 1≤β≤q ) = 0 this a contradiction with (NN). We set ∀i ∈ {1, ..., l}, θ ′ i = θi θj , ∀α ∈ {1, ..., m}, λ ′ α := λα θj , ∀β ∈ {1, ..., q}, µ ′ β := µ β θj and p ′ := 1 θj p.

Therefore, thanks to (St1-bis), we have

). Hence, (x, u) is a solution of the following single-objective optimization problem :

Now, we assume that (θ i ) 1≤i≤l = 0. We want to prove that (x, u) is a weak Pareto optimal solution. We proceed by contradiction, we assume that (x, u) is not a weak Pareto optimal solution i.e. there exists (x, u) ∈ Adm(M) such that for all i ∈ {1, ..., l}, J i (x, u) > J i (x, u). Consequently, we have

But this contradicts the optimality of (x, u) for the problem (P θ ). Next, we assume that for all i ∈ {1, ..., l}, θ i = 0. We want to prove that (x, u) is a Pareto optimal solution. We proceed by contradiction, we assume that (x, u) is not a Pareto optimal solution i.e. there exists (x, u) ∈ Adm(M) such that for all i ∈ {1, ..., l}, J i (x, u) ≥ J i (x, u) and for some i 0 ∈ {1, ..., l}, J i0 (x, u) > J i0 (x, u). Hence, we obtain that l i=1 θ i J i (x, u) > l i=1 θ i J i (x, u) which contradicts the optimality of (x, u). 7.2. Proof of the Theorem 5.2. Notice that (Shm2) implies (Shm1). Indeed, let (x, u) ∈ Adm(M). For all t ∈ [0, T ], for all ε > 0 small enough, we have x(t) + ε(x(t)x(t)) ∈ Ω, therefore by using (MP.M)

ε (H M (t, x(t) + ε(x(t)x(t)), u(t), p(t)) -H M (t, x(t), u(t), p(t))). Therefore, since (Shm2) and (Sv2), when ε → 0 we have D G,2 H * M (t, x(t), p(t)) • (x(t)-x(t)) ≥ D G,2 H M (t, x(t), u(t), p(t))•(x(t)-x(t)). Therefore, by using (AE.M), we have

Besides, for all ε > 0 small enough, we have x(t) + ε(x(t)x(t)) ∈ Ω, therefore by using (MP.M) and (Shm2), we have

Hence, we have

). Consequently, when ε → 0 and thanks to (AE.M) and (7.2), we have H M (t, x(t), u(t), p(t)) -H M (t, x(t), u(t), p(t)) ≥ dp(t) • (x(t)x(t)). Hence, the assumptions of the Theorem 5.1 are verified and the conclusions follow. 7.3. Proof of the Theorem 5.3. Notice that (Shm3) implies (Shm1). Indeed, let (x, u) ∈ Adm(M), let t ∈ [0, T ], since [(ξ, ζ) → H M (t, ξ, ζ, p(t))] is Gâteaux differentiable and concave at (x(t), u(t)), we have H M (t, x(t), u(t), p(t)) -H M (t, x(t), u(t), p(t)) ≤ D G,(2,3) H M (t, x(t), u(t), p(t)) • (x(t)x(t), u(t)u(t)). Therefore, by using (x, u) ∈ Adm(B) such that for all i ∈ {1, ..., l}, J i (x, u) ≥ J i (x, u) and there exists i 0 ∈ {1, ..., l}, J i0 (x, u) > J i0 (x, u). Hence, we obtain that l i=1 θ i J i (x, u) > l i=1 θ i J i (x, u). This is a contradiction with (x, u) is a solution of (P θ ). 7.5. Proof of the Theorem 5.6. Notice that (Shb2) implies (Shb1). Indeed, let (x, u) ∈ Adm(B). We set θ = (θ i ) 1≤i≤l . For all t ∈ [0, T ], for all ε > 0 small enough, we have x(t) + ε(x(t)x(t)) ∈ Ω, therefore by using (MP.B)

. Hence, by using (AE.B), we have

Besides, for all ε > 0 small enough, we have x(t) + ε(x(t)x(t)) ∈ Ω, hence by using (MP.B) and (Shb2), we have ). Consequently, when ε → 0, from (7.4), we have H B (t, x(t), u(t), p(t), θ) -H B (t, x(t), u(t), p(t), θ) ≥ dp(t) • (x(t)x(t)). Hence, the assumptions of the Theorem 5.5 are verified and the conclusions follow. 7.6. Proof of the Theorem 5.7. Notice that (Shb3) implies (Shb1). Indeed, let (x, u) ∈ Adm(B), let t ∈ [0, T ], since [(ξ, ζ) → H B (t, ξ, ζ, p(t), (θ i ) 1≤i≤l )] is Gâteaux differentiable and concave at (x(t), u(t)), we have H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l ) -H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l ) ≤ D G,(2,3) H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l ) • (x(t)x(t), u(t)u(t)). Therefore, by using (AE.B) and (MP.B), we have D G,(2,3) H B (t, x(t), u(t), p(t), (θ i ) 1≤i≤l ) • (x(t)x(t), u(t)u(t)) = -dp(t) • (x(t)x(t)). Hence, (Shb1) is verified. Therefore,the assumptions of the Theorem 5.5 are verified and the conclusions follow.


Eran Shuali

Keywords: Christian Hebraism, Neo-Latin poetry, Biblical Hebrew, Renaissance, Humanism, translation studies, New Testament reception studies

production of his translation as well as on the major linguistic and stylistic choices

sources he used and on his main choices of language and style. Through this investigation of Thúri's method of work, I will seek to describe his conception of the intellectual activity in which he engaged, that of translating the New Testament into Hebrew.

György Thúri

György Thúri was born in Belgrade in the middle of the sixteenth century. [START_REF] Szinnyei | Magyar írók élete és munkái[END_REF] His father, Pál Thúri Farkas, was a well-known Calvinist preacher. [START_REF] Dötsch | Respublica literaria: Zu den humanistischen Beziehungen Ungarns zur Kurpfalz un 1600[END_REF] György Thúri was Hungarian, and according to his poems identified as such. [START_REF] Dötsch | The subject matter for me is to put to writing the deeds of the invincible Emperor in poor Pannonian song", he says of Kaiser Rudolf II in one of his poems[END_REF] He studied Theology at the University of Wittenberg until 1597, [START_REF] Vilmos | A Hazai és külföldi iskolázás a XVI. században[END_REF] and like many students of Calvinist denomination he subsequently moved west to continue his studies at the University of Heidelberg, 10 from which he graduated on July 29, 1600. [START_REF] Vilmos | A Hazai és külföldi iskolázás a XVI. Században[END_REF] After his studies, he was a school principal in Tolna, Hungary. [START_REF] Szinnyei | Magyar írók[END_REF] During his studies abroad, Thúri was supported by his patron Zsigmond Rákóczi, who would later become the prince of Transylvania. [START_REF] Dötsch | Respublica literaria[END_REF] Rákóczi is indeed mentioned several times in Thúri's poems, a fact which receives attention in a congratulatory poem written to Thúri:

Then by your singing of the praise of the supreme hero, May the Pannonian land grow in devoutness.

And by your mouth, may the virtue of the Rakoczis be celebrated.

In 1615, Thúri's friend, the Hungarian poet Johann Philipp Pareus, [START_REF] Pareus | Pareus clearly speaks of his friendship to Thúri[END_REF] published a collection of Thúri's poems under the title: Musa Posthuma -that is, a Muse born after the death of its father, the poet György Thúri. Thúri must have died then before the date of publication. [START_REF] Flood | poem entitled: In Effigiem Georgii Thurii Poëta Panninij[END_REF] György Thúri was a part of a group of Hungarian poets who wrote mainly in Latin. [START_REF]For a general presentation of this poetical circle, see A. TARNAI[END_REF] As stated above, his poems were published in David Pareus and Johann Philipp Pareus' book Musae Fugitivae, [START_REF] Pareus | [END_REF] and later on in Johann Philipp Pareus' Delitiae Poetarum Hungaricorum nunc primum in hac Germania exhibitae -"Delights of Hungarian poets: presented now for the first time here in Germany." [START_REF] Pareus | Delitiae Poetarum Hungaricorum[END_REF] The collection of his published poems is composed of thirteen elegies, several short commemorative poems and epitaphs, and a number of epigrams. [START_REF] Tarnai | Latin Poetry in Hungary in the 16 th and 17 th Centuries[END_REF] Stylistically, Thúri's poems are classicizing. They are written in meter, almost all in elegiac couplets. They make frequent mention of Greco-Roman mythology: e.g. of the Muses, Zeus, Apollo, Venus, Minerva, Pallas and Mercury. [START_REF] Amor | Delitiae Poetarum Hungaricorum, p. 341 three times)[END_REF] What stands out when looking through the corpus of Thúri's poems is the social functionality of a great many of them. Most of his poems seem to have been written either for some specific occasion or to a specific addressee. Thus, one finds in the collection, poems for the occasion of birthdays, marriages, graduations of classmates at University, laureation of fellow Neo-Latin poets, and deceases. The poems are most often addressed to friends, poets, scholars and patrons. Thúri's poems as well as those of other students and teachers around him at the Universities of Wittenberg and Heidelberg, 22 suggest that the writing of Latin poetry was a common part of the social and intellectual life led in Thúri's circle. Thúri received the title of Poeta Laureatus together Johann Philipp Pareus, from the German poet Paul Schede Melissus at Heidelberg on December 20, 1600. 23 A last biographical issue that is of importance for this study is Thúri's knowledge of Hebrew.

In the title of one of the congratulatory poems addressed to him, Thúri is called: "a man most learned in the science of the three cardinal languages". 24 And this would, of course, attest that he was known for his mastery of Latin, Greek and Hebrew. It is reasonable to assume that he acquired his knowledge of Hebrew in the course of his University studies in Theology. 25 In addition to what he might have learned in class, his knowledge of the language most probably benefited from his personal acquaintance with Valentin Schindler. Schindler was a renowned Hebraist and philologist of Semitic languages. He was a professor of Hebrew at the University of Wittenberg from 1571 until he was compelled to leave his post in 1592 as he was suspected of Calvinist tendencies. 26 Thúri addresses two very short epigrams to Schindler. The first one runs as follows:

You are a true Hebrew, Chaldean, Arab and Syrian.

The homeland and the faith have however made [of you] a German. 27 The second epigram is rather more cryptic:

I

Does this poem suggest that Thúri and Schindler corresponded in poems?

Besides Thúri's translation of the Epistles to the Galatians and the Ephesians into Hebrew which will be examined in detail below, we also possess one original Hebrew composition of his that may enable us to assess his mastery of the language. 29 The booklet containing his translation of Galatians and Ephesians ends with an obituary poem to his deceased friend, compatriot and classmate at the University of Wittenberg, Stephen Szikszai. 30 A few lines from the poem (lines 13-18) may serve as an example of Thúri's qualities as a writer of Hebrew and as a poet: In the way of God he is learned.

And he walked in his teaching with desire.

Knowledgeable in every wisdom, he was also knowledgeable in the Sacred Language.

Thúri's Translation of Paul's Epistles

28 "Mitto tibi falcem pro falce sed exigo iuncum, / An non ista tibi conditio aequa sat est?" Ad eundem, in J. P. PAREUS, ed., Delitiae Poetarum Hungaricorum, p. 349. 29 One of his poems in Latin suggests that, together with his classmate at Heidelberg, Petrus Aluincius, he wrote an unpublished book in Latin and Hebrew. Ad eundem [Petrum Aluincium] discedentem in J. P. PAREUS, ed., Delitiae Poetarum Hungaricorum, p. 352. 30 Stephen Szikszai (Stephanus Sixai) is listed alongside Thúri among Hungarian students who finished their studies at Wittenberg in 1597; F. VILMOS, A Hazai és külföldi iskolázás a XVI. században, 319. As mentioned in note 22, in 1598 appeared a collection of poems written in his memory. This collection includes a poem written by Thúri in Latin. This poem is different in its content from the Hebrew poem mentioned here: "Sequitur Carmen in Obitum Dn. Stephani The pages in this book are not numbered. 31 The Hebrew text of Thúri's book is quoted as it appears there, including the occasional errors in spelling or pointing.

Thúri's Hebrew translation of the Epistles to the Galatians and the Ephesians was published by Johannes Crato in Wittenberg in 1598, a year after Thúri had finished his university studies there. 32 The book contains 23 printed pages, including the title page. Except titles which are in Latin, the book comprises only Hebrew text, attesting that it was intended for rather advanced students of Hebrew. The title page includes a dedication of the book to Thúri's patron (mecoenatus) Zsigmond Rákóczi. Before and after the text of the translation itself, there is a poem in Hebrew. The book begins with a poem by the Conrad Neander of Bergen, entitled:

Tetrastichon in duas Paulinas Epistolas à D. Georgio Thurio in Hebraeam linguam translatas. 33 The poem presents the translation to the reader, hoping he will learn from it how to write in Hebrew by himself: He has translated incense from the Greek language.

He has taken the truth about everything from the Hebrew language.

Learn from it [how] you too [can] compose in this manner.

In this poem, Neander encourages the reader of this translation to learn from the work of György "the peasant" 34 Thúri, how to compose similar works. What this poem teaches us most of all is that the readers in view for this translation were advanced Christian students of Hebrew able to read and write in Hebrew.

As stated above, the book ends with Thúri's poem in memory of his friend Stephen Szikszai.

32 Robert Dán supposes the translation was Thúri's doctoral exercise; R. DÁN, "Thury's Hebrew Galatians and Ephesians", p. 73. 33

Compare this poem to Conrad Neander's Latin Tetrastichon on the first page of his own Epistolae Anniversariae, quae Dominicis diebus ac Sanctorum festis praecipuis in Ecclesia praeleguntur EBRAEAE iam recens ex Graeco textu ac Syra Paraphrasi factae; breuibusque scholijs, sed tamen haud poenitendis illustratae, à M. Conrado Neandro Bergensi. Et nunc demum τετραγλώττως EBRAICE, GRAECE, LATINE, ac GERMANICE. Editae opera ac cum praefatione clarissimi ac doctißimi viri D. M. Michaelis Neandri Soraviensis, Leipzig, Georg Defner, 1586.

34 This is a play of words on György Thúri's first name, which comes from the Greek word γεωργός -"peasant" / "husbandman". Neander renders this using the Biblical expression

ע ֹ ֵב ד א ֲ ָד מ ָ ה
-"worker of the soil"; see notably Gen 4:2, but also Gen 2:5; 3:23; 4:12; Prov 12:11; 28:19. I owe this observation to David Benka, whom I greatly thank for it.

The Hebrew text of the translation is pointed and punctuated using the three following disjunctive accents: Atnaḥ, Silluq and Sof Pasuq. Verse numbers of every fifth verse are marked in Hebrew letters in the margin. The Hebrew text is altogether correctly printed, but certain typographical errors do appear every so often. 35

The Sources Used

There is every reason to believe the claim stated in the title of the book that the Epistles were translated from the Greek text and not from that of the Latin Vulgate like some of the first translations of New Testament writings to Hebrew: Epistolae Pauli Apostoli ad Galatas et Ephesios: è Graecâ in purè Hebraeam linguam translatae. 36 First, Thúri most probably knew Greek, as is stated by the poet who wrote that he was learned in the three cardinal languages. 37 This can also be attested by his use of Greek words written in Greek characters in his poems. 38 Second, in the few cases where the Greek text of the Epistle to the Galatians clearly differs from that of the Latin Vulgate, Thúri's Hebrew follows the Greek: 39 ἕτερον δὲ τῶν ἀποστόλων οὐκ εἶδον (…) (Gal 1: 19) alium autem apostolorum vidi neminem (...)

ְו ַא ֵח ר ִמ ִצּ י ִר י ם ל ֹ א ְר ִא י ִ֑ת י ו ) . . . (
The Greek and Hebrew texts say: "I have not seen another (one) of the Apostles", whereas the Vulgate says: "I have not seen any other of the Apostles". Thúri most probably used an edition based on Erasmus' so-called Textus Receptus, which had become the dominant form of the Greek text of the New Testament in the sixteenth century. 40 A hint about a second source used by Thúri is provided on the very first page of the book, which begins, as we have seen, with a poem by the Lutheran Hebraist Conrad Neander. 41 Indeed, the only Hebrew translation of Pauline writings published prior to Thúri's, is the fragments of Paul's Epistles comprised in Neander's Epistolae Anniversariae 42 -a collection of short biblical passages, mainly from the New Testament Epistles, intended for liturgical use. Among these passages we find several ones from the Epistles to the Galatians and the Ephesians. 43 And when comparing Thúri's translation of these passages with that of Neander, it becomes clear that Thúri has made use of his Lutheran predecessor's work, without however copying it word for word. Several examples may illustrate Thúri's practice.

In the following comparative tables significant similarities between the two translations are marked by bold characters, and significant differences by a darker background.

Gal 3:15-17 42 For full title, see note 33. 43 The passages of Gal and Eph included in Neander's book are the following: Gal 3:15-22; 3:23-29; 4:1-7; 4:21-31; 5:16-24; 5:25-6:10; Eph 2:17-22; 3:13-21; 4:1-6; 4:7-13; 4:23-28; 5:1-9; 5:15-20; 6:10-17.

The rather particular example of György Thúri, a vir trilinguis who mastered Greek, Latin and Hebrew, wrote and published in the last two languages and translated from the first, shows in an especially neat manner, I think, how Hebrew learning was conceived in the late sixteenth as part and parcel of the interest, which had become widespread, in the classical languages and literatures considered as the foundations of European culture.

Furthermore, the example of Thúri is useful for illustrating at least two features of the way in which these ancient languages were approached: the classicizing tendency and the importance of intellectual networks. Hence, Thúri's writing in both Latin and Hebrew attests to his wish to approximate the classical idiom and style of each of these languages, that of metrical mythologically inspired poetry in Latin and that of the Old Testament in Hebrew. In addition, not at all the work of a solitary poet, Thúri's Latin poetry bears the marks of its production within a circle of Neo-Latin poets who read their poems in social events, corresponded with one another in poems and published together in collective volumes. Similarly, his translation of Paul's Epistles shows his involvement in the network of Christian Hebraists of the time, whose works he read and used for the composition of his own, and with whom he was in personal contact. The existence of these networks is both a proof of the magnitude of the study of Greek, Latin and Hebrew in Thúri's time and an important factor that enabled the growth of Classical and Hebrew scholarship.

Is this making not rather easy for you? 28 22 See for instance the small collection of poems in Latin and Greek written in commemoration of Stephen Szikszai by Thúri himself and professors and comrades of his at Wittenberg; Iusta doctissimo, ornatissimo

  am sending you a sickle for a sickle, but I demand a cane. Die Octobr. honeste sepulto: Rectore Academiae M. Antonio Evonymo, Senjore coetus Ungarici, XLIX. Balthasare G. Miscolcino: Persoluta ab Amicis, & clarißimis viris, Wittenberg, Simon Grönberg, 1598. 23 J. DÖTSCH, "Respublica literaria", p. 10; J. L. FLOOD, Poets Laureate in the Holy Roman Empire, p. 2095. 24 Bartholomaus Keckermannus ad Georgium Thurium Vngarum, Virum trium Cardinalium linguarum peritia cultißimum, nuper Laurea coronatum, in J. P. PAREUS, ed., Delitiae Poetarum Hungaricorum, p. 313. 25 Robert Dán presumes that Thúri studied Hebrew at Wittenberg under Lauritius Fabricius who occupied the chair of Hebrew in this University from 1593 to 1629; R. DÁN, "Thury's Hebrew Galatians and Ephesians", p. 72.

	, vitae
	inculpatae, morumque integritate praestanti, & in artibus ac linguis optime versato viro, Dn. Stephano K.
	Sixovio, Ungaro, in celeberrimam Witebergensium Academiam, augendorum studiorum caussa, Anno D. 1597,
	die 14 Martii concedenti; Anno vero subsequente 1598, die 6. Octobris, ad horam 11. meridianam, in Domino
	pie placideque obdormienti, ibidemque in coemeterio Templi, ut vocant, parochialis, juxta tumulos reliquorum
	quatuor popularum, olim demortuorum, 8.

26 S. G. BURNETT, Christian Hebraism in the Reformation Era (1500-1660), p. 63, 110, 255. 27 "Verus es Hebraeus, Chaldaeus, Arabsque Syrusque / Patria Germanum sed facit, atque Fides." Ad Valentinum Schindlerum, V. C., in J. P. PAREUS, ed., Delitiae Poetarum Hungaricorum, p. 349.

  Sixai Amici et Contubernali charissimi", in G. THÚRI, Epistolae Pauli Apostoli ad Galatas et Ephesios: è Graecâ in purè Hebraeam linguam translatae: gratique animi ac sempiternae memoriae testificandae ergo Spectabili ac Magnifico Domino Domino Sigismundo Rakotzi, Comiti Comitatuum Thorna & Munkats, noc non Sacr: Caes: Regiaeque Maiest: Consiliario & c.Mecoenati studiorum suorum benignissimo dicatae à Georgio Thurio Alumno humilimo, Wittenberg, Iohannes Crato, 1598, [last two pages].

quae erant in Christo

  ἤµην δὲ ἀγνοούµενος τῷ προσώπῳ ταῖς ἐκκλησίαις τῆς Ἰουδαίας ταῖς ἐν Χριστῷ (Gal 1:22)eram autem ignotus facie ecclesiis IudaeaeThe Greek and Hebrew texts say: "… which (are) in Christ", whereas the Vulgate has: "which were in Christ". Examples of Christian translations from a Latin original are Sebastian Münster's translation of the Gospel of Matthew; see E. SHUALI, "Les deux versions de l'Évangile de Matthieu en hébreu publiées par Sebastian Münster (1537) et par Jean du Tillet et Jean Mercier (1555)"; and admittedly Giovanni Battista Iona's Quator Evangelia Novi Testamenti Ex Latino in Hebraicum sermonem versa, Rome, Propaganda Fide, 1668. 37 See note 24. 38 See several examples in his epigrams; J. P. PAREUS, ed., Delitiae Poetarum Hungaricorum, p. 349, 351, 353, 354. 39 The differences between the Greek and Latin texts of Gal are taken from M.-J. LAGRANGE, "La Vulgate latine de l'Épître aux Galates et le text grec", Revue biblique 14 (1917), p. 428-431. τὸ γὰρ Ἁγὰρ Σινᾶ ὄρος ἐστὶν ἐν τῇ Ἀραβίᾳ (…) (Gal 4:25)Greek and Hebrew: "For Hagar is Mount Sinai in Arabia". Vulgate: "For Mount Sinai is in

					Sina enim mons est in Arabia (…)
																									) . . . (	ָ֑ר ב	ֲﬠ	ַבּ	ָה ר	ַנ י	ִס י	ִה י א	ָג ר	ָה	ִכּ י
	Arabia."
																									׃	ַח	ִֽשׁ י	ָמ	ְב	ֲא שׁ ֶ ר	ָ֑ד ה	ְי ה וּ	ֵל י	ֵה	ְק	ַמ	ְל	ִנ י ם	ָפ	ְב	ִתּ י	ְﬠ	ַד	נ וֹ	ְו ל ֹ א
	35 See for instance the following cases: ָא ד וֹ ן ָח	ִח י	ֲא	ֲﬠ ק ֹ ב	ַו	ִ֑ת י	ְל	ִבּ	ִ֑ת י ו	ִא י	ְר	ל ֹ א	ִר י ם	ִצּ י	ִמ	ֵח ר	ַא	ְו	(Gal 1:19);	ָבּ א וּ	ֲא שׁ ֶ ר	ֶק ר	ֵח י שׁ ֶ	ַא
	? (	ָלּ ט	ַב	)	ס	ָלּ	ַב	(2:4);	ִתּ י	ְס	ַר	ָה	? (	ֶשׁ ר	ֲא	)	מ	ֲא שׁ ֶ	ֶא ת -	(2:18).

36

  The Text of the New Testament: Its Transmission, Corruption, and Restoration, 4 th edition, New York -Oxford, Oxford University Press, 2005, p. 149. 41 As a proof of Conrad Neander's confession, see his citation of Luther in his Epistolae Anniversariae, p. [24]. See also S. G. BURNETT, Christian Hebraism in the Reformation Era, p. 124, 134.

	Textus Receptus										Neander																			Thúri
	15 Ἀδελφοί κατὰ ἄνθρωπον λέγω•			ָד ם	ָא	ְכּ	ֵבּ ר	ַד	ְל	ִל י	ֶר ְך	צ ֹ	ֵכּ ן	ִר י ם	ָק	ְי	ַה	ַח י	ַא		ִנ י	ָ֑א	ֵבּ ר	ַד	ְמ	ָד ם	ָא	ָה	ֶר ְך	ֶד	ְכּ	ִר י ם	ָק	ְי	ַה	ַח י	ַא
	ὅµως ἀνθρώπου κεκυρωµένην	ל וֹ	ֱא נ וֹ שׁ	ָת ה	א ֹ	ִק י ם	ֵה	ֲא שׁ ֶ ר		ִר י ת	ְבּ	ַה		ֶא ת			ִכּ י	ל וֹ	ֱא נ וֹ שׁ	ָת הּ		א ֹ	ִק י ם	ֵה	ֲא שׁ ֶ ר	ִר י ת	ְבּ	ַה	ֶא ת -	ִכּ י
	διαθήκην
	οὐδεὶς ἀθετεῖ ἢ ἐπιδιατάσσεται	ַר ע	ְג	ִי	ְו ל ֹ א	ָה	ֶל י	ָﬠ		ִס י ף	י וֹ	א וֹ	ַא ס	ְמ	ִי	ל ֹ א	ִא י שׁ					׃	ָה	ֶֽל י	ָﬠ	ִס י ף	י וֹ	ְו ל ֹ א	ִא י שׁ	ֵפ ר	ָי	ל ֹ א
																							׃	ָנּ הּ	ֶמּ		ִמ
	16 τῷ δὲ Ἀβραὰµ ἐρρήθησαν																	ָה ם	ָר	ְב	ַא	ְל	ִה י	ְתּ	ַו						ָה ם	ָר	ְב	ַא	ְל	ֵמ ר	ָא	ֵתּ	ַו
	αἱ ἐπαγγελίαι																					ָﬠ ה	ַה שּׁ ְ ב וּ						ָﬠ ה	ֻב	ַה שּׁ ְ
	40 B. M. METZGER, B. D. EHRMAN,

Although they are similar, the two Hebrew versions are manifestly not identical. One notes that the two translators have made different choices of vocabulary, as in the translation of the Greek verb ἀθετεῖ -"declares invalid", rendered in verse 15 by Neander as ִי ְמ ַא ס -"will reject", and by Thúri as ָי ֵפ ר -"will break". One also remarks that Neander is, in general, less faithful than Thúri to "the letter" of the Greek text. He, for instance, has no problem rendering the Greek verb ἐρρήθησαν -"have been said", as ַו ְתּ ִה י -"has been" (vs. 16). He does also not hesitate to add in his translation elements that are completely absent from the Greek text, as in writing "no one adds to it or subtracts from it", whereas the Greek has only "no one adds to it." (vs. 15) 44 Certain similarities between the two versions cannot however be a matter of coincidence. In verse 15, both translators add the adjective ַה ְי ָק ִר י ם to the address ַא ַח י -"My dear brothers", although this adjective is not in the Greek text. This addition seems to go back to Luther's common rendering of the address ἀδελφοί -"Brothers", into "liebe Brüder". 45 Both translators render the first occurrence of the noun ἄνθρωπος in verse 15 as ָא ָד ם , and the second as ֱא נ וֹ שׁ , surely for the sake of variatio. In verse 17, both translators make a considerable effort to render the difficult Greek sentence intelligible in Hebrew. They change the passive clause: διαθήκην προκεκυρωµένην ὑπὸ τοῦ θεοῦ -"a covenant ratified by God", to an active one. Both place the verb meaning "to annul" before its subject "the Torah", and not after it as 44 These two verses have been translated by Neander and Thúri in an extremely similar way. However, several identical renderings are particularly revealing. Both render ὅσον χρόνον -"all the time that…", very idiomatically by the Hebrew phrase ָכּ ל - ע וֹ ד

. 46 Both interpret πάντων -"all (things)" / "everything", as

-"all possession", probably under the influence of Luther's rendering: "aller Güter". Both add the word ע ֹ ל -"burden" to the phrase: "those under the burden of the Torah". This word is not in the Greek text. And finally, both render the Greek phrase meaning: "… so that we might receive adoption as sons", in the exact same way by a Hebrew phrase signifying: "so that he will consider us as his sons".

Language and Style

In general, Thúri attempts to render the Epistles into Biblical Hebrew broadly defined. Hence, he uses the so-called converted verbal forms Wayyiqtol and Weqatal, which are very frequent in the Hebrew Bible and completely absent in the Hebrew language from the time of the 46 See 2 Sam 1:9; Job 27:3. Mishnah onwards. [START_REF] For | The Verbal System of Biblical Hebrew: A New Synthesis Elaborated of the Basis of Classical Prose[END_REF] He imitates the syntax of Biblical Hebrew, for example, by introducing the apodosis of conditional phrases with the conjunction ְו -, as in the following sentence:

Thúri's confusion between is ִא ֶמּ ָך

-"your mother" and ִﬠ ֶמּ ָך -"with you" is surprising. He also points the verb כ ב ד incorrectly, making it a past tense and not an imperative. The errors in this sentence are odd, mainly since it is a citation from the Hebrew Bible. 56 Most notable are Thúri's difficulties in using particles meaning "also". Thúri does not seem to be sure where in the Hebrew phrase the particle ַגּ ם should be placed. Thinking that the Greek καὶ applies to both the personal pronoun ὑµεῖς and to the participle ἀκούσαντες, he renders it twice: once by ַגּ ם , and once by ְו -.

What stands out most when studying Thúri's translation choices closely, is his constant effort to render Paul's Greek into idiomatic Hebrew, even though he does not always succeed in this. The correctness and beauty of the language seem to have been of chief importance for Thúri. In his translation of Paul's Epistles, we seem to identify the workings of Thúri the poet.

A few examples will serve to illustrate his translation technique.

Ἔπειτα ἦλθον εἰς τὰ κλίµατα τῆς Συρίας καὶ τῆς Κιλικίας• (Gal 1:21)

Then I went into the regions of Syria and Cilicia (NRSV) Thúri paraphrases the Greek adverb ἔπειτα -"then", using an idiomatic expression taken from the Hebrew Bible which means "after these things". 58 He also prefers here toponyms from the Hebrew Bible over a transcription of the Greek ones. He renders the Greek relative pronoun ὃ -"that which" by the Hebrew common noun ַח ִיּ י ם -"life". Thus, he forms an expression comprising a verb and a direct object derived from the same root, such as is very common in Hebrew: 59 "and the life that I now live in the flesh".

Often, Thúri uses in the translation pairs of words with similar sounds, thus creating an effect of assonance. In many cases, the Greek words that he renders in this way do not present a similar assonance. And in some cases, we can see that Thúri goes to great lengths to choose words that "sound nice" in Hebrew, since some of them are clearly not the most evident equivalents of the Greek words that they render. Several examples may show this: