J D Djida

G M Mophou

I Area

OPTIMAL CONTROL OF DIFFUSION EQUATION WITH FRACTIONAL TIME DERIVATIVE WITH NONLOCAL AND NONSINGULAR MITTAG-LEFFLER KERNEL

In this paper, we consider a diffusion equation with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. Existence and uniqueness of solution are proved by means of a spectral argument. The existence of solution is obtained for all values of the fractional parameter α ∈ (0, 1). Moreover, by applying control theory to the fractional diffusion problem we obtain an optimality system which has also a unique solution.

Introduction

In some problems related with transport dynamics in complex systems, the anomalous diffusion appear in a natural way [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]. As a consequence, some classical differential equations have been studied in the context of fractional derivatives, such as diffusion equation, diffusion-advection or Fokker-Plank type [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]. In [START_REF] Schneider | Fractional diffusion and wave equations[END_REF] diffusion and wave equations together with appropriate initial condition(s) have been rewritten as integro-differential equations with time derivatives replaced by convolution with t α-1 /Γ(α), α = 1, 2, respectively. These equations have been solved numerically by using different approaches (see e.g. [START_REF] Lei | A circulant preconditioner for fractional diffusion equations[END_REF][START_REF] Hao | A fourth-order approximation of fractional derivatives with its applications[END_REF][START_REF] Wang | A direct O(N log 2 N ) finite difference method for fractional diffusion equations[END_REF] and references therein). In [START_REF] Mophou | Optimal control of fractional diffusion equation[END_REF] the classical control theory has been applied to a fractional diffusion equation in a bounded domain, where the fractional time derivative was considered in a Riemann-Liouville sense.

We are interested in the analysis of the initial boundary value problem for a fractional time diffusion problem. Let Ω be an open and bounded subset of R d (d ≥ 1), with sufficiently smooth boundary ∂Ω. Given α ∈ (0, 1), a well behave function f ∈ L 2 (Q), and an initial datum y 0 ∈ L 2 (Ω), we seek y such that (1.1)    abc 0 D α t y(x, t) -(Ay)(x, t) = f, for (x, t) ∈ Q = Ω × (0, T ), y(x, 0) = y 0 (x), for x ∈ Ω, y(x, t) = 0, on ∂Ω × (0, T ),

where abc a D α t for α ∈ (0, 1) denotes the Atangana-Baleanu fractional derivative of order α in the sense of Caputo [START_REF] Atangana | New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model[END_REF] with respect to the time t, which is formally defined in section 2. In the cases of single and multiple fractional time Caputo derivatives, similar problems to (1.1) have been already studied. For the Caputo fractional derivative there exists a large and rapidly growing number of publications (see e.g. [START_REF] Bazhlekova | Fractional Evolution Equation in Banach Spaces[END_REF][START_REF] Diethelm | Numerical solution of linear multi-term initial value problems of fractional order[END_REF][START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF][START_REF] Mophou | Optimal control of fractional diffusion equation[END_REF][START_REF] Luchko | Maximum principle for the generalized time-fractional diffusion equation[END_REF][START_REF] Luchko | Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation[END_REF] and references therein). Moreover, in [START_REF] Mophou | Optimal control of fractional diffusion equation[END_REF] the Caputo fractional derivative was used to obtain an optimality system for the optimal control problem.

We would like to emphasize here that the existence and uniqueness of solutions for the fractional diffusion equation of type (1.1) when the Caputo fractional-time derivative is considered, have been obtained for α ∈ (1/2, 1) [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF][START_REF] Mophou | Optimal control of fractional diffusion equation[END_REF][START_REF] Baleanu | Low regret control for a fractional wave equation with incomplete data[END_REF]. In this work using the fractional-time derivative with the nonsingular Mittag-Laffler kernel, we have obtained the existence and uniqueness results for the fractional diffusion equation of type (1.1) for all the values α ∈ (0, 1). Furthermore, as for as the optimal control of the problem different authors (see e.g. [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF][START_REF] Mophou | Optimal control of fractional diffusion equation[END_REF][START_REF] Baleanu | Low regret control for a fractional wave equation with incomplete data[END_REF][START_REF]A general formulation and solution scheme for fractional optimal control problems[END_REF][START_REF]Formulation of Euler-Lagrange equations for fractional variational problems[END_REF][START_REF] Frederico | Fractional optimal control in the sense of Caputo and the fractional Noether's Theorem[END_REF] and references therein). We are interested in finding the control û ∈ L 2 (Q) that minimizes the cost function

J (v) = y(v) -z d 2 L 2 (Q) + N 2 v 2 L 2 (Q) , z d ∈ L 2 (Q), N > 0,
subject to the problem (1.1) where the the Atangana-Baleanu fractional-time derivative is involved.

In other words we are interested in finding the control parameter û ∈ L 2 (Q) such that the functional

J (û) = inf v∈L 2 (Q) J (v), holds true.
This manuscript is structured as follows: In section 2 we collect notation, definitions, and preliminary results regarding the solution representation and we compute the integration by parts involving the Atangana-Baleanu fractional-time derivative. Consequently the weak formulation of the (1.1) is obtained. Section 3, is dedicated to the sketch of proof of the existence and uniqueness of solution of the weak formulation problem. More precisely we prove Theorem 3.1 as well as some corollaries. Moreover, section 4 is devoted to an optimal control problem given by Theorem 4.2.

Preliminaries

In this section, we collect some notations on the functional spaces, useful properties on the Mittag-Leffler function and the fractional time derivatives, regularity results for the fractional diffusion (1.1), and basic estimates.

2.1. Notations. Let L 2 be the usual Hilbert space equipped with the scalar product (•, •), and H ℓ (Ω), H m 0 (Ω) denote the usual Sobolev spaces. In order to make things clear, we define the second order elliptic operator as (2.1)

Ay := -∆y, for x ∈ Ω, D(A) = H 2 (Ω) ∩ H 1 0 (Ω). Since A is an uniformly elliptic operator, the spectrum of A is composed of eigenvalues λ n ∞ n=1
and its corresponding orthogonal eigenfunctions ϕ n

∞ n=1 which belong to H 2 (Ω) ∩ H 1 0 (Ω), such that Aϕ n = λ n ϕ n .
2.2. Atangana-Baleanu fractional times derivatives: definitions and some properties. We first start by recalling the Mittag-Leffler function E α,β (z) which will be used extensively throughout this work and is defined below

E α,β (z) = ∞ k=0 z k Γ(kα + β) , z ∈ C,
where Γ(•) denotes the Gamma function defined as

Γ(z) = ˆ∞ 0 t z-1 e -t dt, ℜ(z) > 0.
The Mittag-Leffler function is a two-parameter family of entire functions of z of order α -1 [7, pp. 42]. The exponential function is a particular case of the Mittag-Leffler function, namely E 1,1 (z) = e z , [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF][START_REF] Samko | Fractional Integrals and Derivatives[END_REF]. As presented in [START_REF] Jin | Error Estimates for a Semi-discrete Finite Element Method for Fractional Order Parabolic Equations[END_REF], two important members of this family are E α,1 (-λt α ) and t α-1 E α,α (-λt α ), which occur in the solution operators for the initial value problem (1.1).

Next we also recall the generalized Mittag-Leffler function [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF][START_REF] Samko | Fractional Integrals and Derivatives[END_REF] defined as

(2.2) E ρ α,β (z) = ∞ k=0 (ρ) k z k Γ(αk + β)k! , ∀t ∈ C,
where (ρ) k is the Pochhammer symbol defined as (ρ

) k = Γ(ρ + k)/Γ(ρ) = ρ(ρ + 1) • • • (ρ + k -1).
Remark. Once could notice that for ρ = 1 we have that

E 1 α,β (z) = E α,β (z)
. Furthermore we recall the following Lemma from [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF].

Lemma 2.1. Let α, β, ρ ∈ C such that ℜ(α) > 0 and ℜ(β) > 0. Then we have that

(2.3) d dz k E ρ α,β (z) = (ρ) k E ρ+k α,αk+β (z), z ∈ C, k ∈ N,
and in the other hand,

(2.4) αρ E ρ+1 α,β (z) = (1 + αρ -β)E ρ α,β (z) + E ρ α,β-1 (z), z ∈ C.
Let us recall some useful definitions of fractional derivatives in the sense of Atangana-Baleanu [START_REF] Atangana | New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model[END_REF][START_REF] Djida | Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel[END_REF].

Definition 2.2. For a given function u ∈ H 1 (a, T ), T > a, the Atangana-Baleanu fractional derivative of u of order α in Caputo sense with base point a is defined at a point t ∈ (a, T ) by

(2.5) abc a D α t u(t) = B(α) 1 -α ˆt a u ′ (τ )E α [-γ(t -τ ) α ] dτ,
where γ = α/(1α), E α (•) stands for the Mittag-Leffler function, and

B(α) = (1 -α) + α/Γ(α).
So from the definition on [START_REF] Abdeljawad | Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel[END_REF] we recall the following definition Definition 2.3. For a given function u ∈ H 1 (a, T ), T > t > a, the Atangana-Baleanu fractional derivative of u of order α in Caputo sense with base point T is defined at a point t ∈ (a, T ) by

(2.6) abc T D α t g(t) = - B(α) 1 -α ˆT t g ′ (τ )E α [-γ(τ -t) α ] dτ, Definition 2.4. Let u ∈ H 1 (a, T ), T > a, α ∈ (0, 1
). The Atangana-Baleanu fractional derivative of order α of u in Riemann-Liouville sense with base point a is defined at a point t ∈ (a, T ) as

(2.7) abr a D α t u(t) = B(α) 1 -α d dt ˆt a u(τ )E α [-γ(t -τ ) α ] dτ.
For α = 1 in (2.5), we consider the usual classical derivative ∂ t .

Furthermore the Atangana-Baleanu fractional integral of order α with base point a is defined as (2.8)

I α t u(t) = 1 -α B(α) u(t) + α B(α)Γ(α) ˆt a u(τ )(t -τ ) α-1 dτ.
Notice that if α = 0 in (2.8) we recover the initial function, and if α = 1 in (2.8) we obtain the ordinary integral.

Next we recall the following lemma which has been stated and proved in [8, Theorem 1.6] Lemma 2.5. Let α ∈ (0, 1) and β ∈ R be arbitrary, and µ is such that απ 2 < µ < min(π, απ). Then there exists a real constant C = C(α, β, µ) > 0 such that

(2.9) |E α,β (z)| ≤ C 1 + |z| , µ ≤ |arg(z)| ≤ π.
Lemma 2.6. Let set η : [0, T ] → R, then for all α ∈ (0, 1) we the equivalence relation hold true

(2.10) abc T D α t η(T -t) = abc 0 D α t η(t).
Proof. The well-posedness of (2.10) follows from a change of variables. Let us set η : [0, T ] → R, and let us define by η(t) := η(Tt). Notice that

η ′ (t) = -η ′ (T -t).
Next by making change of variables t → Tt in (2.6), we get

abc 0 D α T -t η(T -t) = abc T D α t η(t) = - B(α) 1 -α ˆT T -t η ′ (τ )E α [-γ(τ -(T -t)) α ] dτ, setting ω = T -τ , we get that abc T D α t η(t) = B(α) 1 -α ˆT T -t η ′ (T -ω)E α [-γ(t -ω) α ] dω = B(α) 1 -α ˆt 0 η ′ (T -ω)E α [-γ(t -ω) α ] dω abc T D α t η(t) = - B(α) 1 -α ˆt 0 E α [-γ(t -ω) α ] η ′ (ω)dω, This means that abc T D α t η(t) = -abc 0 D α t η(t)
As a consequence, the backwards in time with the fractional-time derivative with nonsingular Mittag-Leffler kernel at the based point T is equivalently written as a forward in time operator with the fractional-time derivative with nonsingular Mittag-Leffler kernel -abc 0 D α t . 2.3. Solution representation. Under the functional spaces introduced in subsection 2.1, we can associate with u(x, t) a map u(•)

: (0, T ) → L 2 (Ω) by u(t)(x) = u(x, t), 0 < t < T, x ∈ Ω.
Let us now multiply (1.1) by a function ϑ ∈ H 1 0 (Ω) and by integration by parts on Ω, we get that

ˆΩ abc 0 D α t u(x, t)ϑ(x)dx -ˆΩ ∆u(x, t)ϑ(x)dx = ˆΩ f (x, t)ϑ(x)dx.
By Green's formula, we have

(2.11) ˆΩ abc 0 D α t u(x, t)ϑ(x)dx + ˆΩ ∇u(x, t)∇ϑ(x)dx = ˆΩ f (x, t)ϑ(x)dx. Now we set for all ϕ, ψ ∈ L 2 (Ω), (ϕ, ψ) L 2 (Ω) = ˆΩ ϕ(x)ψ(x)dx.
The scalar product in L 2 (Ω), with endowed norm ϕ L 2 (Ω) . In the same spirit, we define the bilinear form in H 1 0 (Ω) as (2.12) a(ϕ, ψ) = ˆΩ ∇ϕ(x)∇ψ(x)dx, ∀ϕ, ψ ∈ H 1 0 (Ω), so that we have ϕ 2

H 1 0 (Ω) = a(ϕ, ψ).
Under the assumption that the operator A is an uniformly elliptic operator, the spectrum of A is composed of eigenvalues {λ k } ∞ k=1 , and its corresponding orthogonal eigenfunctions {λ

w k } ∞ k=1 which belongs to D(A), such that a(w k , p) = λ k (w k , r) L 2 (Ω) , ∀r ∈ H 1 0 (Ω). Hence we have in this setting (2.13) ϕ 2 H 1 0 (Ω) = ∞ k=0 λ i (ϕ, w i ) 2 L 2 (Ω) , ∀ϕ ∈ H 1 0 (Ω).
As an immediate consequence, equation (2.13) can be written as follows

( abc 0 D α t u(x, t), ϑ) L 2 (Ω) + a(u(t), ϑ) L 2 (Ω) = (f (t), ϑ) L 2 (Ω)
. In this way, the problem (1.1) takes the representation as

(2.14)      abc 0 D α t u(x, t), ϑ L 2 (Ω) + a u(t), ϑ L 2 (Ω) = f (t), ϑ L 2 (Ω) in Ω, ∀ ϑ ∈ H 1 0 (Ω), u(x, 0) = u 0 for x ∈ Ω, u(t) = 0 on ∂Ω.
To discuss the existence of uniqueness results of (1.1) or (2.14), we shall need some informations over the various time. We are interested in the solution of the fractional differential equation involving the Atangana-Baleanu fractional derivative, in the form of (2.14). To this end, we propose to solve the fractional integro-differential equation involving the Atangana-Baleanu fractional derivative in the form (2.15)

   abc 0 D α t y i (t) + λ i y i (t) = f i (t), t ∈ (0, T ), y i (0) = y 0 i .

This type of fractional differential equation involving the fractional-time derivative with nonsingular

Mittag-Leffler kernel has been solved in [START_REF] Djida | Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel[END_REF]Proposition 3.3] by the means of Laplace transform. By following the same procedure, we state the the next proposition Proposition 2.7. Let f ∈ L 2 (Q), such that the Atangana-Baleanu fractional derivative exists. Then, the solution of differential equation (2.15), is given by

(2.16) y i (t) = ζ i E α -γ i t α y 0 i + (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α E α,α -γ i (t -s) α f i (s)ds,

with

(2.17)

γ i = αλ i B(α) + (1 -α)λ i , ζ i = B(α) B(α) + (1 -α)λ i
, and

K i = αζ i B(α)Γ(α) + (1 -α)γ i ζ i B(α) . 2.
+ ˆΩ ˆT 0 y(x, t) -abc T D α t φ(x, t) -∆φ(x, t) dt dx - B(α) 1 -α ˆΩ ˆT 0 y(x, 0)E α,α [-γt α ]φ(x, t)dt dx + B(α) 1 -α ˆΩ φ(x, T ) ˆT 0 y(x, t)E α,α [-γ(T -t) α ] dt dx.
Proof. For a given φ, y ∈ C ∞ ( Q), we have that

ˆΩ ˆT 0 abc 0 D α t y(x, t) -∆y(x, t) φ(x, t)dx dt = ˆΩ ˆT 0 abc 0 D α t y(x, t)φ(x, t)dx dt - ˆΩ ˆT 0 ∆y(x, t)φ(x, t)dx dt = (A) + (B)
Let us compute each part.

• computation of (A): (Integration by parts)

Using the definition of the Atangana-Baleanu fractional derivative (2.5) at the based point a = 0 we have the following

(2.19) ˆT 0 abc 0 D α t y(x, t)φ(x, t)dt = ˆT 0 B(α) 1 -α ˆt 0 y ′ (τ )E α [-γ(t -τ ) α ] dτ φ(t)dt.
We define by

A 1 := ˆt 0 y ′ (τ )E α [-γ(t -τ ) α ] dτ
Then by computing A 1 by integration by parts, we get that

A 1 = y(t) -y(0)E α [-αt α ] - ˆt 0 γ(t -τ ) α-1 E α,α [-γ(t -τ ) α ] y(τ )dτ.
Replacing A 1 by its value in (2. [START_REF] Luchko | Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation[END_REF]) we get that (2.20)

ˆT 0 abc 0 D α t y(x, t)φ(x, t)dt = B(α) 1 -α ˆT 0 y(x, t)φ(x, t)dt - B(α) 1 -α ˆT 0 y(x, 0)E α [-αt α ]φ(x, t)dt - B(α) 1 -α ˆT 0 ˆt 0 γ(t -τ ) α-1 E α,α [-γ(t -τ ) α ] y(x, τ )dτ φ(x, t)dt := A 2 + A 3 + A 4 ,
with

A 2 := B(α) 1 -α ˆT 0 y(x, t)φ(x, t)dt; A 3 := B(α) 1 -α ˆT 0 y(x, 0)E α [-αt α ]φ(x, t)dt A 4 := B(α) 1 -α ˆT 0 ˆt 0 γ(t -τ ) α-1 E α,α [-γ(t -τ ) α ] y(x, τ )dτ φ(x, t)dt.
Let us compute A 4

A 4 := - B(α) 1 -α ˆT 0 ˆt 0 γ(t -τ ) α-1 E α,α [-γ(t -τ ) α ] y(x, τ )dτ φ(x, t)dt = B(α) 1 -α ˆT 0 y(x, τ ) - ˆT τ γ(t -τ ) α-1 E α,α [-γ(t -τ ) α ] φ(x, t)dt dτ = B(α) 1 -α φ(x, T ) ˆT 0 y(x, t)E α,α [-γ(T -t) α ] dt - B(α) 1 -α ˆT 0 y(x, t)φ(x, t)dt + ˆT 0 y(x, t) - B(α) 1 -α ˆT t E α,α [-γ(t -τ ) α ] φ ′ (x, τ )dτ dt = B(α) 1 -α φ(x, T ) ˆT 0 y(x, t)E α,α [-γ(T -t) α ] dt - B(α) 1 -α ˆT 0 y(x,
+ B(α) 1 -α φ(x, T ) ˆT 0 y(x, t)E α,α [-γ(T -t) α ] dt - B(α) 1 -α y(x, 0) ˆT 0 E α [-αt α ]φ(x, t)dt • computation of (B)
By a simple integration by parts we have that

(2.22) (B) := - ˆΩ ˆT 0 φ(x, t)∆u(x, t)dx dt = ˆT 0 ˆ∂Ω u ∂φ ∂σ φdσ dt - ˆT 0 ˆ∂Ω φ ∂u ∂σ dσ dt - ˆΩ ˆT 0 u(x, t)∆φ(x, t)dt dx.
Hence adding (2.21) to (2.22), we obtain the desired result (2.18). Now using the conditions that φ(x, T ) = 0, we have the following Corollary.

Corollary 2.9. Let y be the solution of (1.1). Then, for any φ, y

∈ C ∞ ( Q) such that φ(x, T ) = 0 in Ω,
+ ˆΩ ˆT 0 y(x, t) -abc T D α t φ(x, t) -∆φ(x, t) dt dx - B(α) 1 -α ˆΩ ˆT 0 y(x, 0)E α,α [-γt α ]φ(x, t)dt dx.

Existence and uniqueness of solution

Let us consider the following problem, for α ∈ (0, 1). Assume f ∈ L 2 (Q) and y 0 ∈ L 2 (Ω). We want to find y ∈ L 2 (0, T ),

H 1 0 (Ω) , y(x, 0) ∈ C [0, T ]; H 1 0 (Ω) , such that ∀ ϑ ∈ H 1 0 (Ω), we have (3.1)    abc 0 D α t y(t), ϑ L 2 (Ω) + a y(t), ϑ L 2 (Ω) = f (t), ϑ L 2 (Ω) for all t ∈ (0, T ), y(x, 0) = y 0 for x ∈ Ω, Theorem 3.1. Let α ∈ (0, 1). Assume f ∈ L 2 (Q) , y 0 ∈ L 2 (Ω)
and the bilinear form a(•, •) defined as in (2.12). Then the problem (1.1) or (3.1) has a unique solution y ∈ L 2 (0, T );

H 1 0 (Ω) ∩ C [0, T ]; L 2 (Ω) given by y(x, t) = +∞ i=1 ζ i E α -γ i t α y 0 i + (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds w i ,
where the constants γ i and ζ i are defined as in (2.17). Moreover y satisfies the bounded conditions

(3.2) y L 2 (0,T );H 1 0 (Ω) ≤ Λ 1 y 0 H 1 0 (Ω) + f L 2 (Q) ,
where

Λ 1 = max (C 1 (α, λ 1 , T ), C 2 (α, λ 1 , T )), with (3.3) C 1 (α, λ 1 , T ) = CB(α) (1 -α) 6T λ 1 , C 2 (α, λ 1 , T ) = 6B 2 (α) (1 -α) 2 λ 1 + 12C 2 T 2 Γ 2 (α) Γ 2 (α) + 1 λ 1 ,
and

(3.4) y L 2 (Ω) ≤ Λ 2 y 0 L 2 (Ω) + f L 2 (Q) ,
where

Λ 2 = max (C 3 (α, λ 1 , T ), C 4 (α, λ 1 , T )), with C 3 (α, λ 1 , T ) = CB(α) λ 1 (1 -α) √ 6T , C 4 (α, λ 1 , T ) = 6 λ 2 1 + 12C 2 T 2 Γ 2 (α) Γ 2 (α) + 1 λ 2 1 ,
providing that y 0 ∈ L 2 (Ω).

Proof. We first assume that the series defines in (2.13) converges uniformly. Let ϑ = w i in (3.1) then we have that

(3.5)    abc 0 D α t y i (t) + λ i y i (t) = f i (t), t ∈ (0, T ), y i (0) = y 0 i , where we have used a(y(t), w i ) = λ i (y(t), w i ) L 2 (Ω) = λ i y i .
We then recognize the system given by (3.5) which has been solved in subsection 2.3. So the solution of the fractional differential equation takes the form

(3.6) y i (t) = ζ i E α -γ i t α y 0 i + (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds.
Notice that the λ i are contained in ζ i , γ i and K i previously defined in subsection 2.3.

Now the next step is to look at the uniqueness of the solution. We will make it in two steps. We shall start by showing that the solution belongs to the space L 2 ((0, T ); H 1 0 (Ω)) and in the second step, we will show that the solution also belong to C([0, T ]; L 2 (Ω)), so that y belongs to L 2 ((0, T );

H 1 0 (Ω)) ∩ C([0, T ]; L 2 (Ω)).
Let defined by V m a subspace of H 1 0 (Ω) generated by w 1 , w 2 , . . . , w m . We want to find y m : t ∈ (0, T ] → y m (t) ∈ V m solution of fractional differential equation

(3.7)    abc 0 D α t (y m (t), ϑ) L 2 (Ω) + a y m (t), ϑ L 2 (Ω) = f (t), ϑ L 2 (Ω) for all ϑ ∈ V m , y m (x, 0) = y 0 m for x ∈ Ω.
Since y m (t) ∈ V m , we have

y m (t) = m i=1 y(t), w i L 2 (Ω) w i = m i=1 y i (t)w i ,
which could still be written in an explicit form as (3.8)

y m (x, t) = m i=1 ζ i E α -γ i t α y 0 i + (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds w i .
The next step is to show y m (t) is a Cauchy sequence in the space L 2 (0, T ); H 1 0 (Ω) . Let m and r be two natural numbers such that r > m. Then

y r (t) -y m (t) = r i=m+1 y i (t)w i .
But we also have that (3.9)

a y r (t) -y m (t), y r (t) -y m (t) = r i=m+1 λ i y i (t) 2 = r i=m+1 λ i ζ i E α -γ i t α y 0 i + (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds 2 ≤ 3 r i=m+1 λ i ζ i E α -γ i t α y 0 i 2 + 3 r i=m+1 λ i (1 -α)ζ i B(α) f i (t) 2 + 3 r i=m+1 λ i K 2 i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds 2 .
Hence

ˆT 0 y r (t) -y m (t) 2 L 2 (0,T );H 1 0 (Ω) dt = ˆT 0 a y r (t) -y m (t), y r (t) -y m (t) dt ≤ 3(Z 1 + Z 2 + Z 3 ),
where Z 1 , Z 2 and Z 3 are defined respectively as

Z 1 := r i=m+1 λ i ˆT 0 ζ i E α -γ i t α y 0 i 2 dt, Z 2 := r i=m+1 λ i ˆT 0 (1 -α)ζ i B(α) f i (t) 2 dt, Z 3 := r i=m+1 λ i K 2 i ˆT 0 ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds 2 dt.
Now the goal is to estimate each term Z 1 , Z 2 and Z 3 .

• Estimate of Z 1 . Using the estimate of the Mittag-Leffler function provided in Lemma 2.5 we have that (3.10)

Z 1 = r i=1+m λ i ˆT 0 ζ i E α -γ i t α u 0 i 2 dt ≤ 2C 2 r i=m+1 λ i ζ 2 i |u 0 i | 2 ˆT 0 1 1 + γ i t α 2 dt ≤ 2C 2 T r i=m+1 λ i ζ 2 i |u 0 i | 2
Now we find the estimate λ i ζ 2 i as

λ i ζ 2 i = λ i B 2 (α) B(α) + (1 -α)λ i 2 ≤ λ i B 2 (α) (1 -α) 2 λ 2 i ≤ B 2 (α) (1 -α) 2 λ i ≤ B 2 (α) (1 -α) 2 λ 1 ,
since λ i are increasing. We finally get the estimate of Z 1 as

Z 1 ≤ 2C 2 B 2 (α) λ 1 (1 -α) 2 T r i=m+1 |u 0 i | 2 .
• Estimate of Z 2 .

(3.11)

Z 2 = 2 r i=m+1 λ i ˆT 0 (1 -α)ζ i B(α) f i (t) 2 = 2(1 -α) 2 B(α) 2 r i=m+1 λ i ζ 2 i ˆT 0 f i (t) 2 dt ≤ 2B 2 (α) (1 -α) 2 λ 1 r i=m+1 ˆT 0 f i (t)
2 dt .

• Estimate of Z 3 . By applying the Cauchy-Schwarz inequality and by setting δ = ts we have that

Z 3 = 2 r i=m+1 λ i K 2 i ˆT 0 ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α-1 f i (s)ds 2 dt ≤ 2 r i=m+1 λ i K 2 i ˆT 0 ˆt 0 δ α-1 E α,α -γ i δ α-1 2 dδ ˆt 0 f i (s) 2 ds dt
Now from Lemma 2.1 we have that

d dδ E α,1 -γ i δ α = d dδ E 1 α,1 -γ i δ α = E 2 α,α+1 -γ i δ α = -γ i δ α-1 E α,α -γ i δ α = 1 α E α,α -γ i δ α dδ.
Furthermore the estimate of λ i K 2 i is giving by

λ i K 2 i = λ i αζ i B(α)Γ(α) + (1 -α)γ i ζ i B(α) 2 ≤ 2 α 2 B 2 (α)Γ 2 (α) λ i ζ 2 i + (1 -α) 2 B 2 (α) γ 2 i λ i ζ 2 i ≤ 2α 2 λ 1 (1 -α) 2 1 + 1 Γ 2 (α)
Then we get that (3.12)

Z 3 ≤ 4α 2 λ 1 (1 -α) 2 1 + 1 Γ 2 (α) r i=m+1 1 γ 2 i ˆT 0 ˆt 0 E α,α -γ i δ α-1 2 dδ ˆt 0 f i (s) 2 ds dt ≤ 4C 2 λ 1 1 + 1 Γ 2 (α) T r i=m+1 ˆT 0 ˆT 0 f i (t) 2 dt dt ≤ 4C 2 T 2 Γ 2 (α) Γ 2 (α) + 1 λ 1 r i=m+1 ˆT 0 f i (t) 2 dt
Finally, adding together the estimates Z 1 , Z 2 , and Z 3 we get

ˆT 0 y r (x, t) -y m (x, t) 2 L 2 (0,T );H 1 0 (Ω) dt ≤ 6C 2 B 2 (α) λ 1 (1 -α) 2 T r i=m+1 |y 0 i | 2 + 6B 2 (α) (1 -α) 2 λ 1 r i=m+1 ˆT 0 f i (t) 2 dt + 12C 2 T 2 Γ 2 (α) Γ 2 (α) + 1 λ 1 r i=m+1 ˆT 0 f i (t) 2 dt
Thus we obtain

ˆT 0 y r -y m H 1 0 (Ω) ≤ C 1 (α, λ 1 , T )   r i=m+1 |y 0 i | 2   1/2 + C 2 (α, λ 1 , T )   r i=m+1 ˆT 0 f i (t) 2 dt   1/2 , with C 1 (α, λ 1 , T ) = CB(α) (1 -α) 6T λ 1 , C 2 (α, λ 1 , T ) = 6B 2 (α) (1 -α) 2 λ 1 + 12C 2 T 2 Γ 2 (α) Γ 2 (α) + 1 λ 1 .
As in the previous case, the norm in L 2 (Ω) is computed as

(3.13) ˆT 0 y r -y m 2 L 2 (Ω) dt = ˆT 0 r i=m+1 y i (t) 2 dt = ˆT 0 r i=m+1 ζ i E α -γ i t α y 0 i + (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds 2 dt ≤ 3 ˆT 0 r i=m+1 ζ i E α -γ i t α y 0 i 2 dt + 3 ˆT 0 r i=m+1 (1 -α)ζ i B(α) f i (t) 2 dt + 3 ˆT 0 r i=m+1 K 2 i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds 2 dt.
From the previous estimations in the first step of the proof, we have that

(3.14) ˆT 0 y r -y m 2 L 2 (Ω) dt = 6C 2 B 2 (α) (1 -α) 2 λ 2 1 T r i=m+1 y 0 i 2 + 6 λ 2 1 r i=m+1 ˆT 0 f i (t) 2 + 12C 2 T 2 Γ 2 (α) Γ 2 (α) + 1 λ 2 1 r i=m+1 ˆT 0 f i (t) 2 dt
Thus we obtain

sup t∈[0,T ] y r -y m L 2 (Ω) ≤ C 3 (α, λ 1 , T ) r i=m+1 |y 0 i | 2 1/2 + C 4 (α, λ 1 , T )   r i=m+1 ˆT 0 f i (t) 2 dt   1/2 , with C 3 (α, λ 1 , T ) = CB(α) λ 1 (1 -α) √ 6T , C 4 (α, λ 1 , T ) = 6 λ 2 1 + 12C 2 T 2 Γ 2 (α) Γ 2 (α) + 1 λ 2 1 .
From the fact that y 0 ∈ H 1 0 (Ω) and f ∈ L 2 (Q), we have that lim

m,r→+∞   r i=m+1 |y 0 i | 2   1/2 = 0, lim m,r→+∞   r i=m+1 ˆT 0 f i (t) 2 dt   1/2 = 0.
Hence we have that lim m,r→+∞ ˆT 0 y ry m 2 L 2 ((0,T );H 1 0 (Ω)) = 0, and

lim m,r→+∞ ˆT 0 y r -y m 2 L 2 (Ω) = 0, which means that the sequence (y m ) is Cauchy sequences in L 2 (0, T ); H 1 0 (Ω) and C [0, T ]; L 2 (Ω) . Therefore we have that y m ⇀ y in L 2 (0, T ); H 1 0 (Ω) , y m ⇀ y in C [0, T ]; L 2 (Ω) .
Now proceeding as in [START_REF] Baleanu | Low regret control for a fractional wave equation with incomplete data[END_REF], one can prove by interpretation that the solution y is equivalent to the initial problem (1.1). Furthermore, setting Λ 1 = max (C 1 (α, λ 1 , T ), C 2 (α, λ 1 , T )) and Λ 2 = max (C 3 (α, λ 1 , T ), C 4 (α, λ 1 , T )), an using the estimates obtained at the beginning of the proof, we have that

y L 2 ((0,T );H 1 0 (Ω)) = ˆT 0 y H 1 0 (Ω) dt 1/2 ≤ C 1 (α, λ 1 , T ) y 0 L 2 (Ω) + C 2 (α, λ 1 , T ) f L 2 (Q) ≤ Λ 1 y 0 H 1 0 (Ω) + f L 2 (Q) , and sup t∈[0,T ] y L 2 (Ω) ≤ C 3 (α, λ 1 , T ) y L 2 (Ω) + C 4 (α, λ 1 , T ) f L 2 (Q) ≤ Λ 2 y 0 L 2 (Ω) + f L 2 (Q)
Hence we get the desired result, which ends the proof.

Remark. We would like to draw the attention of the reader that in the case for the Caputo fractionaltime derivative, the existence and uniqueness of solutions for the fractional diffusion equation of type (1.1), were obtained under the conditions that α ∈ (1/2, 1) [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF][START_REF] Mophou | Optimal control of fractional diffusion equation[END_REF][START_REF] Baleanu | Low regret control for a fractional wave equation with incomplete data[END_REF]. But here in our case using the Atangana-Baleanu fractional-time derivative, we got the existence and uniqueness of solutions for the fractional diffusion equation of type (1.1) for all α ∈ (0, 1).

Next we state two corollaries which give the estimates of the solution y(t) in L 2 (0, T ), H 2 (Ω) ∩ H 1 0 (Ω) for the case f = 0 and in the case where the initial data y 0 = 0 respectively in the problem (1.1) or (3.1).

• First case: f = 0. For f = 0 in the problem (1.1) or (3.1) we have the following Corollary. Corollary 3.2. Let α ∈ (0, 1). Assume f = 0, y 0 ∈ L 2 (Ω) and the bilinear form a(•, •) defined as in (2.12). Then the problem (1.1) or (3.1) has a unique solution y ∈ L 2 (0, T ),

H 2 (Ω) ∩ H 1 0 (Ω) given by y(x, t) = +∞ i=1 ζ i E α -γ i t α y 0 i w i ,
where the constant ζ i is defined as in (2.17). Moreover y satisfies the bounded conditions

(3.15) ˆT 0 y 2 H 2 (Ω) dt = y 2 L 2 ((0,T ),H 2 (Ω)) ≤ Λ 3 y 0 H 2 0 (Ω) ,
where

Λ 3 = CB(α) (1-α) √ T .
Proof. Following the same procedure as in the first step for the proof of the Theorem 3.1, but now in H 2 0 (Ω), the solution of the problem (1.1) or (3.1) for f = 0 is given by

y(t) = +∞ i=1 ζ i E α -γ i t α y 0 i w i ,
where the constant ζ i is defined as in (2.17). Furthermore the estimate of y can be obtained as follows (3.16)

ˆT 0 y 2 H 2 (Ω) dt = ˆT 0 r i=m+1 λ 2 i y i (t) 2 dt = ˆT 0 r i=m+1 λ 2 i ζ i E α -γ i t α y 0 i 2 dt ≤ r i=m+1 λ 2 i ζ 2 i ˆT 0 E 2 α -γ i t α 2 y 0 i 2 dt ≤ C 2 B 2 (α) (1 -α) 2 T r i=m+1 y 0 i 2 ≤ C 5 (α, T ) r i=m+1 y 0 i 2 .
• Second case: y 0 i = 0. For y 0 i = 0 in the problem (1.1) or (3.1) we have the following Corollary. Corollary 3.3. Let α ∈ (0, 1). Assume y 0 i = 0, and f ∈ L 2 (Q). Then the problem (1.1) or (3.1) has a unique solution y ∈ L 2 (0, T ), H 2 (Ω) ∩ H 1 0 (Ω) given by

y(x, t) = +∞ i=1 (1 -α)ζ i B(α) f i (x, t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds w i ,
where the constants γ i , ζ i and K i are defined as in (2.17). Moreover y satisfies the bounded conditions

(3.17) y L 2 ((0,T ),H 2 (Ω)) ≤ T Λ 4 f L 2 (Q) ,
where

Λ 4 = 2 + 4C 2 T 2 1 + 1 Γ 2 (α) ,
Proof. Let α ∈ (0, 1). Assume y 0 i = 0, and f ∈ L 2 (Q). Then the problem (1.1) or (3.1) has a unique solution y ∈ H 2 (Ω) given by y(x, t)

= +∞ i=1 (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds w i ,
where the constants γ i , ζ i and K i are defined as in (2.17). Furthermore we compute the estimate of y in the space H 2 (Ω) as

(3.18) ˆT 0 y 2 H 2 (Ω) dt = ˆT 0 r i=m+1 λ 2 i (1 -α)ζ i B(α) f i (t) + K i ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α f i (s)ds 2 dt ≤ 2 r i=m+1 λ 2 i ˆT 0 (1 -α)ζ i B(α) f i (t) 2 + 2 r i=m+1 λ 2 i K 2 i ˆT 0 ˆt 0 (t -s) α-1 E α,α -γ i (t -s) α-1 2 ds ˆt 0 f i (s) 2 ds dt ≤ 2 r i=m+1 ˆT 0 f i (t) 2 + 4C 2 T 1 + 1 Γ 2 (α) r i=m+1 ˆT 0 f i (t) 2 dt ≤ C 6 r i=m+1 ˆT 0 f i (t) 2 dt
Now, we consider the following fractional differential equation:

(3.19)    -abc 0 D α t η(t) -Aη(t) = g(t), t ∈ (0, T ), η(T ) = 0,
where α ∈ (0, 1), g ∈ L 2 (Q). We state the following Lemma.

Lemma 3.4. Let α ∈ (0, 1), g ∈ L 2 (Q). Then problem (3.19) has a unique solution η ∈ L 2 (Q) given by

(3.20) η i (x, t) = +∞ i=1 (1 -α)ζ i B(α) g(t) + K i ˆt 0 (t -s) α E α,α -γ i (t -s) α g(s)ds w i .
with the same constants as defined in (2.17). Moreover

(3.21) η i L 2 (0,T );H 1 0 (Ω) ≤ C 2 g L 2 (Q) .
Proof. Using Lemma 2.6, we found that the backwards in time with the Atangana-Baleanu fractionaltime derivative at the based point T is equivalently written as a forward in time operator with the Atangana-Baleanu fractional-time derivative at a based point zero time a negative sign. More precisely, abc T D α t η(t) = -abc 0 D α t η(t). Using this relation and by making the change of variable t → Tt in (3.19), we obtain the fractional differential equation

   abc 0 D α t η(t) -A η(t) = g(t), T -t ∈ [0, T ], η(0) = 0,
with g(t) = g(Tt). This fractional differential equation can still be written in the short form as

(3.22)    abc 0 D α t η(τ ) -Aη(τ ) = g(τ ), τ ∈ [0, T ], η(0) = 0,
Next to show that (3.20) holds, we just apply the Laplace transform on (3.22) and get the inverse following the steps as in [START_REF] Djida | Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel[END_REF]Proposition 3.3], and by taking into account the fact that η(0) = 0. Hence one get

η(t) = (1 -α)ζ i B(α) g(t) + K i ˆt 0 (t -s) α E α,α -γ i (t -s) α g(s)ds,
where ζ i , γ i and C 2 were defined respectively in (2.17) and in (3.3) Finally proceeding as for the proof of (3.2) in Theorem 3.1 once can obtain

η i L 2 (0,T );H 1 0 (Ω) ≤ C 2 g L 2 (Q) .

Optimal control problem

In this section, we want to approach the state y(v) of the problem (1.1) by a desired state z d in controlling v.

Let v ∈ L 2 (Q).
As it has been shown in the existence and uniqueness part, the problem (1.1) have a unique solution y = y(v) which belongs to L 2 (Q). In this way, we can define can define the functional as (4.1)

J (v) = y(v) -z d 2 L 2 (Q) + N 2 v 2 L 2 (Q) ,
where z d ∈ L 2 (Q) and N > 0. We are interested on finding the control parameter, say, û ∈ L 2 (Q) such that the functional (4.2)

J (û) = inf v∈L 2 (Q) J (v).
Now we state the following Proposition.

Proposition 4.1. Let consider the problem given by (1.1). Then there exists a unique optimal control u such that (4.2) holds true.

Proof. The functional J is continuous, coercive and convex. Therefore, there exist a control called û. By calling ŷ = ŷ(û), the associated state of û then its come that, ŷ satisfies

(4.3)    abc 0 D α t ŷ -∆ŷ = û for (x, t) ∈ Q, ŷ(x, 0) = ŷ0 (x) for x ∈ Ω, ŷ(x, t) = 0 on ∂Ω × (0, T ),
where y ∈ L 2 ((0, T );

H 1 0 (Ω)) ∩ C([0, T ]; L 2 (Ω)).
Remark. Another way of proving this result is to use minimizing sequences. The reader interested could follows the idea proposed by the second author in [START_REF] Mophou | Optimal control of fractional diffusion equation[END_REF].

In the following we are concern with the interpretation of the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of the backward Atangana-Baleanu fractionaltime derivative. As a result, we obtained an optimality system for the optimal control given by the following Theorem.

Theorem 4.2. If û is a solution of (4.2), then there exist η such that

(û, ŷ, η) ∈ L 2 (Q) × L 2 (0, T ); H 1 0 (Ω) × L 2 (0, T ); H 2 (Ω) ∩ H 1 0 (Ω) , satisfies the following optimality system (4.4)    -abc 0 D α t ŷ -∆ŷ = û in Q, ŷ(x, 0) = ŷ0 in Ω, ŷ(x, t) = 0 on ∂Ω × (0, T ). (4.5)    -abc T D α t η -∆η = ŷ -z d in Q, η(T ) = 0 in Ω, η = 0 on ∂Ω × (0, T ). (4.6) û = - η N in Q.
Proof. In order to prove the theorem, we start by expressing the Euler-Lagrange optimality conditions which characterize the optimal control û. We means The state z (vû) ∈ L 2 (0, T ); H 2 (Ω) ∩ H 1 0 (Ω) associated to the control (vû) ∈ L 2 (Q) is a solution of (4.8)

   abc 0 D α t z -∆z = v -û in Q, z (0, v -û) = 0
inΩ, z = 0 on ∂Ω × (0, T ).

Now we find the following behaviour of our functional when µ converges to 0.

(4.9) lim µ→0 J (û + µ(vû)) -J (û) µ = 0, for all v ∈ L 2 (Q).

Using the functional defined in (4.1) we have that In order to make the interpretation (4.10), we consider the adjoint state equation But since ŷz d ∈ L 2 ((0, T ), H 1 0 (Ω)), applying the Lemma 3.4, we deduce that the problem given by (4.11) has a unique solution in L 2 (0, T ); H 2 (Ω) ∩ H 1 0 (Ω) . In this scenario, by multiplying the state equation system (4.8) by the solution η of (4.11), we obtain by 2.9 that We then get finally that û = -η N in Q.

J (û + -û)) = N 2 (û + µ(v -û)(û + µ(v -û) + 1 2 (ŷ(û) -z d + µŷ(v -û)) (ŷ(û) -z d + µŷ(v -û)) = N 2 û 2 L 2 (Q) + µN û(v -û) + µ 2 N 2 v -û 2 L 2 (Q)

Conclusion and remarks

In this work the existence and uniqueness of solution in L 2 (Q) were proved by means of a spectral argument and the control problem has also been studied. We have shown in the first step that the existence of solution were obtained for all values of the fractional parameter α ∈ (0, 1), contrary to the Caputo and Riemann fractional-time derivative where the existence and uniqueness results were obtained for α ∈ (1/2, 1) with y 0 ≡ 0. This reveals one particularity of the fractional-time derivative with the nonsingular Mittag-Leffler function. Moreover, we have also shown that one can approach the state y(v) of (1.1) by a desired state z d by controlling v and compute the control û using the algorithm given by the optimality system and following the method of Lagrange. Let us also mention that by proceeding as in this work, one can also study the same problem by considering the Atangana-Baleanu fractional-time derivative in the Riemann-Liouville sense.

  (1.1), one can refer to the methods of the Lagrange multiplier technique for the Caputo and Riemann fractional time derivative presented by The first author grateful for the facilities provided by the German research Chairs and the Teacher Training Program of AIMS-Cameroon. The first author is also indebted to the AIMS-Cameroon 2017-2018 Tutor fellowship. The second author was supported by the Alexander von Humboldt foundation, under the programme financed by the BMBF entitled "German research Chairs". The work of the third author has been partially supported by the Agencia Estatal de Innovación (AEI) of Spain under grant MTM2016-75140-P, cofinanced by the European Community fund FEDER, and Xunta de Galicia, grants GRC 2015-004 and R 2016/022.

  + µ(vû)) | µ=0 , for all φ ∈ L 2 (Q).

J

  )z d ) 2 L 2 (Q) + µ(ŷ(û)z d ) (ŷ(v -(û + µ(vû)) -J (û) µ = z (ŷ(û)z d ) + N û (vû) = 0, for all v ∈ L 2 (Q).Now integrating over Q we get that (4.10)ˆT 0 ˆΩ z (ŷ(û)z d ) dx dt + N ˆT 0 ˆΩ û (vû) dx dt = 0 for all v ∈ L 2 (Q).

  ∂Ω × (0, T ).

  t z -∆z ηdx dt = ˆT 0 ˆΩ abc T D α t η -∆η zdx dt = ˆT 0 ˆΩ (ŷ(u)z d ) zdx dt.Hence one get the following relation from (4.8) and (4.10), we deduce thatˆT 0 ˆΩ (vû) η dx dt = -N ˆT 0 ˆΩ (vû) û dx dt for all v ∈ L 2 (Q).


J Drader

G Saint-Louis

J M Muller

M-C Charbonnel

P Guilbaud

L Berthon

K M Roscioli-Johnson

C A Zarzana

C Rae

G S Groenewold

B J Mincher

S P Mezyk

K Mccann

S G Boyes

J Braley

Radiation chemistry of the branched-chain monoamide di-ethylhexylisobutyramide

The radiolytic degradation of di-ethylhexyl-isobutyramide (DEHiBA) was examined by subjecting the compound to gamma irradiation, measuring the remaining concentration of the intact compound, identifying the degradation products and measuring uranium distribution ratios.

The combined effects of radiation dose, contact with aqueous solutions of HNO 3 , and aeration were also examined. The DEHiBA displayed significant stability at doses up to 1000 kGy, undergoing a slow exponential concentration decrease that was accompanied by the appearance of multiple degradation products. The most abundant compounds that were formed by radiolysis resulted from cleavage of the C carbonyl -N and C ethylhexyl -N bonds, generating di-ethylhexylamine, and mono-ethylhexyl-isobutyramide. Acid contact did alter the radiolytic pathways, with acid favoring cleavage of the C carbonyl -N bond, while a more diverse array of compounds were formed in the absence of acid. Pulsed radiolysis experiments were also conducted, in which picosecond bursts of energetic electrons were used to irradiate solutions of dodecane containing DEHiBA; formation of the dodecane radical cation was implicated, which serially reacted with DEHiBA to form radical or radical cation species intermediate in the formation of the observed products. The slow degradation kinetics suggests that DEHiBA possesses good potential for selective extraction of uranium in fuel cycle extraction operations.

INTRODUCTION

The dialkyl-monoamide class of extactant molecules has been studied for several decades as alternatives to tributylphosphate (TBP) in PUREX-type separations for the nuclear fuel cycle.

Their advantages over TBP, as summarized in a review by Gasparini and Grossi, [START_REF] Gasparini | Long-chain disubstituted aliphatic amides as extracting agents in industrial applications of solvent-extraction[END_REF] include greater selectivity for uranium over fission products, higher radiolytic stability, generation of inoffensive radiolysis products, and the elimination of phosphorous. This latter point is referred to as adherence to the CHON principle, in which reagents are composed of only carbon, hydrogen, oxygen and nitrogen, making them incinerable and thus preferable to phosphorouscontaining ligands for used nuclear fuel treatment purposes.

An additional advantage of the dialkyl-monoamides over TBP is the ease of their synthetic tunability. This feature allows for metal ion selectivity to be designed into the structure of the monoamide. For example, although n-alkane substituted monoamides have overall higher extraction efficiencies for U(VI), branched-chain alkane-substituted monoamides are used for selection of U(VI) over Th(IV) and Pu(IV). Branching at the α-carbon of the alkane attached to the carbonyl carbon was shown to increase the separation factor for uranium from the tetravalent ions including zirconium, [START_REF] Siddall | Effects of structure of N,N-disubstituted amides on their extraction of actinide and zirconium nitrates and of nitric acid[END_REF] and to minimize third phase formation due to an overall lower concentration of metal in the organic phase. [START_REF] Pathak | Structural effects in N,N-dialkyl amides on their extraction behavior toward uranium and thorium[END_REF] Prabhu et al. reported that di-2-ethylhexylbutyramide (DEHBA, Figure 1) extracts U(VI) and Pu(IV), while di-2-ethylhexyl-isobutyramide (DEHiBA) selectively extracts U(VI), although with a lower D U than for the unbranched monoamide. [START_REF] Prabhu | Di(2-ethyl hexyl)butyramide and di(2-ethyl hexyl)isobutyramide as extractants for uranium(VI) and plutonium(IV)[END_REF] Thus, for U/Pu co-extraction, DEHBA is preferred, while for applications involving selective extraction of U, DEHiBA is a better extractant. In the Gasparini and Grossi review, [START_REF] Gasparini | Long-chain disubstituted aliphatic amides as extracting agents in industrial applications of solvent-extraction[END_REF] the stability of monoamides toward γ-radiolysis was reported to be relatively unaffected by the nature of the alkane groups, though Mowafy [START_REF] Mowafy | The effect of previous gamma-irradiation on the extraction of U(VI), Th(IV), Zr(IV), Eu(III) and Am(III) by various amides[END_REF] reported that unsymmetrically substituted monoamides were more susceptible to radiolytic damage.

Although it has been commonly claimed that the monoamides have good radiation stability, there are few literature reports quantitatively investigating this. They involve mainly postirradiation studies of effects on solvent extraction and attempts to identify degradation products by spectroscopic or GC methods. [START_REF] Mowafy | The effect of previous gamma-irradiation on the extraction of U(VI), Th(IV), Zr(IV), Eu(III) and Am(III) by various amides[END_REF][START_REF] Ruikar | Extraction behavior of uranium, plutonium and some fission-products with gamma-irradiated n,n'-dialkylamides[END_REF][START_REF] Ruikar | Extraction of uranium, plutonium and some fission-products with gamma-irradiated unsymmetrical and branched-chain dialkylamides[END_REF][START_REF] Parikh | Radiolytic Degradation Studies on N,N-dihexyloctanamide (DHOA) under PUREX Process Conditions[END_REF][START_REF] Pathak | Actinides 2009, IOP Conference Series-Materials Science and Engineering[END_REF][START_REF] Ruikar | Extraction behavior of uranium(VI), plutonium(IV) and some fission-products with gamma-preirradiated n-dodecane solutions of N,N'-dihexyl substituted amides[END_REF][START_REF] Pathak | Evaluation of Di(2éthylhexyl)isobutyramide (D2EHiBA) as a process extractant for the recovery of 233 U from irradiated Th[END_REF][START_REF] Manchanda | Amides and diamides as promising extractants in the back end of the nuclear fuel cycle: an overview[END_REF][START_REF] Berthon | Radiolysis of Solvents Used in Nuclear Fuel Reprocessing[END_REF] The major products are invariably reported to be the corresponding amine and organic acid, suggesting rupture of the carbonyl C-N bond. For example, Musikas [START_REF] Musikas | Solvent-extraction for the chemical separations of the 5f elements[END_REF][START_REF] Musikas | Potentiality of nonorganophosphorus extractants in chemical separations of actinides[END_REF] reported that the products of di-2-ethylhexyl-hexanamide radiolysis were di-2-ethylhexyl-amine and hexanoic acid. However, the secondary amine was produced in substoichiometric amounts, suggesting that it hydrolyzed in the presence of acid. Similarly, Parikh et al. [START_REF] Parikh | Radiolytic Degradation Studies on N,N-dihexyloctanamide (DHOA) under PUREX Process Conditions[END_REF] identified the main products of di-hexyl-octanamide radiolysis using GC/GC-MS as dihexyl-amine and octanoic acid. Uranium and plutonium extraction and subsequent stripping were little affected by an absorbed dose of 600 kGy. This was attributed to the innocuous, watersoluble nature of the monoamide degradation products which is in stark contrast to the degradation products of tributylphosphate (TBP) [START_REF] Berthon | Radiolysis of Solvents Used in Nuclear Fuel Reprocessing[END_REF][START_REF] Mincher | Review Article: The Effects of Radiation Chemistry on Solvent Extraction: 1. Conditions in Acidic Solution and a Review of TBP Radiolysis[END_REF] , highlighting one of their advantages over the conventional extractant. Similarly, di-2-ethylhexyl-amine and isobutanoic acid were reported by Pathak et al. as the products of DEHiBA irradiation in the presence of 3-4 M HNO 3 , and the ability to strip the loaded solvent of uranium and plutonium was maintained even at 600 kGy absorbed dose. [START_REF] Pathak | Actinides 2009, IOP Conference Series-Materials Science and Engineering[END_REF] As the monoamides continue to be developed for potential application to the nuclear fuel cycle, research has been focused on the behavior of DEHBA and DEHiBA, (Figure 1). The goal of this work is to provide a rigorous examination of the radiolytic decomposition of the uranium selective DEHiBA by identifying its degradation dose constants and products using mass spectrometry and measuring the impact on the extraction behavior of uranium(VI) by the irradiated solvent.

EXPERIMENTAL

Materials

The DEHiBA used in the preliminary work was prepared at Colorado School of Mines (USA) using a 1.2:1 molar ratio of iso-butyryl chloride diluted in chloroform, added dropwise to a mixture of the secondary amine, bis-2-ethylhexyl-amine, diluted in chloroform and tri-ethylamine. The dropwise addition was done in a nitrogen atmosphere at 5 °C under vigorous stirring.

Once mixed, the solution was refluxed for 2 hours at 60 °C. This solution was then filtered and washed with water, 10 wt % sodium carbonate, and 1 M HCl to remove unreacted tri-ethyl-amine as well as residual iso-butyryl chloride. The solution was then dried with sodium sulfate overnight. To purify the product, chloroform was removed by rotary evaporation followed by vacuum distillation. Characterization by 1 H-NMR showed the product to be 99%+ pure. Upon the need for larger volumes for irradiations and solvent extraction, DEHiBA was purchased from Technocom (UK).

Steady State Irradiations

A 1 M solution of the monoamide was prepared by combining the appropriate amount of weighed DEHiBA in dodecane for irradiation. Samples were irradiated using a 60 Co Nordion Gammacell 220, at a dose rate that varied from 4.73-4.34 kGy•h -1 during the course of the experiments. This dose rate was initially measured using standard Fricke procedures [START_REF] Fricke | The chemical action of Roentgen rays on dilute ferrosulphate solutions as a measure of dose[END_REF] , and then subsequently corrected for 60 Co decay. The sample chamber temperature was approximately 30 o C. The solutions were irradiated either as dry organic solutions, or in contact with either 0.1 M HNO 3 (stripping acidity), or 4 M HNO 3 (extraction acidity). In some cases, the samples were irradiated in static, sealed containers, considered deaerated upon exposure to even low absorbed doses. Some samples, as indicated in the text, were irradiated with air sparging at a rate of 1 mL•min -1 , as delivered from a compressed air bottle through a mass flow controller. This was done to ensure that the samples so treated remained saturated with dissolved oxygen. Following irradiation, the samples were split for analysis by the collaborating laboratories at CEA-Marcoule and INL. Organic-only samples were irradiated to 500, 750, and 1000 kGy, while the 0.1 M and 4.0 M HNO 3 -contacted samples were irradiated to 500, 750, 1000, and 1250 kGy.

Pulsed Irradiations

Picosecond electron pulse radiolysis/transient absorption experiments were performed at the Brookhaven LEAF facility, as described previously. [START_REF] Zarzana | A Comparison of the gamma-Radiolysis of TODGA and T(EH)DGA Using UHPLC-ESI-MS Analysis[END_REF] Samples were irradiated in 1.00-cm Suprasil semimicro cuvettes sealed with Teflon stoppers. The doses per pulse for various experiments were in the range 20-40 Gy. Time-resolved kinetics were obtained using a FND-100 silicon diode detector and digitized using a LeCroy WaveRunner 640Zi oscilloscope (4 GHz, 8 bit). Interference filters (*10-nm bandpass) were used for the wavelength selection of the analyzing light.

Analytical Measurements

Electrospray ionization-Mass Spectrometry (ESI-MS) analysis

The mass spectrometry measurements were recorded in positive ion mode using a Bruker Esquire-LC quadrupole ion trap equipped with an electrospray interface. A Cole Palmer syringe infusion pump delivered the sample at a rate of 90 µL•h -1 to the electrospray source. The capillary voltage was set to -4000 V. Nitrogen was employed as the drying and nebulizing gas. The drying gas flow rate was 5.0 L min -1 and the nebulizing gas pressure was set to 5.0 psi. The source temperature was 250 °C. Spectra were acquired over a mass/charge (m/z) range of 45-2200 with a trap drive setting of 50.0 at various skimmer voltages. Helium gas was used as the buffer gas in the trap to improve trapping efficiency and as a collision gas for fragmentation experiments.

Each sample was prepared by dilution in acetonitrile to a concentration of 1.0 × 10 -4 M. To favor the ionization of compounds and avoid the formation of sodiated adducts, samples irradiated without an aqueous phase were acidified before analysis by addition of 1 drop of 1 M nitric acid. The ESI-MS measurements were taken within a short time after sample preparation to avoid any solvent influences on the speciation of degradation products. Species were identified by comparison with an isotopic pattern calculated using the software DataAnalysis 4.0. Ion trap collision induced dissociation (CID) experiments were used to help identify the structures of various species through fragmentation studies.

UHPLC-ESI-MS analysis

Aliquots of the same samples were also analyzed at INL in positive ion mode, using a Dionex (Thermo Fisher, Sunnyvale, CA) Ultimate 3000 UHPLC coupled to a Bruker (Billerica, MA) micrOTOF-Q II electrospray ionization mass spectrometer (UHPLC-ESI-MS). The UHPLC-ESI-MS was used to identify the degradation products in all samples and also to quantify the DEHiBA concentrations in the two organic-only samples. The mass spectrometer nitrogen drying gas flow rate was 9 mL•min -1 , with a nebulizer gas pressure of 5.8 psi. The mass spectrometer conditions were a capillary voltage of -4000 V and a source temperature of 220 o C.

All solvents used for sample dilution and mobile phases were Thermo Fisher Scientific (Waltham, MA USA) Optima ® LC-MS grade. Samples were diluted by a factor of 10 3 in 2propanol followed by a factor of 4 × 10 2 in 50 % /50 % acetonitrile/water prior to analysis. The LC eluent was isocratic with 65% acetonitrile/35% 0.1% formic acid in water at a flow rate of 400 µL•min -1 . Separations were achieved using 5 µL injections through a flow-through needle onto a Kinetex 1.3 µm particle C18 50 mm × 2.1 mm column (Phenomenex, Torrance, CA, USA). Each sample was analyzed in quintuplicate. DEHiBA was quantified using a calibration curve spanning the equivalent (accounting for dilution) of 2 M to 0.1 M DEHiBA. No standards were available for the degradation products, so their response is the integrated LC-MS peak area.

GC-FID and GC-MS analysis

Samples of DEHiBA contacted with 0.1 M and 4.0 M HNO 3 , de-aerated and aerated, were analyzed using an Agilent (Santa Clara, CA, USA) 7890 gas chromatograph with Flame Ionization Detection (GC-FID) and an Agilent 7890A gas chromatograph with a 5975C MSD electron-impact quadrupole mass spectrometer (GC-EI-MS). Samples were diluted into the mM concentration range in Thermo Fisher Scientific (Waltham, MA USA) Optima ® LC-MS 2propanol. Each sample was analyzed in triplicate. Both systems used a Restek (Bellefonte, PA, USA) Rtx-5 (30m × 0.32 mm ID × 0.25 µm d f ). For the GC-FID analysis, the injector and detector were held at 300 °C throughout the run. The oven was held at 100 °C for 1 minute, followed by a ramp at 15 °C/min to 275, and a final hold at 275 °C for 1 minute. For the GC-EI-MS analysis, the injector and ion source were held at 250 °C, while the quadrupole was held at 200 °C for the duration of the run. The oven was held at 100 °C for 1 minute, followed by a ramp at 10 °C/min to 300 °C, with a final hold at 300 °C for 1 minute. Quantification of DEHiBA by GC-FID and GC-EI-MS produced statistically indistinguishable results, though the uncertainties for GC-FID were lower than those for GC-EI-MS.

FT-IR analysis

Organic phase samples were analyzed with a Bruker Vertex 70 spectrometer equipped with an attenuated total reflectance cell. All spectra were collected between 400-4000 cm -1 , using 32 scans and a resolution of 4 cm -1 .

Uranium (VI) Solvent Extraction

Uranium solvent extraction distribution ratios, D U , were measured for both irradiated and nonirradiated solutions at INL. The aqueous phase contained 1 × 10 -5 M natural uranium at the appropriate acidity. Preliminary experiments during this study showed that neither acid preequilibration of the organic phase, nor various contact times over the range of 30 sec-10 min affected the D U value. Subsequently the extractions reported here used non-pre-equilibrated, 30sec, equal volume contacts at 23 ± 2 o C. Although 4 M HNO 3 is proposed for process application extractions, the maximum D U value here was obtained at an initial aqueous phase acidity of 6.5 M HNO 3 . Therefore all D U values to examine irradiation effects were measured at this initial acidity. Stripping was performed using 0.1 M HNO 3 with 1-min contacts. These concentrations were determined by mass spectrometry using a Thermo X series 2 ICP-MS with a Teflon sample introduction system and platinum cones. During analysis of organic solutions, 0.2 L•min -1 of 20 % oxygen in argon was added to the spray chamber to aid in combustion of the organic material and to reduce build up on the cones. Organic solutions were emulsified with Triton TX100 into the normal 1 % nitric acid solution used to dilute samples prior to analysis. Mass balances were calculated by comparison to the initial aqueous feed uranium concentration and averaged 98 ± 6 %.

RESULTS AND DISCUSSION

Degradation rates for monoamide radiolysis

The radiolytic decay kinetics of DEHiBA can be determined by monitoring the concentration of the monoamide in organic solution as a function of absorbed dose. The concentration of DEHiBA decreased slowly as absorbed dose increased, as seen in Figure 2. The rate was slow and approximately 70% of the initial monoamide concentration remained intact even at an absorbed dose of 1250 kGy, under all conditions of acidity and aeration. The exponential nature of the decay suggests pseudo-first order kinetics, so a dose constant is the appropriate description of the radiolytic decay process.

Linear regression analysis of plots of the natural log of measured concentration versus absorbed dose was used to determine the dose constants (d) for monoamide degradation as the exponential constants of each equation. These values ranged from (2.0 ± 1.0) × 10 -4 kGy -1 for the 4.0 M HNO 3 -contacted organic phase to (3.5 ± 1.4) × 10 -4 kGy -1 for the de-aerated organic-only samples. The larger uncertainties in the two organic-only samples are due to the larger measurement uncertainties from the UHPLC-ESI-MS measurements.

A comparison of the dose constants (d values, Table 1) shows that rates of DEHiBA radiolysis in the organic phase were unaffected by aeration, or by contact with aqueous HNO 3 (either 0.1 or 4.0 M). This suggests that all conditions had roughly the same degradation kinetics, and the range is attributable to the difficulties inherent in measuring the small concentration change that occurred across these absorbed doses. The mean value for all conditions for both trials in Table 1 is (3.0 ± 0.9) × 10 -4 kGy -1 . These results are similar to those previously reported for diglycolamides, [START_REF] Zarzana | A Comparison of the gamma-Radiolysis of TODGA and T(EH)DGA Using UHPLC-ESI-MS Analysis[END_REF][START_REF] Galan | Gamma-radiolytic stability of new methylated TODGA derivatives for minor actinide recycling[END_REF][START_REF] Roscioli-Johnson | A Study of the γ-Radiolysis of N,N-Didodecyl-N′,N′-Dioctyldiglycolamide Using UHPLC-ESI-MS Analysis[END_REF] where the aqueous phase and aeration had little or no effect on degradation rates. However, the rate of monoamide degradation is about a factor of 10 slower than for the diglycolamides. [START_REF] Zarzana | A Comparison of the gamma-Radiolysis of TODGA and T(EH)DGA Using UHPLC-ESI-MS Analysis[END_REF][START_REF] Galan | Gamma-radiolytic stability of new methylated TODGA derivatives for minor actinide recycling[END_REF] The stability measured here seems to be of the same order of magnitude as that found for malonamides in alkane irradiated after contact with an aqueous phase of water or 1-4 M nitric acid. [START_REF] Berthon | Diamex process for minor actinide partitioning: Hydrolytic and radiolytic degradations of malonamide extractants[END_REF] In that study, approximately 70-80 % of the initial concentration of three different malonamides remained after an absorbed dose of 750 kGy (-G malonamides = 0.52-0.55 µmol•J -1 in presence of 4M nitric acid). It has been hypothesized that for the DGAs, the ether linkage is the most radiolytically vulnerable bond in the molecule. As the monoamides do not have an ether bond, the observation that their radiolytic degradation rates are significantly lower than those for the DGAs is consistent with this hypothesis.

In spite of the large body of literature examining TBP radiolysis, few G-values have been reported for irradiation under conditions of contact with the acidic aqueous phase, and typically the yield of the main decomposition product dibutylphosphoric acid (HDBP) is reported instead of the actual decrease in TBP concentration. These G HDBP values fall in the range of 0.05-0.5 µmol•J -1 for alkane solutions in the absence of the aqueous phase, [START_REF] Mincher | Review Article: The Effects of Radiation Chemistry on Solvent Extraction: 1. Conditions in Acidic Solution and a Review of TBP Radiolysis[END_REF] but obviously underestimate the actual -G TBP . Adamov reported -G TBP = 0.37 µmol•J -1 in the presence of acid. [START_REF] Adamov | Radiolysis of an extraction system based on solutions of tri-n-butylphosphate in hydrocarbon diluents[END_REF] If the mean dose constant reported here of 3.0 × 10 -4 kGy -1 (for all six experiments) is used to calculate a Gvalue [START_REF] Zarzana | A Comparison of the gamma-Radiolysis of TODGA and T(EH)DGA Using UHPLC-ESI-MS Analysis[END_REF] for the radiolytic degradation of initially 1 M DEHiBA, a value of -G DEHiBA = 0.53 µmol•J -1 is obtained, which is comparable to that of TBP.

Radical cation reactions with DEHiBA

The irradiation of non-acid contacted dodecane results in degradation of the diluent through molecular ionization to generate solvated electrons, (e Therefore, in this study, we elucidated the kinetics of this radical cation reaction with DEHiBA in dodecane. These fast, pulsed, electron radiolysis experiments were performed as detailed previously, [START_REF] Zarzana | A Comparison of the gamma-Radiolysis of TODGA and T(EH)DGA Using UHPLC-ESI-MS Analysis[END_REF] using 0.50 M CH 2 Cl 2 /aerated dodecane to isolate the radical cation with its absorbance directly monitored at 800 nm. Typical kinetic data for the decay of this species in the presence of DEHiBA are seen in Figure 3. As the amount of DEHiBA increases, the decay becomes faster, indicating the dodecane radical cation reacting is being consumed by DEHiBA.

From the analysis of the shown decay curves the plot inset shown in Figure 3

This effectively diffusion-controlled rate constant suggests that this reaction would be an important process in the overall organic phase degradation of this ligand, and supports the hypothesis that it may account for the appearance of the m/z 310.2 compound (which is a ligand formed by abstraction of two hydrogen atoms) when it occurs in the organic-only solution (vida infra).

Organic phase analysis-FTIR

To gain information about the nature of the degradation products formed under different experimental conditions, IR analyses were performed. Figure 4 and Figure 5 show a comparison of the sample spectrum before and after irradiation to 1000 kGy, for two conditions: without aqueous phase and with 4 M HNO 3 present, respectively. Figure 4 compares the impact of irradiation alone to air-sparging during irradiation on monoamide degradation without an aqueous phase present. The solid line represents 1 M DEHiBA dissolved in dodecane before irradiation.

Upon irradiation, there appears to be little impact on the nature of the degradation products in the non-air-sparged sample. This could indicate that the degradation products generated by irradiation maintain similar functional groups as the parent monoamide and are thus not unambiguously observable by IR.

In the air-sparged sample, the spectra are slightly different. A shoulder appears at 1175 cm -1 which could represent a C-N stretch in an aliphatic amine. A new peak is also observed at 1730 cm -1 as well as a shoulder peak at 1625 cm -1 . The appearance of the peak at 1730 cm When the sample is irradiated in the presence of acid, slightly different spectra are observed, (Figure 5). The solid line in Figure 5 represents 1 M DEHiBA dissolved in dodecane contacted with 4 M HNO 3 . The spectrum of this sample shows a new HNO 3 -monoamide peak at 1587 cm -1 which has been assigned as the C=O group bonded to HNO 3. [20, 23, 24] Other new bands appear at 1284 cm -1 , 940 cm -1 , 827 cm -1 , and 686 cm -1 , all of which are consistent with the presence of nitric acid in the organic phase. After irradiation, in both the air-sparged and non-air-sparged samples, the same peaks appear as in the air-sparged sample of the organic-only irradiated sample: 1730 cm -1 and 1625 cm -1 . These likely represent bands from the same or similar degradation products that were observed in the aerated, organic-only sample. Interestingly, there is a clear decrease in the absorbance of the bands representing the C=O-HNO 3 stretch (1587 cm -1 and 827 cm -1 ) upon irradiation for both the air-sparged and non-air-sparged samples. This is consistent with a decrease of the nitric acid concentration in the organic phase after irradiation.

The conclusions from this analysis are that under both oxidizing conditions of acid and airsparging, similar degradation products are formed and these are slightly different than those that formed due to irradiation of the de-aerated organic phase alone. To identify these products, the samples were analyzed by mass spectrometry.

Organic phase ESI-MS analysis

Mass spectra of the irradiated and non-irradiated samples show both the protonated and sodiated parent monomer and dimer compounds at m/z = 312.2, 334.2 and m/z = 623.0, 645.5, respectively. Also observed in the spectra of irradiated and non-irradiated samples is a product at m/z = 242.2 corresponding to bis-2-ethylhexyl-amine (DEHA), produced by carboxyl C-N bond rupture. It should be noted that though traces of DEHA were also found in non-irradiated samples, likely as an impurity from synthesis, it was also clearly generated during irradiation.

Supplementary Information Table S1 includes a summary of all degradation products found with an indication of specific experimental conditions under which they were formed. Most products were detected regardless the experimental conditions, though the relative yields of the degradation products were affected by the experimental conditions.

In the low molecular weight region of the mass spectra of the irradiated solutions, several degradation products were detected (see Supplementary Information Table S1). The ESI-MS spectra for two of the aqueous phase conditions (no aqueous phase, 4 M HNO 3 ) are shown in In addition to the sub-monoamide mass region, a new peak is observed at m/z = 310.2 as well as at values slightly higher than the m/z = 312.2 of the monoamide. The product observed at m/z = 310.2 corresponds to the abstraction of 2 hydrogen atoms from the ligand to form a C=C bond within the monoamide framework. This product was observed regardless acid concentration and air sparging conditions. For samples in contact with the aqueous phase the species most likely to react by H-atom abstraction is the • OH radical, which reacts with alkanes with fast rate constants. The same product was found under aqueous-free conditions. In this case, the monoamide may react with dodecane radical cations by electron transfer. This has been reported

previously for the diglycolamides. [START_REF] Zarzana | A Comparison of the gamma-Radiolysis of TODGA and T(EH)DGA Using UHPLC-ESI-MS Analysis[END_REF] If so, the loss of H-atoms may occur through proton elimination by the resulting monoamide radical cation.

Addition of an • OH radical to the m/z = 310.2 ion would generate the ion seen at m/z = 328.0.

However, this product was also observed in the organic-only solution and another oxidative mechanism may account for its appearance. The product at m/z = 258.2 could be produced either by carbonyl C-N bond rupture of the parent monoamide, or by H-atom abstraction from the amine N-H bond of DEHA, followed by addition of OH to generate di-2-ethylhexylhydroxylamine. This probably results from simple • OH radical addition to the produced • N radical. This product was observed only in air-sparged solutions at both acid concentrations. The product at m/z = 200.2 corresponds to a C-N bond rupture resulting in the loss of one of the 2ethylhexyl groups to produce mono-2-ethylhexyl-isobutyramide (EHiBA). This species was detected more at lower absorbed doses in the organic-only samples than in the 4 M HNO 3 contacted samples, suggesting that it is susceptible to acid hydrolysis.

For products of higher molecular weight, Figure 7, new species appeared at m/z = 551.7, 533.5, 480.5, and 410.5. Spectra generated by MS 2 fragmentation suggest that m/z = 551.7 is an addition product of the monoamide with the amine DEHA. Two addition species between the dodecane or degradation products were identified at m/z = 480.5 (monoamide + dodecane) and 410.5 (DEHA + dodecane). These high molecular weight species are likely produced by the addition reactions of the carbon-centered radicals generated by H-atom abstractions from the corresponding reactants. The peak at m/z = 533.5 remains unidentified.

Interestingly, the higher molecular weight products were observed to have the highest intensities in the samples not contacted with an aqueous phase during irradiation (Figure 7). For samples irradiated in the presence of 0.1 M HNO 3 , the abundance of these species was less intense. In the spectra of the samples irradiated in the presence of 4 M HNO 3 these species are completely absent. This indicates that the carbon-centered radicals were scavenged by produced

• NO 3 radicals under those conditions, although it must be noted that no nitro-substituted species were detected. Many of these nitro-or hydroxy-substituted species may have partitioned to the aqueous phase.

It is clear that the presence of acid during irradiation of the monoamide ligand impacts the formation of higher molecular weight species. Thus, there appears to be competitive degradation pathways for DEHiBA degradation; an acid promoted pathway and an acid independent pathway.

Scheme 1 shows the proposed formation of low molecular weight degradation products for all three conditions. Scheme 2 shows the proposed formation for the higher molecular weight products in the absence of acid.

This conclusion correlates well with the FT-IR data collected by supporting the observation that the samples irradiated in the presence of acid resulted in new peaks in the spectrum, whereas the organic only samples showed no difference when compared to an unirradiated monoamide solution. Interestingly, although the product distribution is affected by contact with the acidic aqueous phase, the overall rate of monoamide degradation is not.

Aqueous phase ESI-MS analysis

To aid in the understanding of the impact of acid during irradiation on monoamide degradation, the aqueous phase was next analyzed to determine if degradation products could be observed. All ions detected in the aqueous phase are shown in the Supplementary Information, Table S2. The nature of the species identified in the aqueous phase by ESI-MS appears to depend on nitric acid concentration. Figure 8 shows the spectra of the 0.1 M HNO 3 irradiated DEHiBA aqueous phase and Figure 9 shows the 4 M HNO 3 sample. There are more species present in the 4 and EHiBA to lower molecular weight, highly oxidized species was favored under the oxidizing conditions of aeration in the presence of high HNO 3 concentration. Small amounts of the parent monoamide were also identified in the aqueous phase. In previous studies, authors reported the formation of carboxylic acid by rupture of the carbonyl-N bond. [START_REF] Parikh | Radiolytic Degradation Studies on N,N-dihexyloctanamide (DHOA) under PUREX Process Conditions[END_REF][START_REF] Pathak | Actinides 2009, IOP Conference Series-Materials Science and Engineering[END_REF][START_REF] Musikas | Solvent-extraction for the chemical separations of the 5f elements[END_REF][START_REF] Musikas | Potentiality of nonorganophosphorus extractants in chemical separations of actinides[END_REF] In the case of DEHiBA, the carboxylic acid expected to form is butanoic acid. This species was not detected in either the organic or aqueous phases. This is likely due to the hrdrolytic degradation of the species resulting in the formation of lower molecular weight species which are too low to be detected by the analytical method used.

M

Effects of irradiation on uranium solvent extraction

The D U for the extraction of non-irradiated DEHiBA was consistent with expectations based on literature results. [START_REF] Siddall | Effects of structure of N,N-disubstituted amides on their extraction of actinide and zirconium nitrates and of nitric acid[END_REF][START_REF] Prabhu | Di(2-ethyl hexyl)butyramide and di(2-ethyl hexyl)isobutyramide as extractants for uranium(VI) and plutonium(IV)[END_REF] It can be seen in Figure 10 that within the ± 8% uncertainty, there was no change in uranium extraction efficiency with absorbed dose to the maximum of 1000 kGy under any irradiation condition. This is consistent with the high stability of DEHiBA under all conditions, the generation of decomposition products such as EHiBA, compounds with m/z = 310 and 326 (DEHiBA with a C=C bond and a C=O bond), and high molecular weight compounds (m/z = 480, 551) that are likely also extractants, as well as the excess initial concentration of the ligand in comparison with the tracer metal concentration (0.01 mM). It can also be seen that the stripping D U appears to gradually decrease with absorbed dose. This may be a result of the presence of DEHiBA radiolysis products in the aqueous strip solution, and the corresponding stripping of uranium by these species. Although the decreases in D U are not large, decreased stripping distribution ratios can only be considered to be advantageous to a process separation.

CONCLUSIONS

In contrast to previous literature claiming exceptional radiation stability for the monoamides, the degradation rate of DEHiBA is actually similar to that of TBP and the malonamides. It is slow compared to the DGAs, and unaffected by contact with an aqueous phase or aeration.

However, radiolytic product distributions vary with irradiation conditions. Based on these results, DEHiBA apparently undergoes degradation via two pathways: an acid promoted pathway, and an acid independent pathway. It is clear that the monoamide degrades when irradiated in the presence of an aqueous phase to form a series of lower molecular weight species generated from the cleavage of the C-N amide bond or C-N amine bond. As this is the active site during synthesis, it is not surprising that this is the weak point in the ligand structure. The main degradation products appear to be the secondary amide DEHA and the amine EHiBA. These species, and the smaller fragments produced by their radioysis have increased solubility in the aqueous phase. Another product common to all irradiation conditions was the species at m/z = 310.2, which is identified as an unsaturated derivative of DEHiBA, resulting from the loss of two H-atoms.

In contrast, when an aqueous phase is not present, higher molecular weight products are generated via carbon radical addition reactions under the more reducing conditions. These products have maximum abundance at 750 kGy, and then decrease with increasing absorbed dose. Their significance to a biphasic solvent extraction process is probably inconsequential.

Solvent extraction results show that DEHiBA radiolytic degradation had little effect on uranium distribution ratios even at absorbed doses as high as 1 MGy. The build-up of degradation products in the aqueous phase apparently decreased stripping distribution ratios, which is not adverse to a process application. Thus, these findings for DEHiBA are in agreement with previous work that claimed generation of inoffensive radiolysis products for the monoamides. [START_REF] Parikh | Radiolytic Degradation Studies on N,N-dihexyloctanamide (DHOA) under PUREX Process Conditions[END_REF][START_REF] Pathak | Evaluation of Di(2éthylhexyl)isobutyramide (D2EHiBA) as a process extractant for the recovery of 233 U from irradiated Th[END_REF][START_REF] Berthon | Radiolysis of Solvents Used in Nuclear Fuel Reprocessing[END_REF][START_REF] Gasparini | Application of n,n-dialkyl aliphatic amides in the separation of some actinides[END_REF] This, in addition to their CHON nature, suggests that they are good candidates for the development of advanced fuel cycles. Interesting future work will include a comparison study on the n-alkane monoamide DEHBA that extracts more Pu(IV) in the organic phase and is therefore more likely to undergo radiolysis effects in potential nuclear fuel recycling processes.

Figure 6 .

 6 Figure 6. The 0.1 M HNO 3 sample spectra were not shown because they only give intermediate

  HNO 3 sample than in the 0.1 M HNO 3 sample. The products DEHA (m/z = 242.2) and EHiBA (m/z = 200.2) were found in both the 0.1 M and 4.0 M HNO 3 aqueous phases of the irradiated samples, with maximum signal strength in the samples irradiated between 500-750 kGy. A new product was detected at m/z = 130.3, identified as 2-ethylhexyl-amine, indicating loss of both a butyl and one ethylhexyl group, likely produced by continued radiolysis of either DEHA or EHiBA. The detected signals of DEHA and EHiBA in the aqueous phase decreased at higher absorbed doses, suggesting that these species are susceptible to continued radiolysis in the aqueous phase, and as their signals decreased, the highintensity signal at m/z = 130.3 (2-ethylhexylamine) grew in. This signal continued to increase to a maximum at the highest absorbed dose of 1000 kGy. Other unidentified ions whose signals increase in abundance with increasing dose in the 4.0 M HNO 3 air-sparged samples were m/z = 126.2 (possibly C 8 H 15 N, a doubly unsaturated mono-alkyl amine), and m/z = 142.2 (possibly C 8 H 15 N). The continued degradation of the DEHA

Figure 1 :

 1 Figure 1: Structure of di-2-ethylhexyl-butyramide (DEHBA), (a) and di-2-ethylhexylisobutyramide (DEHiBA), (b).

Figure 2 :Figure 3 :

 23 Figure 2: DEHiBA concentration in the organic phase as a function of absorbed dose, for a) the unsparged organic phase only (solid squares); organic phase with air sparging (open squares); b) unsparged organic phase in contact with 0.1 M HNO 3 (solid circles); organic phase in contact with 0.1 M HNO 3 with air sparging (open circles); c) unsparged organic phase in contact with 4 M HNO 3 (solid triangles); and organic phase in contact with 4 M HNO 3 with air sparging (open triangles). Each datum for the two organic-only samples is the mean of five injections measured by UHPLC-ESI-MS, while each datum for the 0.1 M and 4.0 M HNO 3 -contacted samples is the mean of 3 injections measured by GC-FID. Error bars represent 99% confidence intervals.

Figure 4 :

 4 Figure 4: FTIR spectra of 1 M DEHiBA dissolved in dodecane (solid black line), 1 M DEHiBA dissolved in dodecane irradiated to 1 MGy (dashed line), 1 M DEHiBA dissolved in dodecane irradiated to 1 MGy under air-sparging conditions (dotted line).

Figure 5 :

 5 Figure 5: FTIR spectra of 1 M DEHiBA dissolved in dodecane contacted with 4 M HNO 3 (solid black line), 1 M DEHiBA dissolved in dodecane irradiated to 1 MGy while in contact with 4 M HNO 3 (dashed line), 1 M DEHiBA dissolved in dodecane irradiated to 1 MGy and air-sparged while in contact with 4 M HNO 3 (dotted line).

Figure 7 :

 7 Figure 7: ESI-MS spectra of DEHiBA at 0 kGy (top), 750 kGy organic phase only (middle), 750 kGy contacted with 4 M HNO 3 (bottom). All spectra were recorded in positive ionization mode, trap drive 50, skimmer voltage 30 V, non-air sparged. Note: the intensity scale was chosen to specifically show the lower abundance species.

Figure 8 :

 8 Figure 8: ESI-MS spectra of 0.1 M HNO 3 aqueous phase irradiated with DEHiBA at 0 kGy (top), 500 kGy (middle), 1000 kGy (bottom). All spectra were recorded in positive ionization mode, trap drive 50, skimmer voltage 30 V, air sparged.

Figure 9 :

 9 Figure 9: ESI-MS spectra of 4 M HNO 3 aqueous phase irradiated with DEHiBA at 0 kGy (top), 500 kGy (middle), 1000 kGy (bottom). All spectra were recorded in positive ionization mode, trap drive 50, skimmer voltage 30 V, air-sparged.

Figure 10 :Scheme 1 :

 101 Figure 10: Uranium solvent extraction (D U ) from 6.5 M HNO 3 , (black symbols) and stripping (with 0.1M HNO 3 ), (grey symbols) using irradiated, initially 1 M DEHiBA in dodecane. Irradiated organic phase only (solid squares); sparged organic phase (open squares); contact with 0.1 M HNO 3 (solid circles); sparged contact with 0.1 M HNO 3 (open circles); contact with 4 M HNO 3 (solid triangles); and sparged contact with 4 M HNO 3 (open triangles). Mean error bars of ± 8 % omitted for clarity.

  is derived. The slope of this plot gives the second-order rate constant for the reaction of C 12 H 26 +• with DEHiBA as k = (1.52 ± 0.11) x 10 10 M -1 •s -1 : C 12 H 26 +• + DEHiBA → C 12 H 26 + [DEHiBA] + •

  Figure 6: ESI-MS spectra of DEHiBA at 0 kGy (top), 1000 kGy organic phase only (middle), 1000 kGy contacted with 4 M HNO 3 (bottom). All spectra were recorded in positive ionization mode, trap drive 50, skimmer voltage 30 V, non-air sparged. Note: the intensity scale was chosen to specifically show the lower abundance species.

							Dow nloads: +MS
					ORG 4 M HNO3
			242.2		ORG 4 M HNO3 1 MGy ORG 4 M HNO3 1 MGy sparged 314.2 335.2
			230.1	254.2	272.3	295.9
	200.2					314.2	Dow nloads: +MS
		214.2	242.2 256.3 270.2	296.2	328.2	338.2
							Dow nloads: +MS
			242.3			310.2	335.2
			251.2			328.2
	200.1 211.4	228.2		272.2 284.2	300.2
	1,800	1,600	1,400	1,200	1,000	800

Scheme 2: Proposed DEHiBA degradation pathway for the formation of high molecular weight species in the absence of acid.

Table 1: Dose constants (d, kGy -1 ) for the degradation of DEHiBA. DEHiBA exhibited pseudofirst order kinetics, so dose constants were calculated from linear fits of plots of the natural log of concentration vs. absorbed dose. Uncertainties are 99% confidence intervals calculated from the standard error of regression based on 5 injections per absorbed dose for the two non-acid contacted conditions and three injections per absorbed dose for the four acid-contacted experiments.

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