Mathieu Bajodek

Alexandre Seuret

Frédéric Gouaisbaut

Insight into the stability analysis of the reaction-diffusion equation interconnected with a finite-dimensional system taking support on Legendre orthogonal basis

The stability analysis of the reaction-diffusion subject to dynamic boundary conditions is not straightforward. This chapter proposes a linear matrix inequality criterion which ensures the stability of such infinite-dimensional system. By the use of Fourier-Legendre series, the Lyapunov functional is split into an augmented finite-dimensional state including within it the first Fourier-Legendre coefficients and the residual part. A link between this modelling and Padé approximation is briefly highlighted. Then, from Bessel and Wirtinger inequalities applied to the Fourier-Legendre remainder and using its orthogonality properties, a sufficient condition of stability expressed in terms of linear matrix inequalities is obtained. This efficient and scalable stability condition is finally performed on examples.

Introduction

The wide class of infinite-dimensional systems [START_REF] Curtain | Transfer functions of distributed parameter systems: A tutorial[END_REF][START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF][START_REF] Evans | Partial Differential Equations[END_REF] generates issues to be analyzed numerically. In most of the cases, when the eigenvalue decomposition of the infinite-dimensional operator cannot be given analytically, the stability properties of these systems remain unknown. For instance, the stability analysis of a system coupled with a partial differential equation is tough task, which has been infrequently studied. Most of recent studies bypass the problem by focusing on the stabilization of such infinite-dimensional systems thanks to the design of infinite-dimensional control laws [START_REF] Aarsnes | Delay robust control design of underactuated PDE-ODE-PDE systems[END_REF][START_REF] Prieur | Feedback Stabilization of a 1-D Linear Reaction-Diffusion Equation with Delay Boundary Control[END_REF]. In the field of numerical analysis, researchers approximate the solutions from spacial discretization [START_REF] Lasieka | Control Theory for Partial Differential Equations: Continuous and Approximation Theories[END_REF] or the tau method [START_REF] Gottlieb | Numerical analysis of spectral methods[END_REF]. Under dissipativity conditions [START_REF] Ito | Legendre-Tau approximations for functional differential equations[END_REF], convergence properties of the numerical schemes emerge. But, they never deal with M. Bajodek 7 avenue du Colonel Roche 31400 Toulouse Tel.: +33-567524243 e-mail: mathieu.bajodek@laas.fr the stability of the original system. Therefore, to rule on the stability of interconnected ordinary-partial differential equations, input-to-state [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF][START_REF] Logemann | Circle criteria, small-gain conditions and internal stability for infinitedimensional systems[END_REF][START_REF] Mironchenko | Input-To-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions[END_REF] or Lyapunov [START_REF] Prieur | Stability of switched linear hyperbolic systems by Lyapunov techniques[END_REF][START_REF] Wang | An LMI approach to stability analysis of reaction-diffusion Cohen-Grossberg neural networks concerning Dirichlet boundary conditions and distributed delays[END_REF] approaches are privileged. For generic cases, quadratic constraints and complete Lyapunov functionals lead to criteria solved be semi-definite programming [START_REF] Das | Robust analysis of uncertain ODE-PDE systems using PI multipliers, PIEs and LPIs[END_REF][START_REF] Das | Representation and stability analysis of PDE-ODE coupled systems[END_REF][START_REF] Valmorbida | Stability Analysis for a Class of Partial Differential Equations via Semidefinite Programming[END_REF]. To refine, adjust and better understand this generic results, works have been pursued on the model and the inequalities involved in the sufficient condition of stability. For instance, with transport [START_REF] Safi | Tractable sufficient stability conditions for a system coupling linear transport and differential equations[END_REF], heat [START_REF] Baudouin | Stability analysis of a system coupled to a heat equation[END_REF] or wave [START_REF] Barreau | Input/output stability of a damped string equation coupled with ordinary differential system[END_REF] equations, an augmented system enriched by Legendre polynomials coefficients of the partial-differential part can be built. Applying Bessel-Legendre inequality, hierarchical stability results are expressed in terms of linear matrix inequalities and dedicated to each equation. Concerning the reaction-diffusion equation, the work is slighty modified since the Wirtinger inequality has to be invoked. In this chapter, one aims at simplifying the linear matrix inequality condition obtained enlightening the Fourier-Legendre series, as done in [START_REF] Bajodek | Stability analysis of an ordinary differential equation interconnected with the reaction-diffusion equation[END_REF] with other boundary conditions. To sum up, a numerical approach to state on the stability of an ordinary differential equation interconnected with a reaction-diffusion equation is proposed. The novelty of this contribution comes from the model transformation which eases the expression and understanding of the stability analysis. Indeed, by defining signals derived from Fourier-Legendre remainder, the Bessel-Legendre and the first Wirtinger inequalities can be rewritten in a nice formalism. Around these signals, the dynamical model can be constructed and related to Padé ( -1| ) approximant. It includes the extended finite-dimensional based on the first Fourier-Legendre polynomials coefficients. Taking a quadratic Lyapunov function based on this augmented system, a tight linear matrix inequality condition of stability is obtained. Numerical results are finally performed to show the effectiveness of the approximation and of the sufficient stability criterion.

Notations :

In this paper, the set of natural numbers, real numbers, real positive numbers, matrices of size × and of symmetric positive definite matrices of size are respectively denoted N, R R ≥0 , R × and S + , respectively. The identity matrix of dimension is denoted by and diag( 0 , ..., ) stands for the diagonal matrix whose coefficients are ( 0 , ..., ). For any matrix ,

, refers to the coefficient located on the ℎ row and ℎ column. For any square matrix , the transpose matrix is denoted , H ( ) = + and 0 means that is symmetric positive definite. For any square matrix , ( ) denotes the spectrum of . Furthermore, if is symmetric, ( ) and ¯ ( ) stands for its lower and larger eigenvalues, respectively. For any analytic function 1 and 2 , 1 ( ) = → 2 ( ) means that the ratio 1 2 ( ) is finite for tends to ∈ R. Moreover, the space of square integrable functions L 2 ( , ; R) is associated to the scalar product

1 | 2 = ∫ 1 ( ) 2 ( )d and the induced norm 2 = ∫ 2 ( )d . Set H 1 ( , ; R)
stands for the set of functions , such that and are in L 2 ( , ; R). With a light abuse of notations, the notation for inner product 1 | 2 will be used when 1 and 2 are vector functions.

Presentation of the system 2.1 Interconnected system

Consider the following system composed of an ordinary differential equation interconnected with a reaction-diffusion partial differential equation with cross type boundary conditions

                 ( ) = ( ) + ( , 1), ∀ ∈ R ≥0 , ( , ) = ( + ) ( , ), ∀( , ) ∈ R ≥0 × (0, 1), ( ,0)
( ,1) = 0 ( ) , ∀ ∈ R ≥0 , ( (0), (0, •)) = ( 0 , 0 ). ( 1a
) (1b) (1c) (1d)
Coefficients > 0, ∈ R, matrix ∈ R × and vectors , ∈ R are supposed to be constant and known.

Existence and uniqueness of solutions

As a first step and before studying stability of such class of system, it is important to show that system (1) is well-posed despite unbounded input and output operators applied on the partial differential equation [START_REF] Lhachemi | Boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and state-delay[END_REF]. This is formulated in the next proposition.

Proposition 1 Assuming that ( 0 , 0 ) belongs to R ×L 2 (0, 1; R), system (1) admits a continuous and unique solution ( , ) ∈ R × L 2 (0, 1; R). Proof Let us define the energy of system (1) by E ( ) = ( ) ( ) + (2 ) -1 ( ) 2 . Taking its derivatives, calculations lead to

d d E ( ) = ( ) ( + ) ( ) + 2 ( ) ( , 1) + ( ) 2 - ( ) 2 + ( ) ( , 1).
Getting rid of the cross terms by application of Young's inequality, one obtains

d d E ( ) ≤ ( ) H ( ) +4 + ( ) + ( ) 2 - ( ) 2 + 1 2 ( , 1) 2 .
However, from Jensen's inequality, one gets to

( ) 2 ≥ ( , 1) 2 , (2)
which simplifies the previous inequality on E ( ), leading to

d d E ( ) ≤ E ( ) -1 2
( ) 2 ≤ E ( ),
where is a generic constant depending on , , , , . From Grönwall's inequality, there exists a unique solution ( , ) in the continuous space from R ≥0 to R × L 2 ( , ; R).

Remark 1 In Section 5, to deal with stability of system (1), a Lyapunov functional is chosen to be equivalent to the energy E ( ) in the sense of the R × L 2 (0, 1; R).

In parallel with the calculations carried out here, the positive term in is upper bounded by Wirtinger inequality and Jensen's inequality [START_REF] Bajodek | Stability analysis of an ordinary differential equation interconnected with the reaction-diffusion equation[END_REF] is improved thanks to Bessel-Legendre inequality.

Equilibrium point

As a second step, it is also important to characterize the equilibrium of (1). More particularly, one has to understand under which conditions, system (1) admits a unique equilibrium. This is formulated below.

Proposition 2 System (1) admits a unique equilibrium (0, 0) if and only if matrix Ω given by (3) has a full rank.

Ω =                    sinh( ˜ ) -˜ cosh( ˜ ) , if < 0, -1 , if = 0, sin( ˜ ) -˜ cos( ˜ )
, if > 0.

(

) 3
Proof Let ( ¯ , ¯ ) be an equilibrium of system (1), meaning that the following relations hold.

             ¯ + ¯ (1) = 0, ( + ) ¯ ( ) = 0, ¯ (0) = 0, ¯ (1) = ¯ . (4a) (4b) (4c) (4d)
By integration of the differential equation (4b) and under condition (4c), one obtains

¯ ( ) =        sinh( ˜ ), if < 0, , if = 0, sin( ˜ ), if > 0,
where in R to be fixed and ˜ = | |/ . By re-injecting this expression into (4a) and (4d), it yields Ω ¯ = 0 with Ω given by (3). Hence, system (4) admits a unique solution leading to the equilibrium ( ¯ , ¯ ) = (0, 0) if and only if det(Ω) ≠ 0 which is equivalent to have Ω a full matrix.

Expected results

Under the conditions stated above, the aim of this paper is to state accurately on the stability of the equilibrium point (0, 0) using the linear matrix inequality framework. More specifically the following results will be obtained:

• The main result is the proposition of a scalable stability analysis for this coupled ordinary-partial differential equations based on Lyapunov arguments. The analysis is performed thanks to an accurate Lyapunov functional, which is build using Legendre polynomials. The calculations developments are highly inspired by [START_REF] Bajodek | Stability analysis of an ordinary differential equation interconnected with the reaction-diffusion equation[END_REF] and have been simplified thanks to the consideration of easier boundary condition applied to the reaction-diffusion equation.

• Contrary to a previous study provided in [START_REF] Baudouin | Stability analysis of a system coupled to a heat equation[END_REF], the analysis will be performed through the introduction of the remainder of Legendre-Fourier series, allowing to simplify some technical aspects. Thanks to this remainder, it is possible to rewrite the Wirtinger and the Bessel-Legendre inequalities in a simpler manner compared to the formulation presented in [START_REF] Baudouin | Stability analysis of a system coupled to a heat equation[END_REF]. • Moreover, thanks to the introduction of an augmented system, a necessary condition for the stability condition emerges: a certain matrix A has to be stable. In Section 4, as a subsidiary result, it is shown that this matrix is the Redheffer product between a realization of the Padé ( -1| ) approximant of the reactiondiffusion transfer function and the finite-dimensional system 0 .

Fourier-Legendre remainder and inequalities

Based on the nature of the boundary conditions of the reaction-diffusion (1c), we consider a function in L 2 (0, 1; R) such as (0) = 0. To construct a complete description of this function , only odd polynomials on an extended interval [-1, 1] need to be used. Then, after recalling the basics of Legendre polynomials and some of their properties, this section provides the definition of Fourier-Legendre coefficients and reaminder on the interval [-1, 1]. In a last step, several relevant inequalities are presented.

Fourier-Legendre coefficients and remainder

Legendre polynomials, denoted as for any positive integer , are given by

( ) = =0 (-1) ( + )! ( !) 2 ( -)! 1 - 2 , ∀ ∈ [-1, 1]. ( 5
)
The orthogonal family { } ∈N spans L 2 (-1, 1; R). For writing comfort, let introduce the notation ℓ for any ∈ N, which gathers the first odd Legendre polynomials in vector formulation, that is

ℓ ( ) = 1 ( ) 3 ( ) . . . 2 -1 ( ) ∈ R . (6)
Recall also some important properties of Legendre polynomials [START_REF] Gautschi | Orthogonal polynomials, quadrature, and approximation: computational methods and software (in Matlab)[END_REF], that will be useful along the paper, that are

ℓ |ℓ = I -1 , ℓ ( ) = L ℓ ( ), ∀ ∈ [-1, 1], ℓ (-) = -ℓ ( ), ∀ ∈ [0, 1], (7a) (7b) (7c)
where matrices L and I are square matrices of dimension , given by

I = 1 2 diag(3, . . . , 4 -1), L , = (4 -1) -1 = (4 + 1), ∀ , ∈ {1, . . . , }. (8)
Functions ℓ can be easily evaluated at ∈ {0, 1} and is given by

ℓ (1) = [ 1 1 ... 1 ] ∈ R , ℓ (0) = [ 0 0 ... 0 ] ∈ R , ℓ (1) = [ 1 6 ... (2 -1) ] ∈ R . (9a) (9b)
As stated in the previous section, we will define here the main features of this paper in the following definition.

Definition 1 For any function ∈ L 2 (0, 1; R) such that (0) = 0 and for integer in N, we define

• the state on the extended interval [-1, 1] by

( ) = -(-), ∀ ∈ [-1, 0], (10)
• the first Fourier-Legendre coefficients of as

=        ∫ 1 -1 1 ( ) ( )d . . . ∫ 1 -1 2 -1 ( ) ( )d        = ℓ | ∈ R , (11)
• the associated Fourier-Legendre remainder of at order as

( ) = ( ) -ℓ ( )I , ∀ ∈ [-1, 1]. ( 12
)
It is worth noting that is in L 2 (-1, 1; R) and that the norm and scalar product used in the sequel are considered in the interval [-1, 1]. The main interest for introducing this remainder is stated in the two following lemmas.

Lemma 1 For any in N, the Fourier-Legendre remainder given by [START_REF] Gottlieb | Numerical analysis of spectral methods[END_REF] verifies

• | = 0,
for any integer ∈ {0, . . . , 2 }, which means in other words that is orthogonal to the 2 + 1 first Legendre polynomials, • (0) = 0, which guarantee that the boundary condition at = 0 satisfied by is maintained for .

Proof By symmetry of all even Legendre polynomials, one obtains directly that 2 | = 0, for all positive integers . Moreover, thanks to the orthogonality (7a) of the Legendre polynomials, re-injecting the definition of into ℓ | yields

ℓ | = ℓ | -ℓ |ℓ I ℓ | = ℓ | -ℓ | = 0,
which lead to the orthogonality on the first odd Legendre polynomials. Lastly, thanks to the value at = 0 given by (9a), we have

(0) = (0) -ℓ (0)I = (0) = 0.
which concludes the proof.

In the next subsections, Bessel and Wirtinger inequalities are rewritten in an adequate manner to be consistent with the Fourier-Legendre remainder introduced above. The forthcoming Lyapunov analysis could then be pursued on an augmented system which takes aside the Fourier-Legendre remainder of the signal .

Bessel inequalities

Let us first recall the Bessel-Legendre inequality.

Lemma 2 For any function ∈ L 2 (-1, 1; R) and for any integer in N, the following inequality holds:

2 ≥ =0 2 + 1 2 | 2 . ( 13
)
Proof The proof can be found in [START_REF] Bajodek | Stability analysis of an ordinary differential equation interconnected with the reaction-diffusion equation[END_REF][START_REF] Baudouin | Stability analysis of a system coupled to a heat equation[END_REF]. It is directly derived from the positivity of the L 2 (-1, 1; R) norm of the Fourier-Legendre remainder of the function .

The application of Lemma 2 on Fourier-Legendre remainder yields 2 ≥ 0, since the remainder is orthogonal to the first Legendre polynomials, i.e. ℓ | = 0. This means that the information encapsulated in the Bessel inequality is already included in the remainder.

The interest of using the remainder is related to the formulation of this inequality when it is applied to its derivatives that is presented in the next lemma.

Lemma 3 For any function in H 1 (0, 1; R) such that (0) = 0 and any integer in N, the derivatives of the remainder given by (12) verifies

2 ≥ 2 (1), with = 2( + 1) (2 + 1). ( 14
)
Proof Thanks to the Bessel-Legendre inequality at order 2 + 1, the following inequality holds

2 ≥ 2 +1 =0 2 + 1 2 2 . ( 15
)
In addition, performing an integration by parts yields

= (1) + (-1) (1)
-.

Then, we recall that is the remainder of the Fourier-Legendre series, which is consequently orthogonal to the 2 + 1 first Legendre polynomials. Therefore, the last term of the previous equality is zero so that 2 ≥ 2 =0 (4 + 1) 2 (1).

To complete the proof, we have

=0 (4 + 1) = 4 ( + 1) 2 + ( + 1) = ( + 1) (2 + 1).
which yields the result.

Remark 2 It is worth noticing that the error made in this new Bessel-inequality is equal to the L 2 (-1, 1; R) norm of the Fourier-Legendre remainder of . Then, this error converges to zero since belongs to L 2 (0, 1; R).

It is important to note that the previous lemma allows to express a lower bound on the derivative with respect to of the Fourier-Legendre series, which only depends on the evaluation of at the boundary = 1. This bound does not depends on the first Fourier-Legendre coefficients, since we have chosen to consider the remainder only, which is orthogonal to the 2 + 1 first Legendre polynomial. This will simplify many technical calculations in the next developments.

Modified Wirtinger's inequality

In the literature [START_REF] Kammler | A first course in Fourier analysis[END_REF], Wirtinger's inequalities refer to inequalities which estimate the integral of the derivative function with the help of the integral of the function. These inequalities have been widely used in the context of analysis, control and observation of time-delay and reaction-diffusion systems [START_REF] Selivanov | Boundary observers for a reaction-diffusion system under timedelayed and sampled-data measurements[END_REF]. In this paper, one proposes to use Wirtinger's inequality of the first type, stated as follows.

Lemma 4 For any function ˜ in H 1 (-1, 1; R) such that ˜ (-1) = ˜ (1) and that ˜ |1 = 0, the following inequality holds:

˜ ≥ ˜ . ( 16
)
Proof The proof comes along the arguments provided in [START_REF] Kammler | A first course in Fourier analysis[END_REF]. Relying on the Fourier series of ˜ given by

∞ =0 with = ˜ ( ) - , it follows ˜ 2 = ∞ =0 2 .
Moreover, taking also the Fourier series of ˜ ( ) as

∞ =0 with = ˜ ( ) , we also have ˜ 2 = ∞ =0 2 .
The result is then obtained by integration by parts, noticing that = .

The next lemma is an application of the previous Wirtinger inequality to the Fourier-Legendre remainder given by [START_REF] Gottlieb | Numerical analysis of spectral methods[END_REF].

Lemma 5 For any function in H 1 (0, 1; R) and for any in N, the Fourier-Legendre remainder given by (12) verifies

2 -2 2 ≥ 2 (1), with = 2 (2 -1) + 2 2 4 -1 . ( 17
)
Proof In order to apply Lemma 4, let us introduce function ˜ , defined by

˜ ( ) = ( ) -2 -1 ( ) (1), ∀ ∈ [-1, 1]. (18)
First, one has to verify the assumptions of the Wirtinger inequality of the first kind, that is ˜ (-1) = ˜ (1) = 0 and ˜ |1 = 0. Recalling that 2 -1 (1) = 1 and that 2 -1 (-1) = -1, we have

˜ (1) = (1) -(1) = 0, ˜ (-1) = (-1) + (1) = 0.
Furthermore, one directly obtains ˜ |1 = 0 from the orthogonality of the Fourier-Legendre remainder with 0 ( ) = 1 (see Lemma 1). Therefore, Wirtinger's first inequality [START_REF] Krstic | Delay compensation for nonlinear, adaptive, and PDE systems[END_REF] states that, under the equality and integral conditions, the inequality ˜ ≥ ˜ holds. It remains to compute ˜ and ˜ . On the first hand,

˜ 2 = ∫ 1 -1 ( ) -2 -1 ( ) (1) 2 d , = 2 -2 2 -1 (1) + 2 -1 2 -1 2 (1)
.

By integration by parts, we can write

˜ 2 = 2 + 2 2 -1 -[ 2 -1 ] 1 -1 (1) + 2 -1 2 -1 2 (1).
Using Lemma 1 verified by , we can get rid of the second term and say that

˜ 2 = 2 - -1 =0 2(4 + 1) 2 (1),
The last term of the previous expression has already been computed in [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF], and we know that

˜ 2 = 2 -2 (2 -1) 2 (1), (19)
On the other hand, the norm of ˜ can be computed as follows. Due to the orthogonality the Legendre polynomials provided by Lemma 1, we have

˜ 2 = 2 + 2 4 -1 2 (1). ( 20
)
Thus, the proof is concluded by merging the two expressions given in ( 19) and ( 20) into ˜ ≥ ˜ .

The previous lemma extends the Wirtinger inequality, in the situation where is equal to 0 at = 0. The main advantages of using the Fourier-Legendre remainder appears in the simple formulation of the lower bound in [START_REF] Lasieka | Control Theory for Partial Differential Equations: Continuous and Approximation Theories[END_REF]. It is important to stress that the orthogonality condition ℓ | = 0 drastically simplifies the expression and the calculations.

It is worth mentioning that both constants and = -1 + 2 2 4 -1 increase in a quadratic manner with respect to . This information will not be used in this paper but could be a key point in the proof of the convergence of the sufficient stability criterion developed in the last section.

Modeling of an augmented system

In this section, one proposes the design of an augmented model issued from the interconnected system [START_REF] Aarsnes | Delay robust control design of underactuated PDE-ODE-PDE systems[END_REF]. The introduction of such model will allow to perform an assurate stability test.

Transfer function of the reaction-diffusion equation

Take the reaction-diffusion equation with cross type boundary conditions:

       ( , ) = ( + ) ( , ), ∀( , ) ∈ R ≥0 × (0, 1), ( ,0)
( ,1) = 0 ( ) , ∀ ∈ R ≥0 , (21a) (21b)
where ( ) is an input signal.

In the following, system ( 21) is transformed in Laplace domain where ∈ C is the Laplace variable and where variables in capital letters represents the Laplace transform associated with variable in non-capital letters. As an infinite-dimensional system, the irrational transfer function between ( ) = ( , 1) and ( , ) can be expressed as in [START_REF] Krstic | Delay compensation for nonlinear, adaptive, and PDE systems[END_REF] by

( , ) = sh - -ch - , ∀( , ) ∈ C × [-1, 1]. ( 22
)
The main issue is to approximate this infinite-dimensional part in a specific manner and to highlight the link with Legendre approximation methods. As ( , ) is an odd analytical function, it is possible to define Fourier-Legendre remainder ˜ done by truncation at order of the Legendre-Fourier series of on Legendre polynomials coefficients which is, for all ∈ C,

˜ ( , ) = ( , ) - -1 =0 4 + 3 2 2 +1 ( ) 2 +1 | ( ) , ∀ ∈ [-1, 1]. ( 23
)
Remark 3 This remainder ( 23) is well-defined on the segment [-1, 1] with respect to by the fact that ( ) belongs to C ∞ ( [-1, 1], C) for any in C and ( ) converges uniformly on any compact subset of C [START_REF] Webb | Pointwise and uniform convergence of Fourier extensions[END_REF].

Then, error [START_REF] Safi | Tractable sufficient stability conditions for a system coupling linear transport and differential equations[END_REF] can be rewritten as

˜ ( , ) = ∞ = 4 + 3 2 2 +1 ( ) ∫ 1 -1 ( , ) 2 +1 ( ) , ∀ ∈ [-1, 1], (24)
and an estimation of its value around = can be proposed.

Lemma 6 Error (23) is → ( ), for all ∈ [-1, 1].
Proof Lemma 6 means that ˜ ( , ) has its -1 first derivatives with respect to evaluated at = equal to zero and this is proven herafter. From the Maclaurin development of around = with an infinite radius of convergence, for all

∈ [-1, 1], ( , ) = 1 ch - ∞ = 2 +1 (2 + 1)! - .
The successive derivations of with respect to evaluated at = give

( , ) = =0 ! 2 +1 (2 + 1)! .
Regarding [START_REF] Selivanov | Boundary observers for a reaction-diffusion system under timedelayed and sampled-data measurements[END_REF], one obtains zero for < because 2 can be decomposed on the 2 + 1 first Legendre polynomials.

This error is close to zero for near and allows us to expect that it leads to accurate models, which satisfy the definition of the Padé approximants [START_REF] Baker | Padé Approximants, 2 edn[END_REF].

Realization of the ( -1| ) Padé approximant of the transfer function taking support on the first Legendre polynomials

In this part, with the measurement ( ) = ( , 1), the irrational transfer function ( ) of the reaction-diffusion equation from ( ) to ( )

( ) = ( , 1) = th - - , ∀ ∈ C, (25)
is considered. We show the relation between the modeling inspired by tau methods [START_REF] Ito | Legendre-Tau approximations for functional differential equations[END_REF] on Legendre polynomials basis and the ( -1| ) Padé approximated transfer function of the irrational transfer function ( ). Particular state form representation of this nice Padé approximated model is provided. Indeed, the states of this realization are Fourier-Legendre polynomials coefficients of the distributed state ( , ) under the following definition.

Definition 2 For any signal ( ) ∈ L 2 (0, 1; R) satisfying ( 21) and for integer in N, we define,

• the state ( ) on the extended interval [-1, 1] by

( , ) = -( , -), ∀( , ) ∈ R ≥0 × [-1, 0], (26)
• the first Fourier-Legendre coefficients of as

( ) =        ∫ 1 -1 1 ( ) ( , )d . . . ∫ 1 -1 2 -1 ( ) ( , )d        = ℓ | ( ) ∈ R , ∀ ∈ R ≥0 , (27)
• the associated Fourier-Legendre remainder of ( ) at order as

( , ) = ( , ) -ℓ ( )I ( ), ∀( , ) ∈ R ≥0 × [-1, 1]. (28)
Then, we build a finite-dimensional approximation of system (21) taking support on Legendre polynomials coefficients and certify that the approximated model matches with Padé approximants, thanks to Lemmas 6 and 7.

Proposition 3 System A B C 0
is a realization of the ( -1| ) Padé approximant of the transfer function ( ) given by [START_REF] Valmorbida | Stability Analysis for a Class of Partial Differential Equations via Semidefinite Programming[END_REF] with

A = + L + B * C , B = 2 ℓ (1), B * = -2 ℓ (1), C = ℓ (1) I , C * = -ℓ (1)I , ( 29
)
where L , I are given in [START_REF] Das | Robust analysis of uncertain ODE-PDE systems using PI multipliers, PIEs and LPIs[END_REF] and ℓ (1), ℓ (1) recalled in [START_REF] Das | Representation and stability analysis of PDE-ODE coupled systems[END_REF].

Proof By two successive integrations by parts, the dynamics of the coefficients ( ) defined by ( 27) lead to

d d ( ) = ℓ | ( ) + ( ), = -ℓ ( ) + [ℓ ( ) ( , )] 1 -1 + z ( ), = ℓ ( ) + ( ) + ℓ ( ) ( , ) 1 -1 -ℓ ( ) ( , ) 1 -1 , = ( + L ) ( ) + 2 ℓ (1) ( ) -2 ℓ (1) ( , 1).
The signal ( ) = ( , 1) can be approximated by its the truncated Fourier-Legendre series at order and called ( , 1). That implies

       d d ( ) = A ( ) + B ( ) + B * ( , 1), ( ) = ( , 1) = C z ( ) + ( , 1). (30)
In Laplace domain, the error W ( , 1) = ˜ ( ) ( ) with

˜ ( ) = ˜ ( , 1) = ( ) -C ℓ | ( ) . (31)
It is the remainder of the truncated Fourier-Legendre series of evaluated at = 1. System (30) leads to

Z ( ) = ( -A ) -1 (B + B * ˜ ( )) ( ), ( ) = C Z ( ) + ˜ ( ) ( ).
The transfer function ( ) from ( ) to ( ) is

( ) = C ( -A ) -1 B + 1 + C ( -A ) -1 B * ˜ ( ).
Thanks to Lemma 6, we already have ˜ ( ) = → ( ). Then, by application of Lemma 7 given in Appendix with vectors = -B * , = C and matrix = -L which satisfy the expected assumptions, we find

( ) -C ( -A ) -1 B = → ( 2 ). in (28) yields ( , ) = ( , ) -ℓ ( )I d d ( ).
From one side, we need to express using the new system of coordinates, that is reflected in

( , ) = ( , ) + ( , ) = ( + ) ( , ) + ℓ ( ) + ℓ ( ) I ( ).
Applying the differentiation rules of the Legendre polynomials in (7b), the previous expression resumes to

( , ) = ( + ) ( , ) + ℓ ( ) L + I ( ). (35)
On the other side, the expression of d and calcultation details can be found in [START_REF] Bajodek | Stability analysis of an ordinary differential equation interconnected with the reaction-diffusion equation[END_REF]. Thus, collecting (35),(36) and simplifying the term in L + , the partial differential equation verified by is recognized. Proof of (32b): Finally, the first boundary condition is already verified according to Lemma 1 and the second boundary condition is directly obtained by derivation of (28) and evaluation at = 1.

Remark 4 Matrix A given by ( 33) is equal to

A = A B C 0 ⊗ 0 = ( ) ⊗ 0 , (37)
where ⊗ denotes the Redheffer product and ( ) the Padé ( -1| ) approximant of ( ) given by (25) at = .

In the new formulation, the reaction-diffusion equation, which characterizes the dynamics of , is similar to the one of the original system. The only difference relies on the last term in (32b). Even though, it seems at a first sight more complicated, it will appear in the next developments that this new term has no impact on the complexity of the analysis. This is due to the orthogonality of this new term with the Fourier-Legendre remainder, .

Stability analysis

This last section is dedicated to the construction of a numerical tractable stability criterion for system (1), based on Lemmas 3 and 5 and highly related to the properties of the augmented model (32).

Theorem 1 For a given integer in N and any , satisfying < 2 , assume that there exists P in S + +1 + such that matrix By comparison with the Padé approximant of ( ) = ( , 1) = sh( √ ) √ ch( √ ) at = 0 computed by Matlab pade tool, the same transfer function are recognized. Then, according to Remark 4, the approximated model A will be equivalent to the star product of the finite-dimensional model with this Padé ( -1| ) transfer function of the reaction-diffusion equation.

B 2 [ 2 2 ] [ 2 2 2 ] [ 2 2 2 2 ] C 3 2 [ 3 2 7 2 ] [ 3 2 7 2 11 2 ] [ 3 2 7 2 11 2 15 2 ] ( ) 3 +3 5 2 +21 2 +45 +105 21 2 +60 +495 3 +210 2 +4725 +10395 9 4 3 +770 2 +30030 +225225 4 +630 3 +51975 +2027025

On stability results

In this subsection, a standard example is chosen. The stability properties with respect to are investigated with the method proposed in this chapter. The feasibility of linear matrix inequality Ξ ≺ 0 with Ξ given by (38) is determined with feasp function and tested for = 2. The maximal bound which ensures the stability of the system is collected in Table 2 for several values of . One compares this result with expected bounds coming from the stability of an approximated model at order 100 (see tau methods [START_REF] Gottlieb | Numerical analysis of spectral methods[END_REF]). According to Table 2, we can see that an upper bound for parameter has been found to guarantee exponential stability of the origin of system (1). Based on cases < 0, it is important to note that the classical upper bound ( 2 ) 2 can be exceeded thanks to the stabilizable properties of < 0. It is also worth noticing that the proposed test seem to be hierarchical and to converge toward the expected bound of the region of stability in a fast manner (from order 3). At last, the required order seems to be larger as increases (i.e. decreases).

Conclusions

From a Lyapunov approach, a new sufficient condition of stability for a system coupled with a reaction diffusion process is presented in the linear matrix framework.

Our work focused on the rewriting of Bessel and Wirtinger inequalities involved in the Lyapunov analysis of the reaction-diffusion equation and signals linked with Legendre remainder emerged. Then, a dynamical model has been constructed to highlight these signals and the extended state composed of the first Legendre polynomials coefficients. Finally, taking a classical quadratic Lyapunov functional fitted to this new modeling, a linear matrix inequality is directly obtained.

Based on the structure of the result, robust approach could be pursued to construct finite-dimensional controllers stabilizing the original infinite-dimensional system. Observer design by early-lumping might also be investigated.

Derived from the matrix inversion and determinant lemmas, a usefull lemma can also be formulated. ) .

Then, since is strictly lower triangular, we have det(( -) + ) = det(( -) ) = ( -) .

and, because has non-zero under diagonal coefficients and under the hypothesis done on vectors , , matrix + has full rank which means det( + ) ≠ 0. That yields the result for tending to .

Example 1

 1 Let system (1) with = -1, taking values in [-1, 2], = 1 and = 1.

Lemma 7 1 -Proof

 71 For any ∈ R with a non-zero first component, ∈ R not equal to the zero vector and ∈ R × a strictly lower triangular matrix with non-zero under diagonal coefficients (i.e. +1, ≠ 0 ∀ ∈ {1, . . . -1}), one obtains The matrix inversion lemma applied to vectors ∈ R , ∈ R and non singular matrix = + ∈ R × gives, for any ∈ C\{0}, 1 -(( -) + + ) -1 = (1 + (( -) + ) -1 ) -1 , and the matrix determinant lemma leads to 1 -(( -) + + ) -1 = det(( -) + ) det(( -) + +

Table 1

 1 Padé ( -1 | ) approximants of transfer function ( ).

	Order	1	2	3	4	Expected
	= 2 -3.751 -3.668 -3.668 -3.668	-3.668
	= 0 2.461 2.467 2.467 2.467 ( 2 ) 2 = 2.467 = 0.1 2.258 2.263 2.263 2.263 2.263
	= -1 3.176 3.672 3.685 3.685	3.685

Table 2

 2 Maximal allowable reaction coefficient guaranteed by Theorem 1.

According to the definition of Padé approximations given in [START_REF] Baker | Padé Approximants, 2 edn[END_REF], the rational function ( ) = C ( -A ) -1 B is a ( -1| ) Padé approximant of ( ) at = .

Modeling of the interconnected system

Consider now ( , ), solution to system (1) and, resuming Definition 2, denote the first Fourier-Legendre coefficients by and the Fourier-Legendre remainder , for any in N. The objective of this section is to rewrite system (1) by exhibiting a finite-dimensional part composed of the = and an infinite-dimensional part represented by the Fourier-Legendre remainder . This is formulated in the following proposition.

Proposition 4 If ( , ) is a solution of system (1), then ( = , ) defined by [START_REF] Webb | Pointwise and uniform convergence of Fourier extensions[END_REF],(28) verifies the following dynamics

where the matrices that defined this model are given by

with matrices A , B , B * , C and C * are given by (29).

Proof The proof is also split into three parts referring to each equation in (32). Proof of (32a): According to the proof Proposition 3, we already have = ℓ | which satisfies (30). Then, adding the dynamics of the ordinary differential equation given by (1a), the dynamics of the finite dimensional state are

which corresponds to the first equation (32a). Proof of (32b): Let us now focus in the second equation (32b), which refers to the partial differential equation resulting from the changes of variable to . To do so, differentiating with respect to time of the Fourier-Legendre remainder given

is negative definite, where constants , are defined respectively by [START_REF] Kammler | A first course in Fourier analysis[END_REF],( 17) and matrices A , B , C and D are defined by (33). Then, under the conditions prescribed by Propositions 1 and 2, the equilibrium (0, 0) of system (1) is globally exponentially stable, in the sense of the R × L 2 (0, 1; R) norm.

Proof Consider the Lyapunov functional given by

with P in S + +1 + . It is a Lyapunov candidate functional for both systems (1),(32). Differentiation of ( ) = V ( , ) ( ) along the trajectories of the system (1) using the dynamics given by (32) yields

Secondly, from the dynamics of the Fourier-Legendre remainder in (32), denoting

Using integration by parts, this expression is decomposed in L 2 (-1, 1; R) norms of signals and as

Thereafter, the proof is split into two cases. If ≤ 0, the proof is a straightforward application of Bessel-Legendre inequality given by [START_REF] Kammler | A first course in Fourier analysis[END_REF]. Indeed, we have

Assuming Ξ ≺ 0, application of Lyapunov theorem leads to Theorem 1. For the case 0 < < 2 , we apply first the adapted Wirtinger inequality ( 17) on the Fourier-Legendre remainder to obtain

for any sufficiently small > 0 such that (1 -2 -) > 0. By application of the Bessel-Legendre inequality given by ( 14), we finally get to

If the linear matrix inequality Ξ ≺ 0 holds, it is possible to take small enough such that -| ¯ (Ξ )| + < 0. Then, there exists > 0 such that the derivatives d d ( )

One concludes by application of Lyapunov theorem on the exponential stability at the equilibrium point.

Remark 5

Notice that A Hurwitz is a necessary condition for the feasibility of linear matrix inequality Ξ ≺ 0 where Ξ defined by (38). This is promising to the matter of fact that A is an approximated model for the original system (1).

Remark 6 Compared to generalized linear matrix inequality formulations based on sum of squares [START_REF] Das | Representation and stability analysis of PDE-ODE coupled systems[END_REF], the result is condensed, more appropriate to the application and numerical burden are improved. This optimization is due to the transformations made to obtain our linear matrix inequality, highly correlated to the system under study. Nevertheless, to the best of our knowledge, we are not aware of stability condition adressing this particular class of system and this is the reason why no further comparison will be presented.

Application to numerical examples 6.1 On Padé approximations

Without loss of generalization, let take = 0 and = 1. The approximated model A B C 0 leads to a transfer function . For low orders, the corresponding matrices and approximated transfer functions have been stored in Table 1.


Abdellah Kaiba
email: a.kaiba@psau.edu.sa

Mohammed H Geesi

Yassine Riadi

Elmutasim O Ibnouf

Talal A Aljohani

Philippe Guionneau

Keywords: Characterization, Hirshfeld Surface Analysis, DFT Calculation, Vibrational Properties and Biological Evaluation X-ray diffraction, Hirshfeld surface, DFT calculation, Micro-Raman spectroscopy, Antibacterial

HAL is

Introduction

In recent years, organic-inorganic hybrids have become one of the most attractive research topics due to the possibility of integrating the advantageous properties of inorganic and organic molecules, including electrical, optical, magnetic and catalytic properties, as well as luminescence and conductivity. The selection of metals such as Tin (Sn)in medicine for example poses a key issue in terms of its toxicity in comparison with that of lead (Pb) [START_REF] Belhajsalah | Crystal structure, Hirshfeld surfaces computational study and physicochemical characterization of the hybrid material (C7H10N) 2 [SnCl6]• H2O[END_REF][START_REF] Mrad | Elaboration, crystal structure, characterization and DFT calculation of a new Hg (II) inorganic-organic hybrid salt [C6H16N2O] HgCl4[END_REF][START_REF] Bejaoui | DFT calculations, crystal structure, Hirshfeld surface analyses and antibacterial studies of a new tetrachlorocuprate salt:(C6H16N2O)[CuCl4[END_REF][START_REF] Bouacida | Synthesis, crystal structure, hirshfeld surface analysis, DFT calculations and thermal properties of a new anilinium derivative chlorostannate (IV) hybrid compound[END_REF][START_REF] Feddaoui | Synthesis, crystal structure, vibrational spectroscopy, DFT, optical study and thermal analysis of a new stannate (IV) complex based on 2-ethyl-6-methylanilinium (C9H14N) 2 [SnCl6][END_REF][START_REF] Moussa | Structural study, vibrational and optical properties, Hirshfeld surface analysis and DFT investigation of a novel organic cation hexachloridostannate (IV),(C5H8N3) 2 [SnCl6[END_REF]. Owing to their interesting structures, these composites have been intensely studied to better correlate their structure with physical properties [START_REF] Mitzi | Organic-inorganic electronics[END_REF][START_REF] Allan | Atomistic lattice simulations of the ternary fluorides AMF3 (A= Li, Na, K, Rb, Cs; M= Mg[END_REF][START_REF] Papavassiliou | Three-and low-dimensional inorganic semiconductors[END_REF]. These compounds exhibit a perovskite-type structure, where the inorganic layer is composed of an MX 6 octahedra or tetrahedral MX 4 . The cohesion between the interlayers in such compounds in the solid state generally occurs by hydrogen bonding, π-stacking, π-π stacking and van der Waals interactions [START_REF] Lemmerer | Synthesis, characterization and phase transitions of the inorganic-organic layered perovskite-type hybrids [(C n H 2n+ 1 NH 3) 2 PbI 4], n= 7, 8, 9 and[END_REF][START_REF] Abid | Phase transition in organic-inorganic perovskite (C9H19NH3) 2 PbI2Br2 of long-chain alkylammonium[END_REF][START_REF] Vaněk | Phase transitions in (CH3) 4NPbBr3[END_REF][START_REF] Mitzi | Structure and optical properties of several organic-inorganic hybrids containing cornersharing chains of bismuth iodide octahedra[END_REF]. Furthermore, some hybrids of this family exhibit the highest energy conversion efficiency for photovoltaic cells [START_REF] Leguy | Reversible hydration of CH3NH3PbI3 in films, single crystals, and solar cells[END_REF][START_REF] Dang | Bulk crystal growth of hybrid perovskite material CH 3 NH 3 PbI 3[END_REF][START_REF] Heo | Efficient inorganic-organic hybrid heterojunction solar cells containing perovskite compound and polymeric hole conductors[END_REF][START_REF] Liu | Hybrid organic/inorganic nanocomposites for photovoltaic cells[END_REF][START_REF] Zhao | Efficient planar perovskite solar cells based on 1.8 eV band gap CH3NH3PbI2Br nanosheets via thermal decomposition[END_REF][START_REF] Peng | Hybrid organic-inorganic perovskites open a new era for low-cost, high efficiency solar cells[END_REF][START_REF] Prakash | Structural and optical studies of local disorder sensitivity in natural organic-inorganic self-assembled semiconductors[END_REF]. This perovskite sub-family is a fascinating family of functional materials, particularly perovskite based on stannate halide, due to its numerous attractive properties, including photoluminescence, optical, electrical and catalytic properties [START_REF] Aruta | Growth and optical, magnetic and transport properties of (C 4 H 9 NH 3) 2 MCl 4 organic-inorganic hybrid films (M= Cu, Sn)[END_REF][START_REF] Kagan | Organic-inorganic hybrid materials as semiconducting channels in thinfilm field-effect transistors[END_REF][START_REF] Knutson | Tuning the band gap in hybrid tin iodide perovskite semiconductors using structural templating[END_REF].

In addition, interesting aspects of the inorganic and organic chemistry of tin have been discussed in various reviews [START_REF] Chandrasekhar | Large, Switchable Electrochromism in the Visible Through Far-Infrared in Conducting Polymer Devices[END_REF]; these derivatives have attracted considerable attention due to their interesting optical properties [START_REF] Kagan | Organic-inorganic hybrid materials as semiconducting channels in thinfilm field-effect transistors[END_REF][START_REF] Knutson | Tuning the band gap in hybrid tin iodide perovskite semiconductors using structural templating[END_REF], the development of cost-effective electronic devices [START_REF] Kagan | Organic-inorganic hybrid materials as semiconducting channels in thinfilm field-effect transistors[END_REF], agriculture, medicine, anti-fouling paints and biological activities [START_REF] Li | Exogenous stimuli maintain intraepithelial lymphocytes via aryl hydrocarbon receptor activation[END_REF][START_REF] Matela | Reactions of tin-and triorganotin (IV) isopropoxides with thymol derivative: synthesis, characterization and in vitro antimicrobial screening[END_REF][START_REF] Nath | Chemistry and applications of organotin (IV) complexes of Schiff bases[END_REF][START_REF] Mathlouthi | Structural characterization, spectroscopic, thermal, AC conductivity and dielectric properties and antimicrobial studies of (C8H12N) 2 [SnCl6][END_REF].

In this study, phenethylamine (C 8 H 11 N) ligand and tin(II) chloride were used as the organic and inorganic parts, The calculated vibrational spectra were compared to the experimental spectrum to assign the vibration modes and to provide deep insights into the characteristic bonding within the structure. In addition, the antibacterial activity of the synthesized compound was investigated against four bacterial strains.

Experimental section

Synthesis of the title compound

Single crystals of (NH

Instrumental procedures

To examine the purity of the title compound, a K-Alpha XPS instrument (Thermo Fisher Scientific, United Kingdom) equipped with a micro-focused monochromatic Al Ka X-ray source (1486.6 eV) was employed for XPS analysis. A scanning electron microscopy SEM (model Quanta FEG 250, Netherlands) system was employed to observe the grain morphology of this compound. Vibrational modes were investigated on a SENTERRA II Compact Raman Microscope (Bruker, Germany) using a green laser with λ = 532 nm and a power of 0.25 mW in the range 50-4000 cm -1 . Powder XRD patterns as a function of temperature were recorded on an Ultima IV X-ray diffractometer made in Japan operated in the -2 Bragg-Brentano geometry [START_REF] Mittemeijer | Modern diffraction methods[END_REF] equipped with an X-ray source (Cu Kα).

Crystal structure

Single-crystal XRD patterns were recorded on a Nonius Kappa CCD diffractometer with a Mo anticathode. The ϕ and ω scan modes were utilized with a crystal-to-detector distance of 40 mm. The mosaicity degree of the selected crystal is about 0.35°. The structure was solved at room temperature. Data reduction was performed using the DENZO-SMN program. The SIR 97 program was employed to solve the structure [START_REF] Altomare | SIR97: a new tool for crystal structure determination and refinement[END_REF], and the SHELX 97 program was employed for the refinement of all atomic parameters by using the full-matrix least-squares technique based on the squared structure factor F 2 [START_REF] Gm | SHELX-97, release 97-2[END_REF]. Hydrogen atoms were theoretically located and isotropically refined. These programs were utilized within the WINGX package [START_REF] Farrugia | WinGX suite for small-molecule single-crystal crystallography[END_REF]. The DIAMOND [START_REF] Brandenburg | DIAMOND 3.1 a[END_REF] and MERCURY [START_REF] Macrae | Mercury CSD 2.0-new features for the visualization and investigation of crystal structures[END_REF] programs were employed for all graphics.

Minimum inhibitory concentration

The minimum inhibitory concentration (MIC) is referred to as the concentration of the derivative at which bacterial growth is completely inhibited. The MIC was determined by the double microdilution method with sterile flatbottomed 96-well polystyrene plates, as detailed in reference [START_REF] Allan | Atomistic lattice simulations of the ternary fluorides AMF3 (A= Li, Na, K, Rb, Cs; M= Mg[END_REF].

Dimethyl sulfoxide (DMSO) was used as the negative control. The MIC was determined as follows: First, the compound was adjusted to concentrations of 100, 75, 50, 38.5, 25, 18.75 and 12.5 μg/mL in DMSO. Second, 200 μL of the old serially diluted test samples was dispensed into 96-well microtrays. Then, test bacteria were inoculated into the wells, the final volume was adjusted to 400 µL and the plates were incubated at 37°C for 24 h. The concentration at which a marked reduction in colour is observed is referred to as the MIC. All experiments were performed in triplicate.

Computational Details

The unit cell of the crystal structure was directly utilized as the initial molecular geometry of (NH3

-(CH2)2- C6H5)2[SnCl6].
Visualization and exploration of intermolecular contacts of a structure was achieved by Hirshfeld surface analysis [START_REF] Mckinnon | Towards quantitative analysis of intermolecular interactions with Hirshfeld surfaces[END_REF] calculated by using Crystal Explorer software [START_REF] Turner | CrystalExplorer. Version[END_REF]. All calculations were performed using the Gaussian 16w package [START_REF] Frisch | [END_REF]. The Becke's three-parameter exchange functional combined with the Lee-Yang-Parr correlation functional (B3LYP) and LanL2DZ as the basis set were used for optimization and vibrational frequency calculations [START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF][START_REF] Parr | Density functional theory of atoms and moleculesoxford univ[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF][START_REF] Stephens | Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields[END_REF][START_REF] Hay | Ab initio effective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg[END_REF]. An empirical scaling factor of 0.961 [START_REF] Johnson | NIST computational chemistry comparison and benchmark database[END_REF] was utilized to offset the systematic error caused by basis set incompleteness, neglect of electron correlation and vibrational anharmonicity. The Gaussview 6.1.1 program [START_REF] Dennington | [END_REF] was utilized for visualization and presentation as well as the verification of the normal mode assignments.

Results and Discussion

3-1 Powder x-ray diffraction at room temperature:

Powder XRD patterns as a function of temperature were recorded on an Ultima IV X-ray diffractometer operated in the -2 Bragg-Brentano geometry equipped with an X-ray source (Cu Kα). The purity and crystallinity of the sample were examined by powder XRD. Fig. 1 shows the XRD pattern of the title material and the pattern simulated from its structure. Clearly, the powder and simulated diffractograms are superposed that explain that this phase is pure. The Sn-Cl bond lengths of the SnCl 6 dianion ranged from 2.408(2) Å to 2.431(2)Å. The Cl-Sn-Cl bond angles ranged from 88.64 (5)° to 91.67 [START_REF] Moussa | Structural study, vibrational and optical properties, Hirshfeld surface analysis and DFT investigation of a novel organic cation hexachloridostannate (IV),(C5H8N3) 2 [SnCl6[END_REF]° for the chlorine atoms in lateral positions, while those for axial chlorines were close to 180°, exhibiting a small geometrical distortion from the ideal octahedra geometry. The Sn-Cl bonds were in agreement with the corresponding values for similar compounds reported previously [START_REF] Bouacida | Bis (3-hydroxymethylanilinium) hexachloridostannate (IV)[END_REF][START_REF] Rouchene | Synthesis, X-ray diffraction and Hirshfeld surface analysis of two new hybrid dihydrate compounds:(C6H22N4)[SnCl6] Cl-2 center dot 2H (2) O and (C8H24N4)[SnCl6] Cl-2 center dot 2H (2[END_REF]. The distance between neighbouring Snatoms (Sn….Sn) in the inorganic sheet ranged from 6.923 (2) Å to 7.345 (5) Å. The distortion angle of the metal coordination sphere can be estimated by the calculation of the ∑ parameter [START_REF] Guionneau | Crystallography and spin-crossover. A view of breathing materials[END_REF]. In this study, ∑ was equal to 9.46°, indicative of a weak distortion.

Formula weight

For the two organic chains of NH 3 -(CH 2 ) 2 -C 6 H 5 of the unit cell, the carbon atoms exhibited extremely high disorder for the one chain (the equivalent isotropic displacement parameter Ueq values were about 0.102 Å 2 and 0.187 Å 2 ), and a slight disorder for the second chain (i.e. Ueq values were about 0.083 Å 2 and 0.060 Å 2 ), the values of C-C bond lengths distances were of (1.211 Å and 1.526 Å ) and (1.40 Å and 1.50 Å), respectively. The N-C bond lengths were (1.479 Å and 1.482 Å) and C-C-C bond angles ranged from 116° to 125°, respectively. The aromatic ring of the two phenethylaminium cations in the unit cell adopted quite a planar conformation. The torsion angles of N2-C21-C22-C23 and N1-C11-C12-C13 were 174.6° and 178.5°, respectively. The torsion angles of C11-C12-C13-C14, C11-C12-C13-C14, C21-C22-C23-C24 and C21-C22-C23-C28 were 43°, 138.7°, 79° and 102°, respectively, indicating that the ethylaminium group of the two phenethylaminium cations adopts anti-periplanar conformations.

The compound comprised alternating organic and inorganic layers stacked along the b-axis. The inorganic layer was formed by isolated octahedras [SnCl6] (Fig. 5). Hydrogen-bond interactions clearly played a key role in the cohesion between layers and stabilized the crystal structure. Indeed, the N-HCl hydrogen bond was formed between the two layers, with the NHCl distance ranging from 2.686 Å to 2.874 Å. In addition, the CHCl hydrogen-bond distance was about 2.807 Å (Fig. 6).

3-5 Hirshfeld Surface

The analysis of intermolecular interactions using the Hirshfeld surface permitted the understanding of interactions in the crystal packing. The Crystal Explorer software was utilized to generate the Hirshfeld surface [START_REF] Turner | CrystalExplorer. Version[END_REF]. Analysis of the intermolecular contacts with an asymmetric unit (Fig8) reveals all of the interlayer interactions calculated by Hirshfeld The H hydrogen bonds constituted 28.4% of the Hirshfeld surface, corresponding to the second-most frequent contacts due to the presence of hydrogen atoms on the molecular surface. The Cl long interatomic interactions constituted 3.7% in the centre of the 2D fingerprint maps, while the CH contacts represented 13.2% of the total Hirshfeld surface (aromatic ring). From a quantitative viewpoint, HCl contacts were clearly the most abundant contacts in the structure, and they ensured the cohesion of crystal packing. Table 2 summarizes the ratio of their contributions.

Hirshfeld surface analysis is crucial and promising in crystal engineering because it permits quantitative and qualitative analyses by indicating all hydrogen-bond interactions. To obtain more information about the titled compound structure, vibrational modes in the range of 400 and 4500 cm 1

were examined by Raman spectroscopy. To obtain further insights into the structure, Raman spectra were recorded using a green laser at λ = 532 nm and different powers in the range of 0.25 to 25 mW ( Fig 10). Clearly, the peak intensity varied depending on the power. A more resolved spectrum was obtained at P = 6.25 mW. Indeed, the spectra were divided in intervals, i.e. in a low-frequency range of 50-400 cm 1 and a high-frequency range of 400-4250 cm 1 . cm 1 and υa = 3118 cm 1 , which was interconnected by a system of hydrogen bonds in the (N-HCl) crystal [START_REF] Gatfaoui | Synthesis, characterization, Hirshfeld surface analysis and antioxidant activity of a novel organic-inorganic hybrid material 1-methylpiperazine-1, 4-diium bis (nitrate)[END_REF][START_REF] Soudani | Structural, Hirshfeld surface and spectroscopic studies of the noncentrosymmetric 1-ethylpiperazinediium pentachloroantimonate (III) monohydrate[END_REF][START_REF] Kassou | Organic-inorganic hybrid perovskite (C 6 H 5 (CH 2) 2 NH 3) 2 CdCl 4: Synthesis, structural and thermal properties[END_REF].

Symmetric and asymmetric stretching vibrations of (-CH 2 -) were observed between 2800 and 3000 cm 1 . In-and outof-plane deformation bands of the aromatic rings were observed at 706 and 806 cm 1 , respectively. The C-C and C-N stretching modes were observed at 911 and 1007 cm 1 , respectively, while the rocking vibration of the NH

3-7 Antibacterial activity

The antibacterial activity of the synthesized compound was evaluated against two gram-negative bacteria, viz.

Escherichia coli (EC) and Pseudomonas aeruginosa (PA), respectively, and two gram-positive bacteria, viz. Bacillus subtilis (BS) and Methicillin-resistant Staphylococcus aureus (MRSA), respectively, in terms of MIC [START_REF] Wiegand | Agar and broth dilution methods to determine the minimal inhibitory concentration (MIC) of antimicrobial substances[END_REF]. Table 3 summarizes the results.

The as-synthesized compound (NH 3 -(CH 2 ) 2 -C 6 H 5 ) 2 [SnCl 6 ] was active against all bacteria strains, with MIC values comparable to those obtained by using standard antibiotic ciprofloxacin as a reference (Table 3). a MIC in μg/mL.

In addition, morphological changes of the microorganisms by the as-synthesized compound were analysed by SEM.

4-Conclusion

In the summery, the synthesis and characterization of a new single crystal of an organic inorganic hybrid compound (Phenethylaminium) hexachlorostannate In summary, the synthesis and characterization of a new single crystal of an organic-inorganic hybrid compound (phenethylaminium) hexachlorostannate are reported. In addition, its structure is characterized and confirmed by SEM, Raman spectroscopy and XRD. The optimization and assignment of the vibrational modes are calculated by the DFT/B3LYP/LANL2DZ method to obtain clear insights into the examined compound. The compound exhibits an ionic structure, comprising alternating organic and inorganic sheets in the form of a layered perovskite system. 3D Hirshfeld surface analysis reveals that the cohesion and stabilization of the structure predominantly occurs due to HCl and HH hydrogen-bond interactions. The synthesized compound (NH3-(CH2)2-C6H5)2[SnCl6] was highly active against Escherichia coli (EC) and Pseudomonas aeruginosa (PA), while a medium effect was obtained on the rest of the studied bacterial strains.

A

  new Organic-Inorganic hybrid compound (NH3(CH2)2C6H5)2[SnCl6]: Crystal structure, characterization, Hirshfeld surface analysis, DFT calculation, vibrational properties and biological evaluation Abdellah Kaiba, Mohammed Geesi, Yassine Riadi, Elmutasim O. Ibnouf, Talal Aljohani, Philippe Guionneau To cite this version: Abdellah Kaiba, Mohammed Geesi, Yassine Riadi, Elmutasim O. Ibnouf, Talal Aljohani, et al.. A new Organic-Inorganic hybrid compound (NH3(CH2)2C6H5)2[SnCl6]: Crystal structure, characterization, Hirshfeld surface analysis, DFT calculation, vibrational properties and biological evaluation. Journal of Solid State Chemistry, 2021, 304, pp.122587. 10.1016/j.jssc.2021.122587. hal-03435026 1.

  respectively, for the preparation of the (NH3-(CH2)2-C6H5)2[SnCl6] compound. In this study, the main aim was to synthesize a new phase of the (NH3-(CH2)2-C6H5)2[SnCl6] compound, examine its crystal structure and characterize it by spectroscopic techniques. Hirshfeld surface analysis allowed the examination the network of interactions, and density functional theory (DFT) calculations allowed the structure optimization and the vibrational modes attributions.

Figure1:

  Figure1: Comparison of experimental and simulated X-ray diffraction patterns of (NH3-(CH2)2-C6H5)2[SnCl6]

Figure 2 :

 2 Figure 2: The whole XPS spectrum of (NH 3 -(CH 2 ) 2 -C 6 H 5 ) 2 [SnCl 6 ].

  In the SEM micrograph of the (NH3-(CH2)2-C6H5)2[SnCl6] single crystal, a foliated-like structure, as well as enormous crystal fragments with a uniform distribution, and a smooth surface were observed, indicative of good crystallinity for this compound[START_REF] Elmebrouki | Synthesis and characterization of new materials like perovskite [NH3-(CH2) n-NH3] ZnCl4 avec n= 8 et 10[END_REF][START_REF] Saber Lassoued | Structural, vibrational and optical properties of a new self-assembled organic-inorganic crystal (C4H7N2)[END_REF], this result confirms the layered perovskite structure (Fig3).

Figure. 3 :

 3 Figure.3: SEM micrographs of (NH3-(CH2)2-C6H5)2[SnCl6] 3-4 Crystal structure determination The prepared compound crystallized in a monoclinic system with the P 21/c space group and unit-cell dimensions of a = 7.345(5) Å, b = 25.667(5) Å, c = 11.971(5) Å, β = 90.106(5) and Z = 4 parameters. The asymmetric unit comprised two cations of [C6H5-(CH2)2-NH3] + and an isolated hexachlorostannate anion of [SnCl6] 2 (Fig. 4).

Figure 4 :

 4 Figure 4: Asymmetric unit cell of organic-inorganic hybrid (NH3-(CH2)2-C6H5)2[SnCl6]. hydrogen's atoms are omitted for clarity.

Figure 5 :

 5 Figure 5: View of the crystal packing projected along the a-axis. The hydrogen bonds are indicated by dashed lines.

Figure 6 :

 6 Figure 6: View showing the inter layers' cohesion via N-H…Cl hydrogen bonds (dashed lines).

Fig. 7

 7 Fig.7 shows the optimized structure of (NH3-(CH2)2-C6H5)2[SnCl6]. The bond lengths and bond angles were in agreement, particularly for [SnCl6] 2 where deviations were between 0.005 Å and 0.273 Å and between 0.6° and 7.7°, respectively. On the other hand, for the organic chain, the deviations were more notably analysed in detail by the calculated geometry, which was related to the torsion angle of the N-C-C-C group. Indeed, in the organic moiety of the examined structure, the C-C bond lengths in the benzene rings for the cations ranged from 1.406 to 1.413 Å. The C-C-C bond angles ranged from 118.9° to 120.7°. These values were within the range of typical values for single and double bonds. The C-C bond lengths of the + NH3-(CH2)2-group ranged from 1.524 Å to 1.546 Å for the two cations, and their N-C bond lengths were 1.513 Å and 1.518 Å. The torsion NCCC for the two chains of the unit cell were about 170.9° and 57.7°, respectively.

Figure 7 :

 7 Figure 7: Optimized structure of I calculated by the DFT/B3LYP/LANL2DZ method.

  surfaces d norm mapping. Fig 8a clearly shows all of the contributor contacts on the Hirshfeld surface. Red spots represented the highlighted shorter first type of interactions that are rich in N-HCl hydrogen bonds, while the blue and white areas represented the second type of interactions, i.e. HH, CH and ClCl. The red regions on the dnorm surface were in agreement with the observed interactions in the crystal structure. The 2D fingerprint plots permitted the highlighting of the contacts of particular atom pairs to separate the contributions from different overlapping interactions in the full fingerprint [1, 51] ( Fig 8b). Fig 9 shows the fingerprint plot of the hydrogen-bond contacts. The general HCl hydrogen bonds were the most abundant (54.6%) in the crystal packing.

Figure 8 :

 8 Figure 8: Hirshfeld Surfaces mapped with dnorm for the title compound.

Figure 9 :

 9 Figure 9: 2D fingerprint plot (99.9%) of (NH3-(CH2)2-C6H5)2[SnCl6], showing contributions from different contacts.

Figure 10 .Figure11.

 10 Figure 10. Raman spectrum of (NH3-(CH2)2-C6H5)2[SnCl6], using a green laser at 532 nm and at P = 0.25, 2.5, 6.25, 12.50 and 25 mW For the low frequency shown in Fig 11, by referring and essentially comparing with a number of hexachloridostannate(IV) compounds in previous studies[START_REF] Belhajsalah | Crystal structure, Hirshfeld surfaces computational study and physicochemical characterization of the hybrid material (C7H10N) 2 [SnCl6]• H2O[END_REF][START_REF] Bouacida | Bis (3-hydroxymethylanilinium) hexachloridostannate (IV)[END_REF][START_REF] Daszkiewicz | Crystal structure, vibrational and theoretical studies of bis (4-amino-1, 2, 4triazolium) hexachloridostannate (IV)[END_REF][START_REF] Lassoued | Structure, spectroscopic measurement, thermal studies and optical properties of a new hybrid compound of aquapentachloroindoidate (III) complex[END_REF], all vibrational bands corresponding to (SnCl 6 ) 2 were observed at less than 350 cm 1 . An intense band observed at 311 cm 1 corresponded to the asymmetric stretching vibrations of υa(Sn-Cl). The shoulder band observed at 236 cm 1 corresponded to the symmetric stretching

Figure12.Figure 13 :

 13 Figure12. Raman spectrum of (NH3-(CH2)2-C6H5)2[SnCl6] in the range of 500-4250 cm 1 using a green laser of 532 nm at P = 6.25 mW

Fig 14

 14 Fig 14 shows the SEM images of untreated bacteria with a regular rod shape. In contrast, after treatment, bacteria were not visible. Thus, serious damage of the microorganisms is observed.

Figure 14 .

 14 Figure 14. SEM images of K. pneumoniae (a) before treatment and (b) after treatment.

Table 1

 1

	summarizes the results from data measurements and refinement. In 2007, Billing et al. have
	reported another phase of this compound (system (monoclinic), space group (P2 1 /c), unit cell parameters (a =
	12.243(2) Å, b = 7.1124(13) Å, c = 13.777(3) Å) and β = 93.697(3))[35]. Structural data can be obtained free of charge

from the Cambridge crystallographic data centre (No: CCDC 2041258) at www.ccdc.cam.ac.uk/data_request/cif.

Table 1 :

 1 Crystal data and structure refinement details of (NH3-(CH2)2-C6H5)2[SnCl6].

Table 2 :

 2 Summary of the major short contacts and their contribution percentage to the Hirshfeld surface.

	Type of contact	Contribution (%)
	H … Cl/Cl…H	54.6
	H … H	28.4
	H … C/ C … H	13.2
	Cl … Cl	3.7
	3-6 Raman spectroscopy of (NH3-(CH2)2-C6H5)2[SnCl6].

Table 3 .

 3 Antibacterial activity of (NH 3 -(CH 2 ) 2 -C 6 H 5 ) 2 [SnCl 6 ].

			Bacteria
	Derivative		MIC (μg/mL)
		EC	PA	BS	MRSA
	Prepared compound	12.5	12.5	25	25
	Ciprofloxacin	12.5	12.5	12.5	18.75