Daniel Denardi Huff
email: daniel.huff@ufrgs.br

Mirko Fiacchini
email: mirko.fiacchini@gipsa-lab.fr

Jo Ão

Member, IEEE Manoel Gomes Da Silva

D D D D Huff

Huff

D D Huff

J M Gomes Da Silva

Necessary and Sufficient Convex Condition for the Stabilization of Linear Sampled-data Systems under Poisson Sampling Process

Keywords: Sampled-data control, random sampling, mean exponential stability, Linear Matrix Inequalities

This work presents a control design method for linear sampled-data systems whose random sampling intervals form a Poisson process. Unlike a previous result in the literature, the proposed stabilization conditions, based on linear feedbacks of both the state and the past input values, are necessary and sufficient for the mean exponential stability of the system. Moreover, such non-conservative conditions correspond to linear matrix inequalities, implying then that the stabilization problem can be efficiently addressed through semidefinite programming. As a second contribution, the characterization and optimization of the mean exponential convergence rate of the closed-loop system is given in form of a generalized eigenvalue problem. A numerical example illustrates the theoretical results.

I. INTRODUCTION

I N many real-world applications, continuous-time plants are controlled by digital devices in a sampled-data fashion [START_REF] Aström | Computer Controlled Systems. Theory and Design, ser. Information and System Sciences Series[END_REF]. These sampled-data systems are often implemented through a network, where communication protocols are responsible for the transmission of data between computers, actuators and sensors [START_REF] Zhang | Stability of networked control systems[END_REF]. The use of a shared network for different purposes has several advantages, including flexibility and easy of maintenance [START_REF] Zhang | Stability of networked control systems[END_REF]. On the other hand, due to imperfections on the communication channels, the control loop is commonly subject to time-varying, uncertain, sampling intervals, i.e. aperiodic sampling [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF].

In this context, the analysis and design of aperiodic sampled-data systems have been the focus of many works in the last years [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. In [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], for instance, a time-delay method based on Lyapunov-Krasovskii funcionals is considered. A similar idea is developed in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], where the looped-functional method is introduced. Alternatively, a discrete-time approach can be considered, where the behavior of the system's state at the sampling instants is modeled through difference inclusions [START_REF] Hetel | Discrete and intersample analysis of systems with aperiodic sampling[END_REF]- [START_REF] Huff | Stability and stabilization of aperiodic sampled-data systems subject to control input saturation: a set invariant approach[END_REF].

A common feature of the aforementioned references is that they consider a non-stochastic framework, where hard bounds are given for the time-varying sampling interval. However, taking into account that this assumption may not be realistic, some recent works have addressed the stability analysis and stabilization of linear sampled-data systems subject to a random sampling interval, where the corresponding distribution function has possibly unbounded support [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]- [START_REF] Sun | Analysis and synthesis of networked control systems with random network-induced delays and sampling intervals[END_REF]. In particular, the works in [START_REF] Shen | Stabilization for sampled-data systems under noisy sampling interval[END_REF]- [START_REF] Sun | Analysis and synthesis of networked control systems with random network-induced delays and sampling intervals[END_REF] address the stabilization problem (in a stochastic sense) of the discrete-time model that describes the evolution of the system's state at the sampling instants. However, even if closely related, the stability of this discrete-time model is not equivalent to the stability of the corresponding (continuous-time) sampled-data system, as remarked, for instance, in [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]Pg. 222] and [START_REF] Antunes | Volterra integral approach to impulsive renewal systems: application to networked control[END_REF]Pg. 610].

The aim of this work is to propose a control design method that guarantees the stabilization of the continuous-time system, as it is done for instance in [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]. Our approach is, however, based on a convex condition -a Linear Matrix Inequality (LMI). Moreover, compared to [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF], our method provides a stabilization condition that is non-conservative with respect to the considered control law, which is also more general. To derive this condition, we use an instrumental result from [START_REF]Stability of networked control systems with asynchronous renewal links: an impulsive systems approach[END_REF], which deals with the stability analysis of impulsive renewal systems but does not consider the control design problem. As in [START_REF] Tabbara | Input-output stability of networked control systems with stochastic protocols and channels[END_REF] (which does not consider the control design problem either) and [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF], we deal with the case of Poisson sampling, i.e. the sampling intervals form a sequence of independent and identically distributed (i.i.d.) random variables with exponential distribution. Furthermore, as a second contribution, we will show how to adapt the approach to optimize the speed of convergence of the trajectories of the closed-loop system to the origin. This is done through a generalized eigenvalue problem [20, Section 2.2.3], which can be efficiently solved numerically.

The paper is organized as follows. Section II contains basic definitions and the problem formulation. Section III presents the main results related to the necessary and sufficient stabilization condition and to the control design method. A detailed comparison between our approach and the one of [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF] is presented in Section IV. Section V shows a numerical example. At last, some concluding remarks end the paper.

Notation. P[•] denotes probability and E

[•] expectation. For f : R → R n , f (t -) ≜ lim τ →t,τ<t
f (τ) if the limit exists. Given the square matrices A and B, λ max (A) (λ min (A)) is the maximal (minimal) real part of the eigenvalues of A and Diag(A, B) is a block diagonal matrix formed by A and B. The symbol ⋆ denotes a symmetric block when applied as an entry of a matrix and ≻ (⪰) characterizes positive (semi)-definiteness of a symmetric matrix. I n ∈ R n×n is the identity matrix. ∥ • ∥ denotes the induced 2-norm of a matrix or the Euclidean norm of a vector. Re(•) and Im(•) are, respectively, the real and imaginary parts of the argument. Given a matrix (or vector) A ∈ R m×n , A (i) is its i-th row and A T its transpose.

II. PROBLEM FORMULATION

Consider the continuous-time plant described by the following linear model:

ẋp (t) = A p x p (t) + B p u(t) (1)
where x p ∈ R n p and u ∈ R m are the state and the input of the plant, respectively. Matrices A p and B p have appropriate dimensions and are constant. It is assumed that the control input is updated at the time instants t k and kept constant (by means of a zero-order-hold) for all t ∈ [t k ,t k+1 ), being given by:

u(t) = K p x p (t - k ) + K u u(t - k ), ∀t ∈ [t k ,t k+1 ), (2)
where K p and K u are matrices of appropriate dimensions. Notice that the control law is based not only on the sampled value of the state x p but also on the value of the last control input applied to the plant. The term K u u(t - k ) has already showed its benefits in [START_REF] Huff | Stability and stabilization of aperiodic sampled-data systems subject to control input saturation: a set invariant approach[END_REF], where the stabilization problem is solved in a non-stochastic framework.

By convention t 0 = 0 and the difference between two successive sampling instants is denoted by δ k ≜ t k+1t k . It is assumed that {δ k } k∈N is a sequence of i.i.d. random variables with exponential distribution:

F(s) ≜ P[δ k ≤ s] = 1 -e -λ s , ∀k ∈ N, ∀s ≥ 0, (3)
where λ > 0 and E[δ k ] = 1/λ . Thus, the sampling process

N t ≜ sup{k ∈ N : t k ≤ t} (4)
is a Poisson process of intensity λ > 0, where [START_REF] Davis | Markov Models & Optimization[END_REF]Pg. 37]

P[N t = n] = e -λt (λt) n n! . ( 5
)
The Poisson process has been successfully used in the literature to model the stochastic behavior of networked control systems under random sampling [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF], [START_REF] Tabbara | Input-output stability of networked control systems with stochastic protocols and channels[END_REF]. It should be noticed that, since E[N t ] < ∞ (cf. [START_REF] Resnick | Adventures in stochastic processes[END_REF]Pg. 186]) and since

E[N t ] < ∞ only if P[N t = ∞] = 0 (cf. [22, Pg. 2]
), there is zero probability of an infinite number of samplings occurring in finite time (i.e. zero probability of Zeno behavior). Denoting x ≜ [x T p u T ] T ∈ R n , n ≜ n p + m, the dynamics (1)-(2) can be described by the following impulsive system [START_REF] Huff | Stability and stabilization of aperiodic sampled-data systems subject to control input saturation: a set invariant approach[END_REF]:

ẋ(t) = f (x(t)) ≜ A c x(t), ∀t ≥ 0,t ̸ = t k , x(0) = x 0 x(t k ) = g(x(t - k )) ≜ A d x(t - k ), ∀k ≥ 1 (6a) (6b)
where

A d ≜ A r + B r K and A c ≜ A p B p 0 0 ∈ R n×n , A r ≜ I n p 0 0 0 ∈ R n×n , B r ≜ 0 I m ∈ R n×m , K ≜ K p K u ∈ R m×n . (7)
There are alternative impulsive system representations for (1)-( 2). The choice [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], different from the one of [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]Eq. 14], will play a fundamental role to obtain a necessary and sufficient convex condition for the stabilization of the system. Definition 1: The equilibrium point x = 0 of ( 6) is mean exponentially stable (MES) if there exist scalars c, γ 0 > 0 such that for every initial condition

x 0 = x(0) ∈ R n : E[∥x(t)∥ 2 ] ≤ ce -γ 0 t ∥x 0 ∥ 2 , ∀t ≥ 0 ( 8
)
where γ 0 > 0 will be referred to as a decay rate of the trajectories of the system. In particular, it is possible to show that ( 6) cannot be MES if the pair (A p , B p ) in ( 1) is not stabilizable. Note that in this case the states corresponding to the uncontrollable part of (1) will behave deterministically.

We can now state the problem we focus on in this work. Problem 1 (Mean square stabilization): Given the intensity λ > 0 of the Poisson sampling process, characterize in a non-conservative way all the linear feedback gains K such that the resulting closed-loop system is MES.

Additionally, as a second contribution, constructive conditions will be provided to maximize the parameter γ 0 in [START_REF] Lombardi | A predictive control scheme for systems with variable time-delay[END_REF]. Moreover, the case where the intensity λ > 0 of the Poisson process is not perfectly known will also be addressed.

A. Piecewise deterministic Markov processes

System (6) has a deterministic behavior except for the jumps that occur at the random sampling times. From the memoryless property of the Poisson process [START_REF] Davis | Markov Models & Optimization[END_REF]Pg. 36], it follows that x(t) is a Markov process. More precisely, system (6) belongs to the class of piecewise deterministic Markov processes (PDMP) defined in [START_REF] Davis | Markov Models & Optimization[END_REF]Section 24]. This fact allows to establish the following key result, which is a particular case of [18, Theorem 1] and does not require the functions f (x) and g(x) in ( 6) to be necessarily linear.

Theorem 1: Assume that f (x) and g(x) in ( 6) are globally Lipschitz and consider V :

R n → R,V ∈ C 1 , such that E ∑ t k ≤p |V (x(t k )) -V (x(t - k ))| < ∞, ∀p ∈ N, ∀x 0 ∈ R n , (9)
where x(t - 0 ) = x(t 0 ) = x(0) by convention. Then, for t ≥ 0,

E[V (x(t))] = V (x 0 ) + E t 0 UV (x(s))ds , ∀x 0 ∈ R n (10)
where

UV (x) ≜ ∇V T (x) f (x) + λ (V (g(x)) -V (x)) (11)
and ∇V (x) is the gradient of V (x). Relation ( 10) is known as the Dynkin's formula [21, Pg. 31] and can be intuitively interpreted as a stochastic version of the fundamental theorem of calculus. Note that the first term of the right-hand side of ( 11) is the usual time derivative of V (x(t)) along the trajectories of ẋ(t) = f (x(t)) while the second term accounts for the jumps at the sampling instants. Theorem 1 will be later used in the proof of Lemma 2.

III. MAIN RESULTS

In order to solve Problem 1, we will use the following lemma, adapted from [START_REF]Stability of networked control systems with asynchronous renewal links: an impulsive systems approach[END_REF]Theorem 7] to the particular case of interest, i.e. model [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. This result can be proved by applying Theorem 1 to a quadratic function of the state x(t).

Lemma 1: Assume that

2λ max (A c ) < λ (12)
holds. Then system ( 6) is MES if and only if there exists P ∈ R n×n , P = P T ≻ 0, such that

A T c P + PA c -λ P + λ A T d PA d ≺ 0. ( 13
) Remark 1: Since λ > 0, it follows from the particular structure of A c in ( 7) that ( 12) is equivalent to 2λ max (A p ) < λ .

Using Lemma 1, we derive next our main result. Theorem 2: There exists K ∈ R m×n such that the resulting closed-loop system ( 6) is MES if and only if there exist W ∈ R n×n ,W = W T ≻ 0, and Y ∈ R m×n such that

WA T c + A c W -λW ⋆ λ (A r W + B r Y ) -λW ≺ 0 (14)
where K and Y are related by the equation

K = YW -1 .
Proof: The proof is divided into two steps: (I) show that the statement of Lemma 1 still holds even if assumption [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF] is not imposed; (II) show the equivalence between ( 13) and ( 14) using reversible operations. The result of the theorem will then follow directly from (I), (II) and Lemma 1.

Step (I): For the sufficiency part of the result, it suffices to note that (13) directly implies that A T c P + PA cλ P ≺ 0 and, hence, (12) holds automatically.

For the necessity part of the result, we have to show that (13) holds if system (6) is MES. If assumption [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF] holds, then it suffices to apply Lemma 1. Thus, let us focus on the nontrivial case where [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF] does not hold. It turns out, however, that this case will never happen, as we will prove next. In other words, system (6) cannot be MES if 2λ max (A c ) ≥ λ . More precisely, we will show that

E ∥x(t)∥ 2 ̸ → 0 as t → ∞ (15)
for an appropriately chosen initial condition x(0) if 2λ max (A c ) ≥ λ . Denote by σ = a + jb, a, b ∈ R, an eigenvalue of A c (possibly complex but not necessarily) which attains the maximum λ max (A c ), i.e. a = λ max (A c ), and by

v = v 1 + jv 2 , v 1 , v 2 ∈ R n
, a corresponding eigenvector. Consider, as initial condition of (6), x(0) = Re(v), where Re(v) ̸ = 0 without loss of generality. Then, recalling that t 1 is the first sampling instant, it follows that, for t ≥ 0:

E ∥x(t)∥ 2 = E ∥x(t)∥ 2 |t 1 > t P[t 1 > t] + E ∥x(t)∥ 2 |t 1 ≤ t P[t 1 ≤ t] ≥ E ∥x(t)∥ 2 |t 1 > t P[t 1 > t] = E ∥ Re(e σt v)∥ 2 |t 1 > t P[t 1 > t], (16)
where the last equality follows from the fact that x(t) = Re(e σt v) before the first sampling instant t 1 , since it must satisfy (6a). Taking into account that the expression || Re(e σt v)|| 2 is deterministic for a fixed t, the expectation operator can be omitted and, then, after some algebraic manipulations, ( 16) becomes

E ∥x(t)∥ 2 ≥ ∥ Re(e σt v)∥ 2 P[t 1 > t] = e 2λ max (A c )t ∥ cos(bt)v 1 -sin(bt)v 2 ∥ 2 P[t 1 > t],
where we replaced a = λ max (A c ). At last, recalling that t 1 = δ 0 by definition and using (3), one concludes that P[t

1 > t] = 1 -P[δ 0 ≤ t] = e -λt , i.e. E ∥x(t)∥ 2 ≥ e (2λ max (A c )-λ )t ∥ cos(bt)v 1 -sin(bt)v 2 ∥ 2 t→∞ ̸ → 0,
which implies [START_REF] Hu | Stabilization of sampled-data systems with noisy sampling intervals and packet dropouts via a discrete-time approach[END_REF]. Therefore, [START_REF] Lombardi | A predictive control scheme for systems with variable time-delay[END_REF] does not hold for the chosen initial condition and system (6) cannot be MES if 2λ max (A c ) ≥ λ , as we wanted to show.

Step (II): Applying the Schur's complement to (13) and replacing A d = A r + B r K, one gets

A T c P + PA c -λ P ⋆ A r + B r K -(λ P) -1 ≺ 0 (17)
Next, defining W ≜ P -1 and left and right multiplying the resulting inequality by Diag(W, λ I n ), one has

WA T c + A c W -λW ⋆ λ (A r W + B r KW ) -λW ≺ 0 (18)
which corresponds to [START_REF] Shen | Quantized/saturated control for sampled-data systems under noisy sampling intervals: A confluent Vandermonde matrix approach[END_REF] with Y ≜ KW . Remark 2: The use of the feedback term K u u(t - k ) in (2) allows to obtain a LMI condition in [START_REF] Shen | Quantized/saturated control for sampled-data systems under noisy sampling intervals: A confluent Vandermonde matrix approach[END_REF] with the change of variable Y = KW , which would not be possible considering K u = 0 without imposing a particular structure on matrix W . However, this does not mean that the use of K u ̸ = 0 is necessary to stabilize the system, that is, it may exist stabilizing solutions with K u = 0 verifying [START_REF] Shen | Quantized/saturated control for sampled-data systems under noisy sampling intervals: A confluent Vandermonde matrix approach[END_REF].

Next we present some extensions of the main result.

A. Optimization of the decay rate

It is possible to adapt condition [START_REF] Shen | Stabilization for sampled-data systems under noisy sampling interval[END_REF] in order to ensure a decay rate γ 0 in [START_REF] Lombardi | A predictive control scheme for systems with variable time-delay[END_REF], according to the next lemma.

Lemma 2: System ( 6) is MES with decay rate γ 0 > 0 if

∃P = P T ≻ 0 : A T c P + PA c -λ P + λ A T d PA d ⪯ -γP (19)
with γ = γ 0 . Moreover, ( 6) is MES with decay rate γ 0 > 0 only if [START_REF] Tabbara | Input-output stability of networked control systems with stochastic protocols and channels[END_REF] holds for all γ ∈ [0, γ 0 ). Proof: The proof, inspired by the reasoning in [START_REF]Stability of networked control systems with asynchronous renewal links: an impulsive systems approach[END_REF], is shown in the Appendix.

Remark 3: In the result above, if ( 6) is MES with decay rate γ 0 > 0, then [START_REF] Tabbara | Input-output stability of networked control systems with stochastic protocols and channels[END_REF] holds for all γ smaller than and arbitrarily close to γ 0 . The result does not guarantee that [START_REF] Tabbara | Input-output stability of networked control systems with stochastic protocols and channels[END_REF] holds with γ = γ 0 because the matrix P = P(γ) goes unbounded as γ → γ 0 (see the proof of the lemma for the details).

Using Lemma 2, we obtain the following result.

Theorem 3: There exists K ∈ R m×n such that the resulting closed-loop system (6) satisfies [START_REF] Lombardi | A predictive control scheme for systems with variable time-delay[END_REF] with decay rate γ 0 > 0 if there exist W ∈ R n×n ,W = W T ≻ 0, and Y ∈ R m×n such that

WA T c + A c W + (γ 0 -λ )W ⋆ λ (A r W + B r Y ) -λW ⪯ 0 ( 20
)
where K and Y are related by the equation K = YW -1 . Proof: Considering now Lemma 2, it is analogous to the one of Theorem 2.

Using the result above, it is possible to stabilize system (6) and to maximize its decay rate through the following optimization problem: max W,R,γ 0 γ 0 subject to: (20) [START_REF] Davis | Markov Models & Optimization[END_REF] Note that ( 21) can be solved using the bisection method on γ, since ( 20) is a LMI for a fixed value of this parameter and since the values of γ for which [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] can be satisfied correspond to a convex subset (an interval) of the real line.

B. Uncertain sampling rate

In this section we assume that the intensity λ of the Poisson sampling process is not exactly known but satisfies

0 < λ lb ≤ λ ≤ λ ub (22)
for some bounds λ lb and λ ub . Then, it is possible to provide sufficient, though not necessary, stabilization conditions for [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]. Consider the following corollary of Theorem 3.

Corollary 1: Assume that ( 22) holds. Then there exists K ∈ R m×n such that the resulting closed-loop system ( 6) is MES with decay rate γ 0 > 0 if there exist W ∈ R n×n ,W = W T ≻ 0, and Y ∈ R m×n such that

WA T c + A c W + (γ 0 -λ )W ⋆ λ (A r W + B r Y ) -λW ⪯ 0, ∀λ ∈ {λ lb , λ ub } (

IV. COMPARATIVE ANALYSIS

The mean square stabilization problem of system (1) has already been dealt with in [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF] considering the control law

u(t) = K p x p (t - k ), ∀t ∈ [t k ,t k+1 ), (24)
which is a particular case of (2). It should be noticed that the results in [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF] are derived from a different impulsive system, where the error e(t) ≜ x p (t)x p (t N t ) between x p (t) and its last sampled value until time t x p (t N t ) is considered as an additional state variable (see [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]Eq. 14]). In this case, due to a different problem structure, it is not possible to obtain non-conservative convex conditions for the computation of K p . In our method, this problem is overcome by considering a different impulsive representation and a more generic control law, which also depends on the past value of the control input. We briefly recall in this section the method of [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF], which provides only a partial solution to the stabilization problem, since it is based on the assumption below.

Assumption 1: There exist positive definite matrices R = R T and P = P T which solve the algebraic Riccati equation

A T p P + PA p -2PB p R -1 B T p P = -αP, α > 0, (25)
and such that (A p -B p R -1 B T p P) is Hurwitz. Moreover, the matrix P = P(α) has the property that for some p > 2 3 lim sup α↓0 λ max (P)

α p < ∞, (26)
i.e. λ max (P) = O(α p ) as α ↓ 0.

This assumption is rather strong, as recognized in [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]Pg. 239]. Indeed, in general, Assumption 1 is not expected to hold for systems where A p has eigenvalues on the open righthalf complex plane (see Remark 6.4 in [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]). Note that even verifying Assumption 1 is not a simple task. It is necessary not only to solve the Riccati equation ( 25) to obtain the matrix P = P(α) ≻ 0, but also to analyze property (26). It is possible to get some insight about the behavior of the ratio λ max (P)/α p computing it for a grid of values of the pair (α, p) ∈ (0, ∞) × ( 23 , ∞). However, this numerical procedure does not formally guarantee that (26) holds.

In case Assumption 1 is satisfied, the stabilizing control law is given by the result below [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF]Theorem 6.5].

Lemma 3: Suppose that Assumption 1 holds. Then there exists α > 0 (sufficiently small) such that the feedback gain

K p = -R -1 B T p P(α) (27)
where P(α) solves (25) renders the closed-loop system composed by ( 1) and ( 24) mean exponentially stable. Notice that Lemma 3 is non-constructive in the sense that it does not provide a valid value for α > 0, which difficults the use of this result in practice. On the other hand, our method does not have this limitation and can be easily applied, since it relies on semidefinite programming problems. Moreover, the non-conservative result of Theorem 2 is valid in the general case and not only for the class of systems which satisfy Assumption 1.

V. NUMERICAL EXAMPLE

Consider the following system matrices taken from [START_REF] Huff | Stability and stabilization of aperiodic sampled-data systems subject to control input saturation: a set invariant approach[END_REF]:

A p = 0 1 1 0 , B p = 0 -5 , (28)
where λ = 3. Assumption 1 was numerically tested and does not seem to hold, what is in accordance with Remark 6.4 of [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF], since one of the eigenvalues of A p is equal to 1. Thus, the stabilization result of [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF], i.e. Lemma 3, cannot be applied in this case. On the other hand, solving [START_REF] Davis | Markov Models & Optimization[END_REF], one obtains the following feedback gain:

K = 0.2536 0.2574 0.0032 , (29)
which renders the closed-loop system mean exponentially stable with decay rate γ 0 = 0.49. The corresponding Lyapunov matrix P is given by:

P = W -1 =   0.0668 0.0665 -0.221 0.0665 0.0683 -0.222 -0.221 -0.222 0.885   . ( 30
)
Figure 1 shows several trajectories of the closed-loop system in the x p -subspace for some realizations of the sequence {δ k } k∈N , where the initial conditions, depicted by blue circles, satisfy ∥x p (0)∥ = 1. Moreover, the control law ( 2) is initialized with u(0 -) = 0. As one should expect, the trajectories converge to the origin. We recall that x p (t) is continuous. On the other hand, its derivative is discontinuous at the sampling instants (represented by black circles) because of the update of the control input u(t) using the law (2). Trajectories of the closed-loop system (28)-(29) in the x psubspace, where the initial conditions x p (0) are depicted by blue circles and the sampling instants t k by black ones.

VI. CONCLUSIONS

A new control design method for linear sampled-data systems under Poisson sampling, which is based on semidefinite programming, was presented. As discussed above, unlike [START_REF] Tanwani | Stabilization of deterministic control systems under random sampling: Overview and recent developments[END_REF], this approach provides a necessary and sufficient stabilization condition for the system and can be easily implemented numerically. Moreover, unlike [START_REF] Shen | Stabilization for sampled-data systems under noisy sampling interval[END_REF]- [START_REF] Sun | Analysis and synthesis of networked control systems with random network-induced delays and sampling intervals[END_REF], which focus on the discrete-time trajectories of the system, we formally guarantee the stability of the continuous-time system.

An idea of future work consists in considering measurement noise as well as other (and more general) distribution functions for the sampling interval of the system. Besides that, the possibility of packet dropouts can also be explicitly considered, as in [START_REF] Hu | Stabilization of sampled-data systems with noisy sampling intervals and packet dropouts via a discrete-time approach[END_REF], and, in principle, it is also possible to extend the approach to cope with input nonlinearities, like saturation and quantization, as in [START_REF] Shen | Quantized/saturated control for sampled-data systems under noisy sampling intervals: A confluent Vandermonde matrix approach[END_REF].

APPENDIX: PROOF OF LEMMA 2

Since time-varying Lyapunov functions will be applied, it will be convenient to consider the augmented process x(t) ≜ (x(t), v(t)) ∈ R n × R ≥0 , where v(t) = 1, v(0) = v 0 ∈ R ≥0 and v(t) = v 0 +t, which also is a PDMP [START_REF] Davis | Markov Models & Optimization[END_REF]Pg. 84]. In particular, the dynamics of x(t) has the form (6) with f ( x) ≜ ( f (x), 1) = [(A c x) T 1] T and ḡ( x) ≜ (g(x), v) = [(A d x) T v] T . Thus, for a function

W (x, v) = x T Pxe γv , P = P T ⪰ 0, γ ≥ 0, ( 31
)
the Dynkin's formula [START_REF] Fujioka | A switched Lyapunov function approach to stability analysis of non-uniformly sampled-data systems with robust LMI techniques[END_REF] (with the appropriate notation changes) holds with UW (x, v) = x T (PA c + A T c P + γP + λ A T d PA dλ P)xe γv (32) as long as ( 9) is satisfied. To show that (9) holds, notice first that x(t) can be written as

x(t) = Φ(t)x 0 ( 33
)
where Φ(t) ≜ e A c (t-t N t ) A d e A c (t N t -t N t -1 ) . . . A d e A c (t 2 -t 1 ) A d e A c t 1 (34)

is the (random) transition matrix of the system and N t , defined in (4), counts the number of samplings until time t. From these relations, it follows that

∥x(t)∥ ≤ c N t 2 e c 1 t ∥x 0 ∥ (35)
where

c 1 ≜ ∥A c ∥ ≥ 0 and c 2 ≜ ∥A d ∥ ≥ 1. Moreover, for k ≥ 1, ∥x(t k )∥ ≤ c k 2 e c 1 t k ∥x 0 ∥ ∥x(t - k )∥ ≤ c k-1 2 e c 1 t k ∥x 0 ∥ ≤ c k 2 e c 1 t k ∥x 0 ∥. (36)
Thus, given p ∈ N and

(x 0 , v 0 ) ∈ R n × R ≥0 , one has E ∑ t k ≤p |W (x(t k ), v(t k )) -W (x(t - k ), v(t - k ))| ≤ E ∑ t k ≤p e γv(t k ) (x T (t k )Px(t k ) + (x T (t - k )Px(t - k )) ≤ (36) E ∑ t k ≤p e γv(t k ) 2∥P∥(c k 2 e c 1 t k ∥x 0 ∥) 2 ≤ E ∑ t k ≤p e γv(p) 2∥P∥(c k 2 e c 1 p ∥x 0 ∥) 2 = e γ(v 0 +p+2c 1 p) 2∥P∥∥x 0 ∥ 2 E N p ∑ k=0 c 2k 2 ≜ CE N p ∑ k=0 c 2k 2 = C ∞ ∑ j=0 P[N p = j] j ∑ k=0 c 2k 2 = C ∞ ∑ k=0 c 2k 2 ∞ ∑ j=k P[N p = j] = C ∞ ∑ k=0 c 2k 2 P[N p ≥ k] = C ∞ ∑ k=0 c 2k 2 P[t k ≤ p] < ∞
where the order of summation was changed in the third-to-last equality, and the last inequality follows from [

(x, v) ∈ R n × R ≥0 : x T (A T c P + PA c -λ P + λ A T d PA d )xe γv ≤ -γx T Pxe γv , which implies, according to (32), that UW (x, v) ≤ 0, ∀(x, v) ∈ R n × R ≥0 . Thus, applying Theorem 1, one has E[W (x(t), v(t))] = W (x 0 , v 0 ) + E t 0 UW (x(s), v(s))ds ≤ W (x 0 , v 0 ), ∀(x 0 , v 0 ) ∈ R n × R ≥0 . (37) Considering v 0 = 0 (which means that v(t) = t), (37) becomes E[x T (t)Px(t)e γt ] ≤ x T 0 Px 0 , ∀x 0 ∈ R n or, equivalently, E[x T (t)Px(t)] ≤ x T
0 Px 0 e -γt , ∀x 0 ∈ R n . Consequently, using the fact that P is positive definite, one concludes after some algebraic manipulations that E[∥x(t)∥ 2 ] ≤ λ max (P) λ min (P) e -γt ∥x 0 ∥ 2 , ∀x 0 ∈ R n , and the result follows.

(Necessity part) Given γ ∈ [0, γ 0 ), consider W : R n × R ≥0 → R ≥0 defined by W (x 0 , v 0 ) ≜ E (x 0 ,v 0 ) ∞ 0 ∥x(s)∥ 2 e γv(s) ds (38)

where E (x 0 ,v 0 ) emphasizes that the initial condition considered is (x(0), v(0)) = (x 0 , v 0 ) with probability one (the subscript will be omitted from now on). Since ( 8) holds (by assumption, ( 6) is MES with decay rate γ 0 ), W (x 0 , v 0 ) is indeed well defined (i.e. it is finite). More precisely, interchanging expectation with integral operations and, then, replacing v(s) = v 0 + s, one has where we used the fact that x(t) = e A c t x 0 before the first sampling time t 1 and applied Exercise 3.17 of [START_REF] Khalil | Nonlinear Systems[END_REF], which provides a lower bound for ∥x(t)∥. Fix now a constant (deterministic) value t * > 0 and observe that W (x 0 ,v 0 ) ≥ e γv 0 ∥x 0 ∥ where we applied (3) (recall that t 1 = δ 0 by definition) and L ≜ e -λt * t * 0 e -2∥A c ∥s ds > 0. Comparing (40) and (41), it follows that P ≻ 0, as claimed.

W (x 0 , v 0 ) =
Applying Theorem 32.2 of [START_REF] Davis | Markov Models & Optimization[END_REF] to (38), we conclude that UW (x 0 , v 0 ) = -∥x 0 ∥ 2 e γv 0 (42)

Combining (42) (with (x 0 , v 0 ) replaced by (x, v)) and (32) and since these eqs. hold for all x ∈ R n and v = 0, it follows that PA c + A T c P + γP + λ A T d PA dλ P = -I n ≺ 0, that implies [START_REF] Tabbara | Input-output stability of networked control systems with stochastic protocols and channels[END_REF].

  [START_REF] Khalil | Nonlinear Systems[END_REF] where K and Y are related by the equation K = YW -1 .

Fig. 1 .

 1 Fig. 1.Trajectories of the closed-loop system (28)-(29) in the x psubspace, where the initial conditions x p (0) are depicted by blue circles and the sampling instants t k by black ones.

∞ 0 E 0 E 0 E 1 0 1 0 1 0e

 000111 ∥x(s)∥ 2 e γv(s) ds = e γv 0 ∞ ∥x(s)∥ 2 e γs ds.(39)At last, applying (8) one getsW (x 0 , v 0 ) ≤ e γv 0 ∞ 0 ce -(γ 0 -γ)s ∥x 0 ∥ 2 ds = ce γv 0 γ 0γ ∥x 0 ∥ 2 < ∞.Moreover, note from (33) and (39) thatW (x 0 , v 0 ) = e γv 0 x T 0 ∞ [Φ T (s)Φ(s)]e γs ds x 0 ≜ e γv 0 x T 0 Px 0 (40)Let us find a lower bound for W (x 0 , v 0 ) to show that P = P T is positive definite. Consider again (39) and note thatW (x 0 , v 0 ) = e γv 0 E ∞ 0 ∥x(s)∥2 e γs ds ≥ e γv 0 E t ∥x(s)∥ 2 ds = e γv 0 E t ∥e A c s x 0 ∥ 2 ds ≥ e γv 0 ∥x 0 ∥ 2 E t -2∥A c ∥s ds

  Assume that[START_REF] Tabbara | Input-output stability of networked control systems with stochastic protocols and channels[END_REF] holds and define W (x, v) as above, i.e. W (x, v) ≜ x T Pxe γv . Left and right multiplying (19) by x T and xe γv , respectively, one gets, for all

	22, Theorem
	3.3.1]. Consequently, (9) holds, as we wanted to show, and
	Theorem 1 can indeed be applied to function (31).
	(Sufficiency part)

  -2∥A c ∥s ds t 1 > t * P[t 1 > t * ] ≥ e γv 0 ∥x 0 ∥ 2 E

			t 1	e -2∥A c ∥s ds
			0
	≥ e γv 0 ∥x 0 ∥ 2 E	0	t 1 e t * e -2∥A
		0

2 E c ∥s ds t 1 > t * P[t 1 > t * ] = e γv 0 ∥x 0 ∥ 2 t * 0 e -2∥A c ∥s dse -λt * = Le γv 0 ∥x 0 ∥ 2 (41)

This work was supported by the Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -(CAPES, Brazil) -Finance Code 001, Conselho Nacional de Desenvolvimento Científico e Tecnol ógico -(CNPq, Brazil) -PQ307449/2019-0, and by the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02).


Aydar Rakhmatullin

Vasiliy V Brusko

Elmira R Shcherbitskaya

Ilya B Polovov

Rinat Bakirov

Catherine Bessada

Solid State 31 P and 109 Ag CP/MAS NMR as a powerful tool for studying of Silver(I) Complexes with N-Thiophosphorylated Thiourea and Thioamide ligands

Keywords: Solid state NMR, 31 P and 109 Ag NMR, N-Thiophosphorylthiourea, Silver(I) complexes, Polynuclear complexes

A family of three-and four-coordinated silver(I) complexes of formulas [Ag(PPh3)2L], [Ag(PPh3)L] and [AgL]n with N-thiophosphorylated thiourea and thioamide ligands of general formula RC(S)NHP(S)(OPri)2 [R = Ph, PhNH, iPrNH, tBuNH, NH2] have been studied by solid-state 109 Ag and 31 P CPMAS NMR spectroscopy. 109 Ag NMR spectra have provided valuable structural information about Ag coordination, which is in good accordance with the available crystal structure data. The data presented in this work represent a significant addition to the available 109 Ag chemical shifts and chemical shifts anisotropies. The silver chemical shift ranges for different P,S-environments and coordination state were discussed in detail. The 1 J( 31 P-107/109 Ag) and 2 J( 31 P-31 P) values were determined and analyzed.

INTRODUCTION

In spite of the fact that there are many new developments and frequent applications of different silver containing materials, the silver solid-state NMR spectroscopy is rarely used and rarely mentioned. To the best of our knowledge, the more recent review we found was published over fifteen years ago. [1,2] There are three major reasons for this rareness: low gyromagnetic ratios (γ), low receptivity, and the rather large longitudinal relaxation time. Low gyromagnetic ratios leads to the acoustic probe ringing which obliterates FIDs. Low receptivity and the rather large longitudinal relaxation time make the acquisition of silver NMR spectra very challenging. Let us additionally note other experimental difficulties that we encountered in the realization of this work and which were not extensively discussed in previous in works: a wide and unknown for different environments the silver chemical shift range, which associated in solid state with high values of chemical shift anisotropy parameter.

Currently, coper(I) complexes attract a special interest as promising emitting materials for OLED devices, solar cells, photocatalysts and bioimaging agents. [3] Only a limited example of silver(I) complexes exhibit phosphorescence at room temperature. [4] Unlike to the related Cu (I) complexes, the energetically lower d-orbitals of silver(I) lead to a larger band gap and, consequently, cause a change of spectral band position in the absorption and emission spectrum.

Contrastingly, to Cu (I), the higher oxidation potential of Ag (I) due to the greater stabilization of d-electrons leads to the diminution the probability of electronic transitions in Metal-to-Ligand Charge Transfer. Therefore, Ag (I) luminophores exhibit less a delayed fluorescence than their Cu (I) analogs. [5] Herein, we report upon the solid-state 31 P and 109 Ag NMR characterization of a series of metallocomplexes, in which the silver has the coordination number of three or four and surrounded by sulfur and phosphorus atoms. N-(Thio)phosphorylated derivatives of the thioamides and thioureas of general formula RC(S)NHP(X)(OR`)2 (TPT, X = O, S) attract attention due to their versatile complexing properties [START_REF] Sokolov | Transition Metal Chemistry: New Research[END_REF] and easy structure modification by changing the nature of X donor site and structure of the R-moiety. [START_REF] Sokolov | [END_REF] Changing of these key sites result in dramatically changes of the coordination properties of the TPT ligands. Besides the obvious factors concerned the nature of the ligand donor atom X, the structure of such complexes can be determined by: 1) electronic effects of the substituents, even quite far from the coordination core, 2) influence of the intramolecular interactions in the ligand structure, e. g. previously reported studies. [8][9][10] SCHEME 1. The structural formulas of the studded silver complexes.

Numerous silver complexes in solution undergo a rapid ligand exchange on the NMR timescale. [1] This phenomenon hides useful information about spin-spin couplings and gives average chemical shifts. The solid-state NMR technique has proved to be a significant tool for the investigation of local structure around a silver core, without limitations due to a dynamics. Magic angle spinning (MAS) NMR on metal nuclei is very useful because it gives direct information about central atom of a complex: coordination numbers, coordination environments, or the distortion of coordination environments. EXPERIMENTAL 31 P and 109 Ag solid-state MAS NMR were carried out on a Bruker Avance 400 (9.4 T) spectrometer using a 4 mm and 7 mm probe-heads respectively. The 1 H-31 P cross-polarization (CP) MAS NMR spectra were recorded at a spinning frequency of 14 kHz using a ramped cross- polarization [11] with a contact time of 1 ms and a recycle delay of 2 s. 1 H decoupling was achieved using the SPINAL-64 sequence [12] with a 1 H nutation frequency of about 60 kHz.

Quantitative MAS spectra were acquired using a single pulse of 0.5 s duration (/12) and a recycle delay of 1 s. Chemical shift is referred to 85 % H3PO4 ( 31 P) solution. For the compounds 4 and 6, additionally, 31 P MAS spectra were carry out on a Bruker Avance III 750 (17.6 T) using a 4 mm probe. 109 Ag magic angle spinning experiments were performed with sample spinning rates 3 kHz. Cross-polarization experiments used contact times of 20 ms and a recycle delay of 2 s, number of scans was between 10000 and 420000 depending on the signal-to-noise ratio required. 1 H decoupling was achieved using the SPINAL-64 sequence with a 1 H nutation frequency of about 60 kHz. The 109 Ag chemical shift of solid AgSO3CH3 (87.2 ppm) was used as a reference. [13] For the compounds 1, 7, and 8, additionally, the data were collected at rotor frequencies of 2.7 and 3.3 kHz. The different spinning speeds were used to distinguish spinning sidebands (ssb) from isotropic peaks. All MAS NMR spectra were modeled using the Dmfit program. [14] RESULTS

P solid state NMR.

The structures of the eight crystalline complexes were characterized by 31 P and 109 Ag solidstate MAS NMR spectroscopy. Modeling of NMR spectra using the Dmfit program has allowed to obtain chemical shifts, J-coupling constant values, and anisotropy parameters (Table 1 and2).

Crystal structures determined by single crystal X-Ray diffraction are known for some of these complexes namely for 1, [15] 2, [15] 3, [16] 7, [9] and 8. [10,17] In the 1 H-31 P CPMAS spectrum of 1 (Figure 1Sa), the resonance of the P atom of the thiophosphoryl group appears as a single peak. At the same time, the triphenylphosphine group gives a doublet owing to the direct J-couplings with the both 107 Ag and 109 Ag isotopes. Due to relatively small isotope shift of the 31 P signal and the close value of the coupling constants (the ratio J-couplings is proportional to the ratio they gyromagnetic ratios γ 107 Ag/γ 109 Ag=0.87), the subspectra of the molecules with 107 Ag and 109 Ag nucleus demonstrate no splitting in the experimental conditions. The overall 1 J( 31 P-107,109 Ag) values are observed in the most of our samples (the only exception is the complex 4, see later.) and also in the previous studies. [18,19] The 1 H-31 P CPMAS spectra of 2, 3, and 5 show singlet signals for the thiophosphoryl group and two doublet of doublets for the inequivalent PPh3 phosphorus atoms, which are a result of 1 J( 31 P-107/109 Ag) and 2 J( 31 P-31 P) coupling (Figure 1S). The NMR measurements fit with the Xray studies of the mononuclear complex 2, containing three different P-sites. The X-ray data shows that the unit cell has two identical asymmetric units, which contain two molecules of 2 each. [15] The values of the 31 P isotropic chemical shifts, 2 J( 31 P-31 P), and overall 1 J( 31 P-107/109 Ag) coupling constants confirm a similarity of structure of three cores of complexes 2, 3, and 5 (Table 1). Crystal structure of the close analog of these compounds of formula [Ag(HL a )(PPh3)2]OTf (9) was reported in 20. The complex 9 has the similar AgP2S2 tetrahedral environment around the silver(I) cation. The neutral form of the ligand HL a demonstrates the same S,S`-bidentate coordination mode and the similar geometry of the ligand moiety.

1 H-31 P CPMAS spectrum of complex 4 contains two independent sets of signals of the same intensity and similar splitting patterns (Figure 1). Each of the sets has one singlet for the thiophosphoryl group and two doublets of doublets for two independent PPh3 groups. To improve the resolution and to have unambiguous fit parameters, additionally, the compound 4 was measured at second magnetic high field 17.6 T (Figure 2S). Then, the both spectra were modeled using a single consistent set of parameters. These sets of signals could be assigned to the two isomers in the asymmetric unit of the compound 4. This crystal structure can be created due to the formation of the more dense crystal packing as well as different non-valent intermolecular interactions. It should be noted that such structural motif is well known in different organic and metalloorganic compounds, see for example, the review [21] or the recent work about the crystal structure of silver complex. [22] There is another possibility of explaining signal doubling what is the coexistence of two polymorphs in the sample, which is also a common occurrence. Unfortunately, there is no any X-Ray data for this compound up to now so we can't prove our assumption. different isotopes of silver 1 J( 31 P-107 Ag), 1 J( 31 P-109 Ag) and between themselves 2 J( 31 P-31 P).

Resulting multiplet was solved by Dmfit taking into account the natural abundances of the 107 Ag (51.82 %) and 109 Ag (48.18 %) isotopes. The cyclic tetramer [Ag4L a 4] 7 in a crystal possesses S4-symmetry, and all four phosphorus atoms are crystallographically equivalent. [9] The singlet signal is observed in the 1 H -31 P CPMAS spectra (Figure S1e). Therefore, our NMR observation is in exact agreement with the X-ray data.

The another cyclic tetramer [Ag4L4] with L = o-MeO(O)CC6H4NHC(S)NP(S)(OPr-i)2 has a similar core with S4-symmetry. [23] It is interesting to note that the copper analog of 7 in crystal contains a similar cyclic tetramer of the same formula [Cu4L a 4] but with lower symmetry S2. [24] The 1 H-109 Ag CPMAS NMR spectra for the obtained chelates (1-6) represents an expected multiplet patterns in total agreement with the obtained 1 H-31 P CPMAS data. There is doublet of doublets in the spectra of all [Ag(PPh3)2L] chelates (2, 3, 4, and 5) due to two 1 J( 31 P-109 Ag) with the two inequivalent PPh3 phosphorus units (Table 2 and Figure S3). The trigonal complex 1

shows one doublet with a significantly higher 1 J value compared to the tetrahedral analogues.

Unlike in previous experiments, this doublet has a several intense spinning sidebands due to the large chemical shift anisotropy of AgS3 site (Figure S4). The spectrum of 6 exhibits two set of doublet of doublets (Figure S3). All measured 1 J( 31 P-Ag) values (329-380 Hz) are significantly lower in comparison with J values (643-733 Hz) reported for mixed carboxylate bis(diphenylphosphino)methane complexes with AgO2P and AgO3P coordination of the metal atom from solid state and solution 31 P NMR experiments. [25] The closer data (203-330 Hz) were found for the complexes of silver(I) nitrate with 1,2-bis(di-n-pyridylphosphino)ethane (dnpype) with AgP4 coordination core. [26] As noted above, in phosphorus spectra the overall 1 J( 31 P-107,109 Ag) values are observed in the most cases (Table 1). To compare the J coupling constants obtained from 31 P and 109 Ag spectra, the following calculations were made. The 1 J( 31 P-107/109 Ag) coupling constants observables from 31 P spectra can be present as 1 Jover= 1 J( 31 P-107 Ag)×Nat.ab. 107 Ag+ 1 J( 31 P-109 Ag)×Nat.ab. 109 Ag, where Nat.ab. 107 Ag and Nat.ab. 109 Ag are natural abundance for 107 and 109 isotopes, respectively. In consideration of 1 J( 31 P-107 Ag)= 1 J( 31 P-109 Ag)× γ 107 Ag/γ 109 Ag, where γ 107 Ag and γ 109 Ag are gyromagnetic ratios for 107 and 109 isotopes, respectively. We obtain 1 Jover= 1 J( 31 P-109 Ag)×γ 107 Ag/ γ 109 Ag×Nat.ab. 107 Ag+ 1 J( 31 P-109 Ag)×Nat.ab. 109 Ag. And finally by substituting the values of Nat.ab. 107 Ag=51.82 %, Nat.ab. 109 Ag=48.18 % and γ 107 Ag/γ 109 Ag=0.87 in equation, we get 1 Jover=0.932634× 1 J( 31 P-109 Ag). The values 1 J( 31 P-109 Ag) were obtained directly from 109 Ag NMR spectra (Table 2). The calculated values 1 Jover are presented in the Table 1 and they are in a very good agreement with the values obtained from the fitting of the experimental 31 As stated above, 31 P spectrum contains three lines. Thereby, 31 P NMR data are definitely useful for prediction of only [Ag3L3] aggregates formation in the crystal but not the association of the trimeric fragment in the symmetric structures of a higher order. Similar 31 P spectral pattern (like at Figure 3) was observed earlier both for the copper(I) trimer containing in the crystal [Cu3L3], L=3-MorphC(S)NP(S)(OPr-i)2 cyclic moiety [27] and for the copper(I) centrosymmetric hexamer of formula [(Cu3L c 3)2], which consists of two Cu3L c 3 cyclic units bonded by two Cu-S bridges via thiophosphoryl sulfur atoms. [28] It should keep in mind that in the trimer complex all copper atoms are three coordinated, while in the hexamer the copper/silver atoms have coordination number of 3 and 4. At the low MAS frequency (3 kHz) solid-state 1 H-109 Ag CP MAS NMR spectrum showed three set of signals due to the strong effect of silver chemical shielding anisotropy (CSA) (Figure 5). AgS4 site is more symmetrical and it has a different coordination than AgS3 sites. It leads to different values of the CSA and chemical shift, respectively. See, for example, the NMR parameters for the complexes given above 1-6. We assigned the signal with higher values of chemical shift at 1160 ppm and with lower values of shift range: 1378-1468 ppm. [26] More electronegative phosphorus than sulfur leads to deshielding of Ag and signal appears at downfield. Fijolek et al. studied AgS2, AgS3, and AgS4 coordinations in silver thiolates and in silver butanethiolate and they found 826 ppm for AgS2 and range 950-1230 ppm for AgS3. [29,30] Our measurements show chemical shift values from 1007 ppm to 1096 ppm and 1141 ppm for AgS3 and AgS2P planar trigonal coordination, respectively. To summarize, the silver isotropic chemical shifts tend to increase for increasing coordination state and for more electronegative phosphorus than sulfur environments. Narula et al. studied silver-substituted yeast metallothionein and found a wide range of 109 Ag chemical shifts, for a protein in which Ag(I) ligation occurs exclusively through cysteine thiolates. They could only assume the presence of mixed Ag(I) coordination numbers. [31] Based on our and previous studies the assignment of seven resonances ranged from 790 ppm to 1250 ppm to different coordination state can be provided and number of different Ag(I) sites can be quantified.

Conclusions

The 31 P and 109 Ag CPMAS NMR data have shown their undoubted usefulness for the studying of the obtained silver(I) chelates in the solid phase. Silver solid state NMR can suggest the structure of polynuclear aggregates or their key structural fragments in the crystalline phase.

Based on experimental and on literature data, the silver chemical shift ranges for different P,Senvironments and coordination state were established. The overall 1 J( 31 P-107,109 Ag) and 1 J( 31 P-109 Ag) values obtained by both 31

hydrogen bonds, 3 )

 3 steric factors. Moreover, the influence of these secondary factors can have crucial importance. The structural formulas of eight studied silver(I) complexes of formulas [Ag(PPh3)2L], [AgL]n and [Ag(PPh3)L] with TPT ligands of general formula RC(S)NHP(S)(OPri)2 [R = Ph L a , iPrNH L b , NH2 L c , tBuNH L d , PhNH L e ] are presented in Scheme 1. The synthesis are described in

FIGURE 1 .

 1 FIGURE 1. (a) Solid state 1 H-31 P CPMAS NMR spectrum recorded at 9.4 T of the complex 4.

FIGURE 2 .

 2 FIGURE 2. The solid state 1 H-31 P CPMAS NMR spectrum of the complex 6 (a). Expansion of

  -31 P CPMAS spectrum of polynuclear silver complex 8 is shown in Figure3. The spectrum contains three resolved resonances with relative intensities in the ratio 1:1:1. The molecule of centrosymmetric hexanuclear complex [Ag6L b 6] 8 consists of two coaxial [Ag3L b 3]cycles interlinked by two Ag-S bonds. The aggregation is provided by the thiocarbonyl sulfur, all of the thiophosphoryl sulfur atoms in the structure form only one Ag-S bond. These two Ag-S-Ag-S-Ag-S cyclic moieties have an envelope conformation with the deviation of the fourcoordinate S atoms from the main plane of the cycle. All three P-S moieties in each cycle are crystallographically inequivalent. Thus, the 1 H-31 P CPMAS spectrum clearly fits well with a crystal structure and the ratio corresponds to the three inequivalent P-S moieties per molecule.

FIGURE 3 .

 3 FIGURE 3. On the left, 1 H-31 P CPMAS NMR spectrum of complex 8 (blue) and reconstructed

  complexes of general formula [Ag(PPh3)2L] the 109 Ag isotropic chemical shift values keep within 1301-1342 ppm range (Table 2), and even at a low rotation rate of 3 kHz the effect of silver chemical shielding anisotropy is negligible. The δAg signals of compounds 1 and 7 with the trigonal coordinated silver(I) cation are shifted upfield on approximately 200 ppm.

FIGURE 4 .

 4 FIGURE 4. Experimental (bleu) and simulated (red), 1 H-109 Ag solid state CPMAS NMR spectra of complexes [Ag4L a 4] 7.

FIGURE 5 .

 5 FIGURE 5. 109 Ag CPMAS NMR experimental spectrum of complex [AgnL b n] 8 (a) and its

  P and 109 Ag NMR measurements agree. The knowledge obtained in this study will propel the field of silver NMR and provide new insight into the design of new silver(I) complexes. ACKNOWLEDGMENT Financial support from the IR-RMN-THC Fr3050 CNRS for conducting the research is gratefully acknowledged. This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program (Priority-2030).

TABLE 1 .

 1 31 P isotropic chemical shifts and 1 JP-Ag and 2 JP-P coupling constants of Ag complexes.

			1 J( 31 P-			Integral
		δiso, ppm		1 Jover,	2 J( 31 P-31 P), Hz
	compound		107,109 Ag), Hz			intensity, %
		(±0.1 ppm)		Hz	(±2 Hz)
			(±2 Hz)			(± 1%)
		56.8 [AgL a ]	-	-	-	50
	1
		12.8 [PPh3]	497	495.6	-	50
		61.8 [AgL a ]	-	-	-	33
	2	5.4 [PPh3]	353	355.3	95	32
		-1.6 [PPh3]	285	287.3	95	35
		63.6 [AgL b ]	-	-	-	33
	3	5.7 [PPh3]	340	344	123	32
		1.4 [PPh3]	290	295	123	35
		59.1 [AgL c ]	-	-	-	33
			330 ( 107 Ag)
		6.2 [PPh3]		-	108	32
	4		379 ( 109 Ag)
			290 ( 107 Ag)
		4.6 [PPh3]		-	108	35
			331 ( 109 Ag)
		62.1 [AgL d ]	-	-	-	33
	5	2.7 [PPh3]	342	340	91	32
		-4.4 [PPh3]	276	280	91	35
		64.5 [AgL e ]		-	-	17
	6
		9.6 [PPh3]		306	125	15

TABLE 2 .

 2 109 Ag isotropic chemical shifts and 1 J( 31 P-109 Ag) coupling constants of Ag complexes.

							Integral
	Compou	Coordinati	δiso, ppm	1 J( 31 P-109 Ag),	Δcsa, ppm	ηcs
							intensity, % (±
	nd	on	(±0.1 ppm)	Hz (±2 Hz)	(±10 ppm)	(±0.1)
							1%)
	1	AgS2P	1141	531	668	0.83	100
				308	-	-
	2	AgS2P2	1358				100
				380	-	-
				317	-	-
	3	AgS2P2	1331				100
				369	-	-
				330	-	-
	4	AgS2P2	1301				100
				379	-	-
	5	AgS2P2	1342	301	-	-	100

369

--


M V D'angelo
email: vdangelo@fi.uba.ar

L Pauchard

H Auradou

B Darbois Texier

Shaping gels and gels mixture to create helices Shaping gels and gels mixture to create helices

In cooking, food gels such as agar-agar or alginate are often prepared and presented in the form of spheres or spaghetti. While experimenting in our kitchen, we realized that it is quite difficult to make more advanced shapes. In this study, we sought to develop new methods to obtain more complex shapes. Our first challenge was to obtain helices. The best method we selected was to deposit the solutions before their gelation in a thread. The robustness of the method is tested by systematically changing the thread pitch, diameter and depth. From the deformation under its own weight, we propose to deduce the mechanical characteristics of the helix. These values are compared to those obtained in the laboratory using indentation testing. Finally, we experimented with mixed gels obtained by combining agar-agar and alginate.

I. INTRODUCTION

When the first lockdown started, we wanted to take this opportunity to finally get into molecular cooking which mixes culinary art and science. It was the ideal meeting place for science enthusiasts and cooking lovers. It was also an opportunity to exchange with colleagues and to explore the connections between fluid mechanics and the food industry. And perhaps most of all, it gave us the opportunity to keep the children busy while giving them a taste for research and science. Gelation is one of the basic phenomena of molecular cooking [START_REF]Handbook of Molecular Gastronomy[END_REF] . This method allows to control the texture and the flavours of the product and thus opens new possibilities of food manufacturing. In this article we will focus on the shaping of gels into complex shapes. Morphing of food is a common problem encountered by manufacturers [START_REF] Tao | Morphlour: personalized flour-based morphing food induced by dehydration or hydration method[END_REF][START_REF] Tao | Morphing pasta and beyond[END_REF][START_REF] Dick | Influence of drying method and 3d design on the 4d morphing of beef products[END_REF] . The methods developed use physical and chemical phenomena like swelling or deswelling of material with anistropic properties to create three dimensional shapes [START_REF] Falahati | Smart polymers and nanocomposites for 3d and 4d printing[END_REF] . In this study, we explored the possibility to use gels and mixture of gels to create 3D shapes. A quick search on the internet reveals that the most popular methods used to obtain gelatinous structure is by using agar-agar or sodium alginate. With kits bought on the internet and with components used in gastronomy, we quickly started trying to produce gels. We first used sodium alginate which is used in food as a thickener, stabilizer and gelling agent. Sodium alginate, the sodium salt of alginic acid extracted from brown seaweed (Phaeophyta) is soluble in water and forms hydrogels in the presence of divalent cations, like Ca 2+ , through cation binding [START_REF] Funami | Rheological properties of sodium alginate in an aqueous system during gelation in relation to supermolecular structures and ca2+ binding[END_REF] . In the field of molecular gastronomy, sodium alginate is used to create spheres. There are two methods which differ based on the calcium content in the product to be encapsulated in the gel. When the substance contains no calcium, sodium alginate is added and the subtance is "cooked" in a bath of calcium. The reverse (we talk then of external spherification) is preferred when the substance contains calcium ions. In this case substance is cooked in a sodium alginate bath and the gelation reaction stops when the spheres are removed. Using sodium alginate is not the only way to obtain spheres with gels. They are also obtained with agar-agar, a product extracted from certain species of red seaweeds [START_REF] Araki | Studies on the chemical constitution of agar-agar. exhaustive mercaptolyses of agar-agar[END_REF][START_REF] Chapman | Agar-agar[END_REF] . This substance is used in very small quantities and has practically no taste nor color. It is a perfect vegetable binder and gelling agent to replace animal gelatin. There is a multitude of recipes in which agar-agar 9 is used to produce jams, fruit jellies, flans or pastries... The gel is formed when the agar-agar diluted in water is heated to 90 • C and then cooled to a temperature of about 40 • C. To make spheres, drops of hot agar-agar solution are formed and left in a "cold" not water-miscible liquid, typically oil, forming gel spheres.

In cooking, sodium alginate or agar-agar gels are often employed to form spheres and straight fibers (spaghetti). In this study, we investigate whether it is possible to realize more complex shapes such as a helical filament with those gels using the tools in our kitchens. We have a set of utensils to make homemade pasta. To obtain fusilli which has a spindle shape similar to the one we want to make, the dough is extruded through a pasta die with ad-hoc perforations [START_REF] Lucisano | Influence of die material on pasta characteristics[END_REF][START_REF] Agnesi | The history of pasta[END_REF] . Agar-agar and alginate before gelation are far too liquid compared to pasta dough to use the same technique. A second method that can be used at home is based on a combination of pasta paddles used to roll pasta, each giving a unique shape and texture. One of the paddles has a groove that looks like a screw channel. It appeared that placing a strand of alginate solution in the groove prior to immersing in a calcium bath was very promising solution. After a few hours, it was quite easy to peel off the gel. Based on this first observation, we present two new meth-ods to generate flexible helices made of sodium alginate or a mixture of alginate with agar-agar. These methods consist in casting an alginate solution (eventually added with agar) in a helical mold prior to immersing it in a bath of calcium lactate. The first section details the methods to produce helices made of different hydrogels. Then, we present the helices resulting from these methods along with a characterization of their mechanical properties. In particular we measure the deformation of the helices when hanged vertically from which we deduced their Young's modulus. Finally, we discuss how these results compare with previous measurements obtained with same hydrogels and give some perspectives to these methods.

II. EXPERIMENTAL METHODS

A. Mold fabrication

To reproduce and optimize the groove geometry, we 3D printed molds with helicoidals grooves (see Fig. 1) using a polylactide filament, also known as PLA. The typical geometry of the molds is shown in Fig. 1. In this work, we used eleven molds with different geometric parameters which are detailed in table I. In a second step, we improved the device by adding a removable sheath that can be placed around the first mold. The sheaths are also made in 3D printing and have a slightly larger diameter than the diameter D of the mold. This technique allows the use of the most liquid gels.

B. Protocol for the gels

In this work, we focused on two different types of gels: pure sodium alginate and a mixture of sodium alginate with agar-agar. As we have seen in the introduction, sodium alginate can form a gel either by immersing the solution in a calcium ion bath or by immersing a fluid rich in calcium ion in an alginate bath. Both methods were tested but we will present experiments performed with the first method (dipping sodium alginate in a calcium ion bath). The following sections describe the preparations of the different solutions and the method followed to obtain the helices.

Method used to obtain sodium alginate helices

Different concentrations of sodium alginate were used to prepared the solution. We used three concentrations: 20, 30 and 40 g/L. All products are food grade compounds and were obtained from QUIMICA M.G. The sodium alginate powder was first dissolved in water (pH 6) at ambient temperature until all lumps have disappeared. Drops of food coloring were then added to the solution. The colors are used to identify the sodium alginate concentration used in the preparation: green, yellow and blue for respectively 20, 30 and 40 g/L.

For concentrations above 20 g/L, the solution is sufficiently viscous so that the groove can be filled without the liquid leaking out. A syringe is used to place the solution into the groove and once filled, the mold is immersed for 5 minutes into the calcium lactate bath (20 g/L). The mold is removed from the bath and washed with clean water. To prevent the drying of the gel, the mold is then placed in an air-tight box for 10 hours. This duration is chosen to ensure that the gelling process has occurred throughout the whole section of the helix.

For the 20g/L sodium alginate solution the process should be slightly modified, as the solution flows very easily and drains out of the mold before gelation begins. In these cases, the molds are immersed in the alginate solution and placed in the freezer for a time, in such a way as to allow the solution to begin to solidify, but not com-pletely. Once the desired solidification point is reached, the excess sodium alginate is removed from the outside of the mold with the help of a spatula and immediately immersed in the calcium lactate bath.

Helices obtained by mixing agar-agar and sodium alginate

Two protocols can be used to obtain a mixture of agaragar and alginate gels. For the first protocol, the sodium alginate solution has to be heated up to 90°C before adding the agar-agar. The solution is stirred until the agar agar is completely dissolved. This phase takes typically no more than two minutes. When the solution cools down an agar-agar gel forms and embeds the alginate. For the second protocol, a hot agar-agar solution, with twice the desired final concentration, i.e. 20 g/L agar-agar, is mixed with the same volume of a 40 g/L alginate solution. In this way, a final solution of 10 g/L of agar-agar and 20 g/L of alginate is obtained. There are no significant differences between the gels obtained with both methods.

We carried out rheological measurements (rotational rheometer Anton Paar MCR 501 with a cone-plane geometry) to confirm that the sodium alginate molecules are not affected by the heat. Figure 2 shows the results for a solution of 20 g/L of sodium alginate, which is heated for 10 minutes at different temperatures and then cooled again. We see that the rheological property is not affected by the different heating conditions. The alginate solution shows, whatever the treatment, a rheofluidizing behavior for shear rates above 10 s -1 .

When this agar gel is immersed in a calcium bath, the alginate reacts and allows to fix the shape we set. We have used this property to make helices. For this, the still warm agar-agar alginate mixture is placed in a syringe. A tube is connected and the solution is injected into the tube. The tube is then placed under cold water or left on the table. After a few minutes, a gel forms in the tube. The syringe is then emptied and filled with air and the tube reconnected. The injection of air allows the agar gel to come out. The gel keeps the shape of the tube and is then deposited inside the groove. The mold is then immersed in the calcium lactate solution.

The helices that we will characterize in the following were made with an alginate concentration of 20 g/L and agar concentrations of 10 and 20 g/L.

Measurements of the elastic properties using indentation testing

The mechanical response of the sample is examined using nanoindentation testing (Anton-Paar). The method consists of driving a spherical tip of radius R i = 200µm into the sample with a controlled penetration depth, δ. The spherical tip assumed to be perfectly rigid is initially in contact with the surface of the gel. Then the η (mPa

• s) 20g/L -20 • C 20g/L -54 • C 20g/L -70 • C 20g/L -80 • C FIG.
2. Dynamic viscosity η as a function of the shear rate γ for sodium alginate solutions with a concentration of 20 g/L kept at different temperatures during 10 min.

tip is driven inside the sample with a controlled loading speed of 10µm/min until a maximum load is reached. The applied force, F , is measured as a function of the penetration depth, δ. Measurements are reported in figure 3 for several samples. The resulting loading curves are well fitted by the Hertz contact law revealing the elastic behaviour of the material:

F = 4E √ R i 3(1 -ν 2 ) δ 3/2 , ( 1
)
where E is the Young's modulus of the sample and ν is the Poisson's ratio of the sample (ν = 0.4 for alginate gels [START_REF] Banerjee | Mechanical behaviour of alginate film with embedded voids under compression-decompression cycles[END_REF] ). The fit of the indentation-force curve with a Hertz law provides an estimation of the Young's modulus of alginate gels.

III. EXPERIMENTAL RESULTS

Figure 4 presents three helices made with the same mold and pure alginate gels of increasing concentrations. We note that the alginate gel adopts a well-defined helical shape whatever the alginate concentration. A helix obtained with a mixture of alginate and agar-agar gels is shown in Fig. 5. Here again the gel filament is observed to be well-shaped as a helix. One of the main differences between the two methods is that the second produces filaments of circular cross section.

The helices are flexible and may experience large deformations when handled vertically. By assuming that they deformed like elastic springs, we will show that the Young's modulus of the gel can be estimated from the measurement of this deformation. This simple test will allow to compare the mechanical properties of different gels. We will first describe the procedure and the result for alginate gels. The method will be finally extended to gel obtained by mixing agar-agar and alginate.

A. Measurement of ℓ0

In order to measure the length of the helical filament in the absence of stress, the helices were left to sediment slowly in a container filled with water. In this way, contact forces with a solid substrate or stresses due to gravity are reduced and ℓ 0 can be estimated with a better precision. Figure 6 shows the picture of one helix made of alginate along with its mold. We see that the helix matches quite well the groove of the mold: the pitch, the length and the diameter are very close. Figure 7 displays ℓ 0 normalized by the length ℓ m of the mold used to fabricate the helix. In the case of pure alginate, the rest length of the helix is slightly larger than the mold length whatever the concentration in alginate. The inset of Fig. 7 show the same measurements for helices made of alginate (c = 20 g/L) and agar-agar gels with two concentrations. Despite the fact that results are more dispersed with gel mixtures, the ratio ℓ 0 /ℓ m is also generally larger than one in this case.

B. Measurement of ℓ

The helices were hung by one end and pictures were taken. The new length ℓ of the helices deformed by their own weight was them estimated. Figure 8 shows the vertical deformation of hanging helix (ℓℓ 0 )/ℓ 0 made of pure alginate as a function of the helix diameter D and FIG. 8. Normalized vertical deformation of the helix hanging on its own weight as a function of the helix diameter D for different pitches p. The helices are made with a concentration c = 30 g/L in alginate. Solid lines correspond to the predictions of Eq.( 4) for E = 40 kPa.

for different pitches p. Note that the length of the helix trapped by the clamp was subtracted from ℓ 0 . We observe that for a concentration in alginate and a given filament radius, the normalized deformation of the helix increases with its diameter and reduces with its pitch. Thus, diameter and pitch appear to be critical parameters for the mechanical properties of these flexible helices.

C. Model

In order to rationalize our observations, we develop a mechanical description of the helix deformation under its own weight. The helix is assumed to be made of a filament with a cylindrical cross section (diameter d), an elastic material of density ρ and Young modulus E. At rest, the diameter, pitch and length of the helix are respectively D, p and ℓ 0 . The stiffness k of the helix is given by the relation 13

k = Ed 4 p 8D 3 ℓ 0 , (2)
and the total mass M of the helical filament writes

M = ρ πd 2 ℓ 0 4 1 + πD p 2 . (3)
When the helix is submitted to its own weight, it extends by a length ∆ℓ = ℓℓ 0 which expresses ∆ℓ = M g 2k . Combining these equations, we get a prediction for the normalized vertical length ℓ/ℓ 0 = (ℓ 0 + ∆ℓ)/ℓ 0

ℓ ℓ 0 = 1 + πρgℓ 0 D 3 1 + πD p 2 Ed 2 p . (4)
In order to compare this prediction with our measurements, we can plot the vertical deformation (ℓℓ 0 )/ℓ 0 as a function of the quantity

P = πρgℓ 0 D 3 1 + πD p 2 /d 2 p.
A linear variation of the data will indicate an elastic regime. We can then get the elastic modulus as the inverse of the slope of the data.

D. Estimations of Young's modulus

We first present the estimation of the Young's modulus from the vertical deformation of gel helices. Figure 9 shows all the data for the vertical deformation of alginate helices as a function of the parameter P. For pure alginate helices, we observe that the vertical deformation increases with P. When P < 150 kPa, the increase is linear and it becomes sub-linear for larger values of P. The solid line shows the best fit of the experiments for P < 150 kPa with a linear law. The inverse of the slope of the linear fit provides an estimation for the Young's modulus of the alginate E = 36 ± 4 kPa. These data do not reveal any clear dependency of E with the alginate concentration. Results for helices made of a mixture of alginate and agar-agar gels are shown in the inset of Fig. 9. The dispersion of these results reflects the greater difficulty to fabricate these helices. Indeed, the wrapping of a soft and fragile gel filament into a helical mold is a difficult step. Nevertheless, we observe a linear variation of the helices deformation as a function of the parameter P and no strong impact of the agar-agar concentration. We also observe that the elastic limit is reached for higher stresses than before. The best fit of these data with a linear law gives an estimation of the Young's modulus for mixed gels E = 270 ± 20 kPa. The double gelation process produces stiffer helices.

On the other hand, we wanted to see if the Young's modulus value obtained by a macroscopic deformation measurement and based on the assumption that the helices behave mechanically like springs are close to the values that can be measured by local measurements. In this way, indentation testing was performed on a fragment of a pure alginate helix. The fit of the indentation-force curve with a Hertz law (see Fig. 3) provides an estimation of the Young modulus of the alginate gels as: E = 92 ± 40 kPa, and no clear dependencies with the concentration in alginate can be observed. In addition, the Young modulus estimated from indentation-force curve on a fragment of alginate and agar-agar helix gives E = 873 ± 301 kPa. These local measurements performed on different samples are in good agreement with the measurements based on material indentation and the vertical deformation of the helical filament. This proves that this inexpensive and easy to implement method is valid for characterising soft gels. Note that the significant uncertainty in the measurements is due to the low stiffness of the gel and therefore the low value of the Young's modulus obtained at the resolution limit of the instrument.

IV. DISCUSSION

The mechanical properties of alginate hydrogels determined here above can be compared with previous measurements. Compression experiments realized on alginate beads prepared in the same range of concentration provide a Young's modulus between 5 to 300 kPa, in relative agreement with our observations [START_REF] Wang | High-speed compression of single alginate microspheres[END_REF][START_REF] Chan | Effect of formulation of alginate beads on their mechanical behavior and stiffness[END_REF] . However, these experiments show a slight increase of E with the alginate and the cation concentrations in contrast with our experiments where the impact of the gel concentration is not significant. Other studies show that the mechanical properties of alginate hydrogels also depend on the ionic strength of the solutions and the storage environment [START_REF] Ouwerx | Physicochemical properties and rheology of alginate gel beads formed with various divalent cations[END_REF][START_REF] Kuo | Ionically crosslinked alginate hydrogels as scaffolds for tissue engineering: Part 1. structure, gelation rate and mechanical properties[END_REF] . Thus, it would be interesting to explore in further detail how these parameters impact the mechanical characteristics of our hydrogels helices.

The second method allows to obtain helices with a regular section and a very clean shape. The difficulty we have encountered with this method concerns the wrapping phase around the mold. In order not to break the agar filament, it is necessary that this operation be done slowly and carefully, especially for the smaller diameter molds. We have not quantitatively characterized the bending strength of the agar-agar alginate mixture but this appears to be a point of improvement of our method.

As noted in the introduction, it is also possible to create a gel by dipping a solution containing calcium ions in an alginate bath. We then talk of external spherification. We also test the possibility of using this method to stiffen agar-agar and make it take a new shape. First, calcium lactate (30g/L) is dissolved in a warm liquid solution containing 10 g/L of agar-agar. We then use the same method as before. After cooling, the agar-agar is rolled onto a mold and then dipped into solution containing 5g/L of sodium alginate (instead of a calcium lactate bath as previously). With these conditions, we do not obtain a gel strong enough to keep the desired shape. However, we believe that this approach is promising but requires further study to optimize the different concentrations to be used.

V. CONCLUSION AND PERSPECTIVES

In this study, we propose two "recipes" allowing to shape filaments made of hydrogels into helices, the first with pure alginate and the second with a combination of alginate and agar-agar gels. These methods enable the creation of helices with a wide range of length, pitch and diameter and different gel concentrations. We also propose a simple test to characterize the mechanical properties of these helices: measuring their vertical deformation when they hang under their own weight. Assuming that the helix behaves as an elastic spring, we estimate the Young's modulus of these objects and we find estimations in good agreement with independent indentation tests as well as the estimations given in the literature. Finally, this work proves that it is possible to impose a natural curvature to gels which have very low rigidity. This opens new avenues to fabricate complex shapes with soft gels, a fact that might be of interest for soft robotics, bio-encapsulation and 3D printing techniques.

Indeed, soft robotics are now looking for new manufacturing methods in order to shape biocompatible and biodegrable materials and reduce its environmental footprint [START_REF] Heiden | 3d printing of resilient biogels for omnidirectional and exteroceptive soft actuators[END_REF] . On the other end, alginate encapsulation of molecules of biological significance, such as enzymes, drugs, macromolecule, microorganism or stem cells is a well-established method [START_REF] Yeatts | Formation of an aggregated alginate construct in a tubular perfusion system[END_REF][START_REF] Zhang | Protein encapsulation in alginate hydrogel beads: Effect of pH on microgel stability, protein retention and protein release[END_REF] . Our study shows that more complex shapes can be obtained from the mixture of two different gels which could have applications in this field by increasing for example the surface to volume ratio and increasing the exchanges between the capsule and the environment. An interesting perspective of this work would be to transfer these methods to 3D printing. The idea would be to extrude a filament made of alginate and agar gels directly in a removable and porous matrix immersed in a calcium lactate bath. This possibility of shaping the flexible filament into the desired shape will be considered in future work.

FIG. 1 .

 1 FIG.1. Sketch of the molds. They were first drawn with autocad and then printed with a 3D printer. D: diameter of the mold, ℓ: height of the mold, R groove depth and p: pitch.

FFIG. 3 .FIG. 4 .

 34 FIG.3. Elastic behaviour of alginate gels using indentation testing. The load, F , applied to the materials is plotted as a function of the penetration depth, δ, of the indenter tip. Each series of dots was obtained on a specific sample. Lines: fit of the data by the Hertz contact law given by Eq.(1).

FIG. 5 .FIG. 6 .

 56 FIG. 5. Helix made of alginate (20 g/L) and agar-agar (10 g/L) suspended under its own weight. The white bar represents 5 mm.

m

  FIG. 7. (a)Estimated length ℓ0 of pure alginate helices after sedimentation in a water normalized by the length ℓm of the mold. Green squares, yellow circles and blue triangles correspond to helices obtained with respectively a concentration of 20, 30 and 40 g/L in alginate. The indices of the x-axis refer to mold reference given in TableI. (b) Same results for helices obtained with a mixture of alginate at 20 g/L and agar-agar gels. Crosses and diamonds represent measurements made for agar-agar concentrations of 10 and 20 g/L respectively.

1 2 /d 2

 122 FIG. 9. (a) Normalized vertical deformation of alginate helices hanging on their own weight (ℓ -ℓ0)/ℓ0 as a function of the parameter P = πρgℓ0D 3 1 + πD p

ACKNOWLEDGMENT

We are grateful to L. Auffray, and R. Pidoux for their help with molds fabrication. We acknowledge support of the French-Argentinian International Research Project "Ingénieries Vertes par la Mécanique des Fluides" (IRP-IVMF), CNRS-INSIS (France), CONICET (Argentina), and the French State grant 'BOGUS' ANR-19-CE06-0030-02. B.D.T. was supported by ANR PIA funding: Grant No. ANR-20-IDEES-0002.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.