Daniel R Alkhorshid

Eduardo S Tognetti

Irinel-Constantin Morȃrescu

A bilinear systems approach with input saturation to control the agreement value of multi-agent systems

While reaching an agreement in multi-agent systems (MAS) can be ensured by enforcing some connectivity properties between agents, the consensus depends on their initial conditions and the network topology. In this context, our main objective in this paper is to sway the consensus value of multi-agent systems towards a desired value. The asymptotic stability and maximization of the domain of attraction for the bilinear model representing the opinion dynamics in the presence of limited control action for a fixed and connected network are investigated. By using algebraic graph theory and linear matrix inequality (LMI), we provide sufficient conditions guaranteeing the convergence of agents toward the desired consensus. Furthermore, examples illustrate the effectiveness of the proposed method.

I. INTRODUCTION

Multi-agent systems have a wide range of applications such as robotic teams, power grids, telecommunication networks, biology and opinion dynamics. In the MAS setup the agents have only a local limited view of the overall system, which means that they design decentralized control actions governing their behavior. The coherent behavior of the MAS is often described in terms of consensus i.e., the agents reach an agreement on certain variables of interest [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF]- [START_REF] Hegselmann | Opinion dynamics and bounded confidence models, analysis, and simulation[END_REF].

A very rich literature exists on the design of consensus algorithms for agents with linear dynamics, for which the behavior is well understood. In this setting, the literature considers fixed and time-varying communication topologies [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF], [START_REF] Moreau | Stability of multiagent systems with time-dependent communication links[END_REF], directed or undirected graphs [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF], [START_REF] Ren | Distributed Consensus in Multi-Vehicle Cooperative Control: Theory and Applications[END_REF], synchronous or asynchronous information exchange [START_REF] Tsitsiklis | Distributed asynchronous deterministic and stochastic gradient optimization algorithms[END_REF], [START_REF] Fang | Asynchronous consensus protocols: Preliminary results, simulations and open questions[END_REF], delayed or immediate transmissions [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF], [START_REF] Michiels | Consensus problems with distributed delays, with application to traffic flow models[END_REF], etc. It is not worthy that in all these cases the resulted agreement value depends on the initial conditions of the agents and the network topology describing their interactions. While in some applications this is convenient in some other such as power grids, traffic control or opinion dynamics, one needs to reach an agreement as close as possible to a desired a priori fixed value. In this case a supplementary centralized control action is required which in many applications is subject to saturation and global cost constraints.

In the last decade, many works focused on the consensus problem with saturated inputs (see for instance [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF], [START_REF] Varma | Marketing resource allocation in duopolies over social networks[END_REF]) or *This work was supported by the Brazilian agency FAPDF. The work of I.-C. Morȃrescu was supported by ANR under the grants HANDY ANR-18-CE40-0010 and NICETWEET ANR-20-CE48-0009. 1 Daniel R. Alkhorshid and Eduardo S. Tognetti are with the Department of Electrical Engineering, University of Brasilia -UnB, 70910-900, Brasília, DF, Brazil, dralkhorshid@gmail.com, estognetti@ene.unb.br constantin.morarescu@univ-lorraine.fr global control effort constraints (see [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF], [START_REF] Rejeb | Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems[END_REF], [START_REF] Tognetti | Synchronization via output feedback for multi-agent singularly perturbed systems with guaranteed cost[END_REF]).

In this work, we focus on reaching a desired consensus on the network of interest by applying an external centralized control action to it. One considers both saturated inputs and the minimization of an overall cost corresponding to the global effort/budget of the external entity to control the agreement value. While different applications such as power grids or platooning can be modeled in this setup, our main motivation comes from the control of opinion dynamics over social networks.

Emergence of consensus in social networks is a controversial problem. Social studies pointed out that, in general, opinions tend to converge one toward another during interactions. Consequently, it is not surprising that consensus received a particular attention in opinion dynamics literature [START_REF] Axelrod | The dissemination of culture: A model with local convergence and global polarization[END_REF], [START_REF] Galam | Towards a theory of collective phenomena: Consensus and attitude changes in groups[END_REF]. It is also worth noting that some of the first mathematical models naturally lead to consensus [START_REF] Degroot | Reaching a consensus[END_REF], [START_REF] Friedkin | Social influence and opinions[END_REF] while some others lead to network clustering [START_REF] Hegselmann | Opinion dynamics and bounded confidence models, analysis, and simulation[END_REF], [START_REF] Altafini | Consensus problems on networks with antagonistic interactions[END_REF], [START_REF] Morȃrescu | Opinion dynamics with decaying confidence: Application to community detection in graphs[END_REF]. To enforce consensus in social networks, some recent studies propose the control of one or a few agents, see [START_REF] Caponigro | Sparse feedback stabilization of multi-agent dynamics[END_REF], [START_REF] Dietrich | Control via leadership of opinion dynamics with state and time-dependent interactions[END_REF].

Besides these methods of controlling opinion dynamics towards consensus, we also find recent attempts to control the discrete-time dynamics of opinions such that as many agents as possible reach a certain set after a finite number of influences [START_REF] Hegselmann | Optimal opinion control : The campaign problem[END_REF]. The closest work to the present manuscript is [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF] in which the authors consider a discretetime implementation (campaign influence) of the exogenous control action trying to sway the agreement value toward a desired one. Unlike [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF] in this work we are considering a continuous-time influence of the external entity, a timevarying state dependent control input which is saturated and a total budget/energy which is continuously spent over time.

From a practical point of view, the problem considered in this manuscript is related to viral marketing in which sellers attempt to artificially create word-of-mouth advertising among potential customers. Social scientists and economists have well established the effectiveness of this trend [START_REF] Leskovec | The dynamics of viral marketing[END_REF], [START_REF] Arthur | Pricing strategies for viral marketing on social networks[END_REF]. The control limitations imposed in the paper are related, on the one hand, to the fact that sellers have a limited power of influence on any potential customer and, on the other hand, they have a fixed given budget to influence customers' decisions. It is noteworthy that for simplicity, we consider the topology of the network of interest is fixed, meaning that the characteristics of each agent and the network itself would not change during time. The model representing the opinion dynamics is bilinear; hence an estimate of the domain of attraction of the origin is obtained based on a local quadratic Lyapunov function and its resulting (ellipsoidal) level sets [START_REF] Genesio | On the estimation of asymptotic stability regions: State of the art and new proposals[END_REF]. As the states belong to the positive orthant, we propose a convex optimization problem, in terms of LMIs, exploiting properties of positive systems [START_REF] Farina | Positive Linear Systems: Theory and Applications[END_REF] to maximize the invariant region where the state trajectories must belong. To deal with the state dependency in the design conditions, we adopted the parametrization of the states as normbounded uncertainties. This approach is more appropriate than the polytopic approximation of the bilinear term [START_REF] Amato | Stabilization of bilinear systems via linear state-feedback control[END_REF] when dealing with high-order systems (large number of agents).

The paper is organized as follows. In section II we introduce the main problem along with the network rules and properties. Additionally, we introduce lemmas that are essential for further developments. In section III, we propose the main result of this paper in the form of a theorem and its mathematical justification along with it. In section IV we present two examples to display the effectiveness of the proposed method. Finally, the conclusion is made at the section V.

NOTATION

The space of real matrices with dimension n×m is denoted by R n×m , R n + denotes the space of vectors of dimension n with positive entries. For a matrix X, X T and X ⊥ denote the transpose of X and any matrix whose columns form a basis for the null space of X, respectively. X (i) denotes the i-th row. If X is square, X -1 denotes the inverse of X; He {X} stands for X + X T ; and X > 0 (X < 0) indicates that matrix X is positive (negative) definite; X 0 (X 0) denotes all the elements of X are positive (nonnegative). For a vector v ∈ R n , diag(v) = diag(v 1 , . . . , v n ) is a diagonal matrix composed with the elements of v, and for a matrix X, diag(X) is composed with the diagonal elements of X. The identity matrix of order n is denoted by I n and the null m × n matrix is denoted by 0 m,n (or simply I and 0 if no confusion arises), 1 stands for a vector of ones of appropriate dimension. The symbol denotes symmetric blocks in partitioned matrices, ⊗ denotes the Kronecker product, L n 2 [0, ∞) denotes the space of square integrable vectors of n functions over [0, ∞).

II. PRELIMINARIES

A. Problem formulation

In this work, we focus on the exogenous control of the consensus value in multi-agent systems. The study is mainly interpreted in term of opinion dynamics in social networks. Consequently, in the sequel we assume that the agents represent individuals that form a social network interacting over a fixed topology. The network is represented as a graph G in which each agent corresponds to a vertex belonging the set V = {1, . . . , N} associated with the N agents. An edge (i, j) in the graph indicates that agent j influences the opinion of agent i. Let us also consider the associated adjacency matrix

A = [a i j ] ∈ R N×N . The corresponding Laplacian matrix is L = [l i j ] ∈ R N×N defined by l ii = ∑ N j=1 a i j , ∀i = 1, . . . , N l i j = -a i j if i = j, (1)
where a i j = 0 means that agent i is influenced by agent j and a i j = 0 otherwise. For every agent i ∈ V we assign a time varying opinion x i (t) described by a scalar value normalized between 0 and 1, i.e., x i (t) ∈ [0, 1]. We consider that each opinion changes in time under the endogenous influence of neighbors' opinions on one hand, and exogenous control (persuasion) action of an external influencer.

In the sequel, d ∈ {0, 1} represents the opinion promoted by the external entity/influencer. The evolution in time of the states/opinions is described by the following dynamics:

ẋi (t) = N ∑ j=1 a i j (x i (t) -x j (t)) + (x i -d)u i (t), ∀i ∈ {1, . . . , N},
where u i (t) ∈ [ū, ū], ū ∈ (0, 1) is a finite energy and bounded exogenous control action.

Here, we consider the desired opinion as extreme values of individual opinion, meaning expected binary behaviors (consume/ not consume, agree/ disagree, etc). Most problems can be put in this classification, while the case of intermediate desired opinion d ∈ (0, 1) (moderate consuming, for instance) is left for future researches. It is noteworthy that the rate of changes in every agent's opinion relies on both the granted budget J u and their original opinion x 0 (t). The granted budget here embodies the limited bound of the control action to reach the desired opinion. It is evident that in real-world situations, not all agents are tempted equally (in a uniform manner) by the external control action [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF].

Assumption 1 We suppose that the graph G representing the social network is weakly connected in the sense that it contains a directed spanning tree (i.e. a directed graph in which, except the root which is not influenced, each node is influenced by a single other node called parent).

Lemma 1 ( [29]

) Under Assumption 1 L has 0 as a simple eigenvalue associated with the right eigenvector 1 ∈ R N , that is, L1 = 0. All the other N -1 eigenvalues of L have positive real parts. Let x(t) = (x 1 (t), . . . , x N (t)) T ∈ R N and u(t) = (u 1 (t), . . . , u N (t)) T ∈ R N be the vectors collecting the states and the control input of all agents. The collective dynamics is then described by

ẋ(t) = -Lx(t) + B(x(t) -1d)u(t) (2)
where the function B(•) : R N → R N×N is given by B(x) = diag(x 1 , x 2 , . . . , x N ). We define the set of admissible values of the states as

χ = {x ∈ R N : x i (t) ∈ [0, 1], i = 1, . . . , N}. (3)
Let us define x d i (t) = x i (t)-d and x d (t) = x(t)-1d. From Lemma 1, the system (2) is rewritten as

ẋd (t) = -Lx d (t) + B(x d (t))u(t) (4)
where

x d i ∈ [-d, 1 -d], i = 1, . . . , N. For d = 1, one has x d i ∈ [-1, 0].
In the solution proposed in this paper, the case d = 1 is symmetric of d = 0, then we adopt hereafter, without loss of generality, d = 0 and x d = x.

The following problem is addressed in this work.

Problem 1 To design a control law u i that solves the following optimization problem. min

u(t) J x = ∞ 0 x(t) T Rx(t) dt (5)
subjected to

|u i (t)| ≤ ū (6) J u = ∞ 0 u(t) T Qu(t) dt < µ (7)
where ū and µ are given specifications, R and Q are positive definite matrices. The matrix Q penalizes the control effort required for synchronization. The conditions ( 6)-( 7) represent constraints in the amplitude and in the energy of the control signal, respectively.

In simple language, when a system (for example an anti smoking campaign) is interested in encouraging the targeted social network (for example young population) to agree on a desired opinion (a behavior), it attempts to sway the network consensus as close as possible to the desired consensus (5), by using a granted budget which could be interpreted as a limited and bounded control action ( 6)- [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF].

Finally, observe that the functions J x and J u can be rewritten as ∞ 0 z(t) T z(t) dt and ∞ 0 y(t) T y(t) dt, respectively, where

z(t) = R 1 2 x(t) y(t) = Q 1 2 u(t). (8)

B. Representation of the bilinear term and saturation model

The constraint (6) can be incorporated in the dynamics as u(t) = sat(v(t)) using the standard decentralized saturation function sat(v ( ) ) = sign(v ( ) ) min(|v ( ) |, ū), = 1, . . . , N, where v is an unbounded control signal to be designed. Then, one has

ẋ(t) = -Lx(t) + B(x(t))sat(v(t)). (9)
System ( 9) can be rewritten using the decentralized deadzone nonlinearity

ψ(v) = v -sat(v) ẋ(t) = -Lx(t) + B(x(t))v(t) -B(x(t))ψ(v(t)). (10)
We describe the bilinear product in (10) for x ∈ χ using a norm-bounded uncertainty, that is,

B(x) = B 0 + B 1 ∆(t), (11)
with B 0 = B(0.51), B 1 = 0.5, and ∆(t) = diag(δ 1 (t), . . . , δ N (t)) ∈ R N×N , where δ i (t), i = 1, . . . , N, are bounded Lebesgue measurable uncertainties belonging to the set

D = {δ ∈ R : δ (t) T δ (t) ≤ 1}.
Let us consider the following state feedback control law

v(t) = Kx(t), K = diag(k 1 , . . . , k N ) ∈ R N×N . ( 12
)
With the diagonal structure of the gain K, we impose the control action of each agent v i to only depend on its opinion x i . The closed-loop system formed by ( 8), ( 10) and ( 12) is given by

ẋ(t) = (-L + (B 0 + B 1 ∆(t))K)x(t) -(B 0 + B 1 ∆(t))ψ(Kx(t)) (13a) z(t) = R 1 2 x(t) (13b)
y(t) = Q 1 2 Kx(t) + Q 1 2 ψ(Kx(t)). (13c)
Observe that the controllability matrix formed from the pair (-L, B(x)) of system (9) loses rank for x = 0 and the system (9) becomes uncontrollable. To circumvent this problem, we consider the interval x i ∈ [ε, 1], where ε is a positive scalar arbitrarily small, in the modeling of B(x) in [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF]. This implies B 0 = B((0.5 + ε)1) and B 1 = 0.5.

Let us finish this section with the following lemmas which are instrumental for further developments.

Lemma 2 (Petersen's Lemma [START_REF] Petersen | A stabilization algorithm for a class of uncertain linear systems[END_REF]) Let G = G T ∈ R n×n , M ∈ R n×p , and N ∈ R q×n be given matrices. For all ∆(t) ∈ R p×q such that ∆(t) T ∆(t) ≤ I, the inequality G + M∆(t)N + N T ∆(t) T M T ≤ 0 holds if and only if there exists a scalar λ > 0 such that

G + λ MM T + 1 λ N T N ≤ 0.

Lemma 3 ( [31]

) Consider a matrix G ∈ R N×N and define the region

Π = x ∈ R N : |v i -G i x| ≤ ū, i = 1, . . . , N . (14)
If x ⊆ Π then the following relation holds

ψ(v) T T (ψ(v) -Gx) ≤ 0 (15)
for any matrix T ∈ R N×N diagonal and positive definite.

As detailed in the next section, the definition of positive systems will help construct the invariant region to which the initial conditions must belong.

Definition 1 ( [27])

The system ẋ(t) = Ax(t) is called positive (Metzlerian) system if for every initial condition x(0) ∈ R n + , the states satisfy x(t) ∈ R n + for t ≥ 0.

Lemma 4 ( [27])

The system ẋ(t) = Ax(t) + Bu(t) is positive if and only if A is a Metzler matrix, that is, A (i j) ≥ 0 for i = j, and B 0.

III. MAIN RESULTS

In this section we present the main conditions to design the control law [START_REF] Varma | Marketing resource allocation in duopolies over social networks[END_REF] and solve Problem 1. The control strategy must assure the trajectories to remain in χ and Π, where [START_REF] Axelrod | The dissemination of culture: A model with local convergence and global polarization[END_REF], used in the design conditions, holds. In other words, one must find an invariant region S such that if x(0) ∈ S , then x(t) ∈ χ ∩ Π for all t ≥ 0. We adopt the following candidate region for S :

S := x ∈ R N + : x T W -1 x ≤ 1, W = W T > 0 . (16)
In general, S is not invariant even if V (t) = x T W -1 x is a Lyapunov function for the closed-loop system. Moreover, inclusion conditions for S ⊂ χ are not easy to obtain. Consider the following level curve of the Lyapunov function V (t) = x(t) T W -1 x(t),

S a := x ∈ R N : x T W -1 x ≤ 1, W = W T > 0 . ( 17
)
The set S a can be obtained straightforwardly from stability conditions however the ellipsoid region is centered in the origin meaning S a χ, that is, even for x(0) ∈ S a ∩ χ ∩ Π the trajectories could try to escape from χ. To solve this problem we first propose an augmented space χ a defined as

χ a = {x ∈ R N : x i (t) ∈ [-1, 1], i = 1, . . . , N}, (18)
where S a is contained, that is S a ⊆ χ a . Observe that χ a is symmetric in all orthants in the state space allowing the maximization of S a ∩ χ. Finally, we propose to design the control law [START_REF] Varma | Marketing resource allocation in duopolies over social networks[END_REF] such that the closed-loop system is positive (see Definition 1). Note that, -L is a Metzler matrix and B(x) 0 for all x ∈ χ, then, according to Lemma 4, system (2) is positive. Therefore, the set S is invariant. Fig. 1 illustrate sets χ, χ a , S , S a and a trajectory x(t). The control input energy/budget constraint [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF] can prevent the trajectories from initial conditions belonging to S to reach the origin. From a practical point of view, making all individual opinions converge to the desired opinion can be costly and budget constraints will lead the opinions to converge to a value different from d. Therefore, we propose a method to find an invariant region S u ⊆ S such that for all x(0) ∈ S u there is enough budget J u < µ to make all opinions reach the origin. We also propose a method to limit investments (u = 0) for all initial opinions belonging to S \ S u when the budget µ is reached.

We can reformulate Problem 1 as:

Problem 2 To design a state feedback gain K such that the closed-loop system ( 13) is positive and to find (i) an estimation of the domain of attraction S ⊂ (χ ∩ Π) such that for all initial conditions x(0) ∈ S , the trajectories of the closed-loop system converge exponentially toward the origin with guaranteed cost J x for all δ i (t) ∈ D and |u i (t)| ≤ ū; (ii) an estimation of a region S u such that for all initial conditions x(0) ∈ S u ∩S , the trajectories of the closedloop system converge exponentially toward the origin with guaranteed cost J u < µ for all δ i (t) ∈ D and |u i (t)| ≤ ū; (iii) a control law to limit investments (u = 0) for all initial opinions belonging to S \ S u when the budget J u = µ is reached.

For the following results, the polyhedral set χ a is represented by

χ a = x ∈ R N : Ωx ≤ 1, (19)
where

Ω = I N ⊗ -1 1 T ∈ R 2N×N , 1 ∈ R N , and 0 ∈ χ a .
As methodology, following ideas presented in [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF] where the maximum possible investment is used as soon as possible to minimize a cost function related to the convergence of x, we first present a result that solves ( 5)-( 6) (Problem 2 (i)). After that, we propose an stability analysis condition that provides Lyapunov level curves that are upper bounds for the guaranteed cost J u solving Problems 2 (ii) and (iii).

Theorem 1 Suppose that there exist diagonal positive definite matrices W ∈ R N×N and S ∈ R N×N , a diagonal matrix Z ∈ R N×N , a matrix Y ∈ R N×N , and a scalar λ > 0, such that the following inequalities hold:

    He {-LW + B 0 Z} + λ I SB T 0 +Y -2S W 0 -R -1 Z S 0 -4λ I     < 0 (20) W Ω (i) W 1 ≥ 0, ∀i = 1, . . . , 2N (21)
W Z (i) -Y (i) ū2 ≥ 0, ∀i = 1, . . . , N. (22)
Then, the state feedback gain K = ZW -1 makes the closedloop system (13a) asymptotically stable with S ⊆ χ ∩ Π the estimation of the domain of attraction of the origin with guaranteed cost J x ≤ x(0) T W -1 x(0) .

Proof: First, consider the closed-loop system (13) and the Lyapunov function

V (t) = x(t) T W -1 x(t). The integral from 0 to ∞ of V (t) + z(t) T z(t) < 0, ∀x ∈ χ, (23)
implies J x < V (0). Condition ( 23) is equivalent to V (t) < -cx(t) T x(t), for c the maximum eigenvalue of R, and thus it verifies the exponential stability of the origin. Using Lemma 3, the inequality (23) holds if V (t) + z(t) T z(t) -2ψ(v(t)) T T ψ(v(t)) + 2ψ(v(t)) T T Gx(t) < 0, and, considering [START_REF] Rejeb | Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems[END_REF] with B(x) = B 0 + B 1 ∆(t), the last inequality is rewritten as

x ψ(v) T He -W -1 L +W -1 B(x)K + R B(x) T W -1 + T G -2T x ψ(v) < 0.
By pre-and post-multiplying the previous inequality by diag(W, T -1 ), one has

He {-LW + B(x)Z} +W T RW SB(x) T +Y -2S < 0,
where Z = KW , S = T -1 , and Y = GW . Using the Schur complement lemma, one has

  He {-LW + B(x)Z} SB(x) T +Y -2S W 0 -R -1   < 0.
For all x ∈ χ we can replace B(x) by [START_REF] Morȃrescu | Space-time budget allocation policy design for viral marketing[END_REF] in the above inequality yielding

  He {-LW + B 0 Z} SB T 0 +Y -2S W 0 -R -1   + He      ∆ 0 0   0.5 Z S 0    < 0.
Applying Lemma 2 for ∆(t) = diag(δ 1 (t), . . . , δ N (t)), δ i (t) ∈ D, and the Schur Complement, we recover [START_REF] Morȃrescu | Opinion dynamics with decaying confidence: Application to community detection in graphs[END_REF].

Observe that ( 21) is equivalent to

W -1 Ω (i) 1 ≥ 0,
which implies (see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]) that S a ⊆ χ a and, consequently, S ⊆ χ. Finally, by pre-and-post multiplying inequality [START_REF] Dietrich | Control via leadership of opinion dynamics with state and time-dependent interactions[END_REF] by diag(W -1 , I), one has

W -1 K (i) -G (i) ū2 ≥ 0
Considering the set Π in Lemma 3 and S a in [START_REF] Degroot | Reaching a consensus[END_REF], the above inequality verifies S a ⊆ Π [START_REF] Silva | Anti-windup design with guaranteed regions of stability: an lmi-based approach[END_REF] and S ⊆ Π.

Conditions of Theorem 1 are sufficient for the exponential stability of the closed-loop system (13a). However, it does consider the energy (budget) constraint in the control input [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF]. Additionally, the Lyapunov function V (t) = x(t) T W -1 x(t) with W obtained by Theorem 1 does not provide S u and a condition that guarantees J u < µ for all x ∈ S . We propose the following result to compute level curves associated with the control input energy J u and solve Problems 2 (ii) and (iii).

Theorem 2 If there exist diagonal positive definite matrices P ∈ R N×N and S ∈ R N×N , a matrix Y ∈ R N×N , and a scalar λ > 0, such that the following inequalities hold

    He {-LP + B 0 KP} + λ I SB T 0 +Y -2S KP S -Q -1 KP S 0 -4λ I     < 0, (24)
then, the closed-loop system (13a) has guaranteed cost J u ≤ x(0) T P -1 x(0). Proof: Considering the Lyapunov function V (t) = x(t) T P -1 x(t), the integral from 0 to ∞ of V (t)+y(t) T y(t) < 0 with respect to the closed-loop system (13a) implies J u < V (0). The rest of the proof follows the same lines of the proof of Theorem 1.

Remark 1

The estimation of the region of initial conditions such that J u ≤ µ is given by S u = {x ∈ R N + : x(0) T P -1 x(0) ≤ µ}. For initial opinions belonging to S \ S u , the following control law assures J u < µ:

u(t) = sat(Kx), x(t) T P -1 x(t) > x(0) T P -1 x(0) -µ 0, otherwise. (25)
Condition ( 25) makes the trajectories of x(0) ∈ S \ S u be attracted to the boundary of the set {x ∈ S : x(t) T P -1 x(t) ≤ x(0) T P -1 x(0) -µ}. Observe that when u = 0 the system dynamics becomes ẋ(t) = -Lx(t) which has a global uniformly exponentially stable attractor given by the consensus manifold.

Remark 2 A way to indirectly maximize the set S defined in [START_REF] Galam | Towards a theory of collective phenomena: Consensus and attitude changes in groups[END_REF] and minimize J x , as presented in [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF], is to maximize the trace of W (see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]). Hence, ( 5)-( 6) and the maximization of the estimate of the basin of attraction S are obtained by solving the following optimization problem: max Trace(W) [START_REF] Genesio | On the estimation of asymptotic stability regions: State of the art and new proposals[END_REF] subjected to (20)- [START_REF] Dietrich | Control via leadership of opinion dynamics with state and time-dependent interactions[END_REF].

To obtain a tighter approximation for the upper bound for cost J u in Theorem 2, we can also solve the following optimization problem: max Trace(P) [START_REF] Farina | Positive Linear Systems: Theory and Applications[END_REF] subjected to [START_REF] Leskovec | The dynamics of viral marketing[END_REF].

Remark 3 For ε > 0 in B 0 = B((0.5 + ε)1), we have that (23) holds for x ∈ χ \ B ε , where B ε = {x ∈ S : x T x ≤ ε}. This means that we can only assure the convergence of the trajectories to B ε . As long ε is sufficiently small, there is no practical implication since constraint (7) usually prevents x i (t) to reach d.

IV. NUMERICAL EXAMPLE

In this section, we present two numerical examples to illustrate the proposed results. The first example contains three agents that helps us in visualizing the domain of attraction along with the agents' trajectories. The second example involves twenty agents and is presented to show the effectiveness of the proposed method on a larger and realistic network.

Example 1 Consider the communication network from [START_REF] Rejeb | Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems[END_REF] described by a undirected graph G , connected, with N = 3 agents and Laplacian matrix is defined as:

L =   3 -1 -2 -1 3 -2 -2 -2 4   .
We consider Problem 1 with Q = 10 -1 I, R = 10 -1 I, µ = 0.675, and ū = 0.9. The design is performed by Theorems 1 and 2 adopting B 0 = B((0.5 + ε)1), with ε = 0.1. Theorem 1 yields K = diag(-1.4143, -1.4143, -1.3886) and the control law ( 25) is implemented from P obtained by Theorem 2. We observe in Fig. 2, depicting regions S , S a , χ, and χ a , that the estimation of the domain of attraction S (intersection of S a and χ) encompasses most of χ. Fig. 3 illustrates S u contained in S showing that for some initial conditions there is not enough budget to reach the origin due to [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF]. The trajectories of the agents is shown in Fig. 4 for the initial condition x(0) = (0.2673, 0.5345, 0.8018) ∈ ∂ S . We can see all trajectories converge as close as possible to the origin with respect to the energy constraint [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF], and the control signal respects the input saturation (u 3 saturates in the initial instant). Theorem 1 is applied to design the state feedback gain K that solves Problem 1 with Q = 10 -1 I, R = 10 -1 I, µ = 7.5, and ū = 0.9. We consider ε = 0.02 in B 0 to solve Theorem 1and 2. The trajectories of the agents are illustrated in Fig. 6. Note that the total budget is over in t = 2.2 and, from this time, the agents are attracted to the consensus manifold (at t = 7, the maximum value of the agents is 0.0559). We also note that for some agents, the control action saturates in the initial instants showing that the maximum investment is applied for some individuals. It is evident that all the agents opinions are swayed toward the desired consensus over time, but due to budget constraints it is not possible to exactly reach the consensus at d = 0.

V. CONCLUSION

In this paper, the problem of reaching a desired opinion (consensus) in a connected network have been discussed as an optimization problem. The proposed approach considers practical aspects such as finite budget in the form of input saturation and constraint in the energy of the control action. The dynamic representing individual opinions, normalized in the interval [0, 1], is bilinear, yielding extra challenge in designing a stabilizing state-feedback control law. The main results are given as constructive conditions in terms of LMIs by exploiting some properties of positive systems and invariant ellipsoids and can be applied for large networks. Future works involve searching new invariant regions that maximize the domain of attraction of the desired opinion.

Fig. 1 :

 1 Fig.1: Sets χ a (black dashed box), χ (red box), S (gray region), S a (blue curve), and trajectory x(t) (bold blue line).

Fig. 2 :

 2 Fig. 2: Top figure: region S a (blue) contained in χ a (red); Bottom left figure: projection of S a (blue) and χ a (red) in the plane x-y; Bottom right figure: estimation of domain of attraction S (blue) contained in χ (green) for Example 1.

Fig. 3 :

 3 Fig. 3: Ellipsoids S a (blue, left figure) and S (blue, right figure) covering S u (green) for Example 1.

Fig. 4 :Example 2

 42 Fig. 4: Trajectories of the agents for x(0) = (0.0673, 0.5345, 0.8018) and control signal for Example 1.

Fig. 5 :

 5 Fig. 5: The schematic of interested network in the form of a directed graph with 20 agents.

Fig. 6 :

 6 Fig. 6: Trajectories of the agents for some initial condition x(0) ∈ ∂ S and control signal for Example 2.


Bénédicte M Lefèvre
email: benedictelefevre@gmail.com

Stecy Mienanzambi

Michael Lang
email: michael.lang@ijm.fr

Developmental timing of Drosophila pachea pupae is robust to temperature changes running title: Temperature and development in

Keywords: Drosophila pachea, pupal development, temperature, heterochrony

Rearing temperature is correlated with the timing and speed of development in a wide range of poikiloterm animals that do not regulate their body temperature. However, exceptions exist, especially in species that live in environments with high temperature extremes or oscillations. Drosophila pachea is endemic to the Sonoran desert in Mexico, in which temperatures and temperature variations are extreme. We wondered if the developmental timing in D. pachea may be sensitive to differing rearing temperatures or if it remains constant. We determined the overall timing of the Drosophila pachea life-cycle at different temperatures. The duration of pupal development was similar at 25°C, 29°C and 32°C, although the relative progress differed at particular stages. Thus, D. pachea may have evolved mechanisms to buffer temperature effects on developmental speed, potentially to ensure proper development and individual's fitness in desert climate conditions.

Introduction

Poikilotherms animals do not regulate their body temperature contrary to homeotherms [START_REF] Precht | Homeothermy and Poikilothermy[END_REF] and are sensitive to environmental temperature.

Environmental temperature in turn affects their metabolism [START_REF] Hazel | Molecular mechanisms of temperature compensation in poikilotherms[END_REF]. In particular, it seems widespread that developmental speed increases with rearing temperature in poikilothermic species [START_REF] Abril | Effect of temperature on the development and survival of the Argentine ant, Linepithema humile[END_REF][START_REF] Asano | Modeling temperature-mediated fluctuation in colony size in the fire ant, Solenopsis invicta[END_REF][START_REF] Hrs-Brenko | Synergistic effects of lead, salinity and temperature on embryonic development of the mussel Mytilus galloprovincialis[END_REF][START_REF] Ikemoto | Intrinsic optimum temperature for development of insects and mites[END_REF][START_REF] Nair | Effect of temperature on the development, growth, survival and settlement of green mussel Perna viridis (Linnaeus, 1758)[END_REF][START_REF] Nishizaki | Thermal stress increases fluctuating asymmetry in marine mussels: environmental variation and developmental instability[END_REF][START_REF] Pechenik | The influence of food concentration and temperature on growth and morphological differentiation of blue mussel Mytilus edulis L. larvae[END_REF][START_REF] Porter | Impact of temperature on colony growth and developmental rates of the ant, Solenopsis invicta[END_REF][START_REF] Sharpe | Reaction kinetics of poikilotherm development[END_REF][START_REF] Vélez | Effects of temperature and ration on gametogenesis and growth in the tropical mussel Perna perna (L.)[END_REF], including various Drosophila species [START_REF] David | Essai de définition d'une température optimale pour le développement de la Drosophile[END_REF][START_REF] James | Thermal evolution of rate of larval development in Drosophila melanogaster in laboratory and field populations[END_REF][START_REF] Kuntz | Drosophila Embryogenesis Scales Uniformly across Temperature in Developmentally Diverse Species[END_REF][START_REF] Powsner | The effects of temperature on the durations of the developmental stages of Drosophila melanogaster[END_REF]. This phenomenon is proposed to be due to thermodynamics of enzymes responsible for biochemical reactions underlying developmental processes [START_REF] Crapse | Evaluating the Arrhenius equation for developmental processes[END_REF][START_REF] Ikemoto | Intrinsic optimum temperature for development of insects and mites[END_REF][START_REF] Schoolfield | Non-linear regression of biological temperature-dependent rate models based on absolute reaction-rate theory[END_REF][START_REF] Sharpe | Reaction kinetics of poikilotherm development[END_REF]. Thermal-stress can accelerate development and has been shown to result in an increase of developmental instability [START_REF] Kristensen | The use of fluctuating asymmetry and phenotypic variability as indicators of developmental instability: a test of a new method employing clonal organisms and high temperature stress[END_REF][START_REF] Nishizaki | Thermal stress increases fluctuating asymmetry in marine mussels: environmental variation and developmental instability[END_REF][START_REF] Polak | Developmental selection against developmental instability: a direct demonstration[END_REF], measured as deviations of an individual's character from the average phenotype in the population under the same conditions [START_REF] Palmer | Fluctuating asymmetry analyses: a primer[END_REF][START_REF] Zakharov | Population phenogenetics: Analysis of developmental stability in natural populations[END_REF]. This may result in a decreased individual's survival and reproductive fitness. In contrast, a slow development may potentially lead to an increased risk of predation at vulnerable stages, such as immobile pupae in holometabolous insects [START_REF] Ballman | Pupation behavior and predation on Drosophila suzukii (Diptera: Drosophilidae) pupae in Maine wild blueberry fields[END_REF][START_REF] Borne | Drosophila glue protects from predation[END_REF][START_REF] Hennessey | Predation on wandering larvae and pupae of Caribbean fruit fly(Diptera: Tephritidae) in guava and carambola grove soils[END_REF][START_REF] Thomas | Survivorship of the Pupal Stages of the Mexican Fruit Fly Anastrepha ludens (Loew) (Diptera: Tephritidae) in an Agricultural and a Nonagricultural Situation[END_REF][START_REF] Urbaneja | Influence of Ground Predators on the Survival of the Mediterranean Fruit Fly Pupae, Ceratitis capitata, in Spanish Citrus Orchards[END_REF]. Furthermore, a variable timing of development among individuals of a same species might induce intraspecific competition [START_REF] Amarasekare | Effects of temperature on intraspecific competition in ectotherms[END_REF][START_REF] Frogner | Variable Developmental Period: Intraspecific Competition Models with Conditional Age-Specific Maturity and Mortality Schedules[END_REF] as individuals developing faster may reproduce sooner and for a longer period compared to those developing more slowly. Different mechanisms have been found to regulate developmental timing. The so-called heterochronic miRNAs, such as let-7 and miR-125, were originally discovered in Caenorhabditis elegans (Rhabditida: Rhabditidae) [START_REF]MicroRNAs and developmental timing[END_REF][START_REF] Ambros | Heterochronic mutants of the nematode Caenorhabditis elegans[END_REF]. These miRNAs are conserved in a wide range of species, such as Drosophila melanogaster (Diptera : Drosophilidae) [START_REF] Caygill | Temporal regulation of metamorphic processes in Drosophila by the let-7 and miR-125 heterochronic microRNAs[END_REF] or Danio rerio (Cypriniformes: Cyprinidae) [START_REF] Ouchi | The heterochronic genes lin-28a and lin-28b play an essential and evolutionarily conserved role in early zebrafish development[END_REF], as well as in mammals and plants [START_REF]MicroRNAs and developmental timing[END_REF]. They act at post-transcriptional level to regulate cellular mRNA levels, and have been found to control the developmental timing, cell fate and cell differentiation. Hormones are also known to be important regulators of developmental timing.

In D. melanogaster, each of the developmental transitions are regulated by ecdysone pulses, and premature transition from larva to pupa with respect to food conditions or starvation is prevented by juvenile hormone [START_REF] Riddiford | Cellular and molecular actions of juvenile hormone I. General considerations and premetamorphic actions[END_REF][START_REF] Riddiford | Effects of juvenile hormone mimics on larval development and metamorphosis of Drosophila melanogaster[END_REF]. Thus, developmental timing might be regulated to reach an optimal duration with respect to outer environmental factors.

More than 1500 described species of the genus Drosophila [START_REF] Bächli | TaxoDros[END_REF][START_REF] O'grady | Phylogeny of the genus Drosophila[END_REF] occupy a wide range of habitats with various climatic conditions [START_REF] Markow | Reproductive ecology of Drosophila[END_REF]. A dozen of species have been reported to be cosmopolitan species (Markow andO'Grady, 2008, 2005), such as Drosophila melanogaster [START_REF] David | Genetic variation of Drosophila melanogaster natural populations[END_REF][START_REF] Li | Inferring the Demographic History and Rate of Adaptive Substitution in Drosophila[END_REF]) that potentially dispersed with humans from Africa around the globe [START_REF] Mansourian | Wild African Drosophila melanogaster Are Seasonal Specialists on Marula Fruit[END_REF]. These species may be generalists but were also found to be locally adapted to diverse environments [START_REF] Kapun | Genomic Analysis of European Drosophila melanogaster Populations Reveals Longitudinal Structure, Continent-Wide Selection, and Previously Unknown DNA Viruses[END_REF][START_REF] Markow | Reproductive ecology of Drosophila[END_REF]. In contrast, the vast majority of species are restricted to certain continental ranges or are endemic to a specific geographic region that encompasses a unique habitat with specific food and climate conditions (Markow andO'Grady, 2008, 2005). Because of their inability to disperse outside their habitat, these endemic species may have evolved temperature-buffering mechanisms to ensure a constant developmental timing under variable temperature conditions.

Drosophila pachea (Diptera : Drosophilidae) is endemic to the Sonoran desert in Mexico and is an obligate specialist on decayed parts, or rot-pockets, of its single host plant, the Senita cactus (Lophocereus schottii) [START_REF] Gibbs | No place to hide: microclimates of Sonoran Desert Drosophila[END_REF][START_REF] Heed | Unique sterol in the ecology and nutrition of Drosophila pachea[END_REF]Lang et al., 2012;[START_REF] Markow | Drosophila: a guide to species identification and use[END_REF]. The micro-climate of the rot-pockets encompasses important changes of temperature all along the year, with a recorded maximum variation from 5°C to 42°C within 24 h [START_REF] Gibbs | No place to hide: microclimates of Sonoran Desert Drosophila[END_REF]. Living in an environment with large daily and annual temperature changes may require a certain temperature robustness with respect to developmental processes in poikiloterm species. We wondered if the developmental timing in D. pachea may be sensitive to differing rearing temperatures. To test this, we first determined the overall timing of the Drosophila pachea life-cycle. Then, we focussed on pupal development at four different rearing temperatures to investigate differences in the pupal timing. Finally, we compared these durations across closely related sister species Drosophila acanthoptera (Diptera : Drosophilidae) and Drosophila nannoptera (Diptera : Drosophilidae) to investigate potential species-specific developmental timing differences. Flies were maintained in transparent plastic vials (25 x 95 mm, Dutscher) containing about 10 mL of standard Drosophila medium. This medium was composed of 66.6 g/L of cornmeal, 60 g/L of brewer's yeast, 8.6 g/L of agar, 5 g/L of methyl-4-hydroxybenzoate and 2.5% v/v ethanol (standard food). We added 40 μL of 5 mg/mL of 7-dehydrocholesterol (7DHC) (Sigma, reference 30800-5G-F) dissolved in ethanol into the food for D. pachea, as this species need this sterol for proper development [START_REF] Heed | Unique sterol in the ecology and nutrition of Drosophila pachea[END_REF]Lang et al., 2012;[START_REF] Warren | Woc (without children) gene control of ecdysone biosynthesis in Drosophila melanogaster[END_REF]) (standard D. pachea food). As a pupariation support, a piece of paper sheet (1 cm x 4 cm, BenchGuard) was added to each vial. Stocks were kept at 25°C or 29°C at a 12 h light:12 h dark photoperiodic cycle with a 30 min transition between light (1080 lm) and dark (0 lm).

Materials and methods

Drosophila stock maintenance

Cohort synchronisation of D. pachea embryos and time-lapse recording of embryonic development

For collection of cohorts of synchronised embryos, about 250-500 adult flies were transferred into a 9 x 6 cm plastic cylinder, closed by a net on the top and by a 5.5 cm diameter petri-dish lid at the bottom. The petri-dish contained grape juice agar (24.0 g/L agar, 26.4 g/L saccharose, 20% grape juice, 50% distilled water, 12% Tegosept [1.1 g/mL in ethanol] (Dutscher), 4% 7-DHC (Sigma)) and 50-200 μL fresh baker's yeast as food source and egg laying substrate on top. These plates are named hereinafter "food plates". Female flies were let to lay eggs on the food plates for 1 h -2 h (1 h to examine embryos and 2 h to synchronise larvae). Then, eggs were retrieved from food plates by filtering the yeast paste through a 100 μm nylon mesh (BS, Falcon 352360).

For time-lapse imaging the chorion of embryos was removed by a 90 sec incubation of the embryo-containing filter in 1.3% bleach (BEC Javel) under constant agitation until about half of the embryos were floating at the surface of the bleach bath. Embryos were extensively rinsed with tap water for at least 30 sec. Dechorionated embryos were then gently glued on a cover slip (ThermoFisher) coated with Tesa glue. For coating, 50 cm TESA tape was transferred into 25 mL n-heptane (Merck) and glue was let to dissolve overnight at room temperature. A total of 15 µL of dissolved glues was finally pipetted onto a cover slip to form a 5 x 20 mm rectangular stripe and n-heptane was let to evaporate. Embryos were covered with 40 μL of Voltalef 10S halocarbon oil (VWR) to avoid desiccation. Live-imaging was immediately launched inside a temperature and humidity controlled chamber at 25°C ± 0.1°C and 80% ± 1% humidity (Lang and Orgogozo, 2012;[START_REF] Lefèvre | Male genital lobe morphology affects the chance to copulate in Drosophila pachea[END_REF][START_REF] Rhebergen | Drosophila pachea asymmetric lobes are part of a grasping device and stabilize onesided mating[END_REF]. Time-lapse acquisition was performed at an acquisition rate of 1 picture every 7.5 sec using a digital camera (Conrad 9-Megapixel USB digital microscope camera) and Cheese software, version 3.18.1, on a computer with an ubuntu 16.04 linux operating system. Movies were assembled with avconv (libav-tools).

In Drosophila melanogaster and closely related species, females were reported to hold fertilized eggs inside the reproductive tract for >12 hours [START_REF] Markow | Egg size, embryonic development time and ovoviviparity in Drosophila species: Ovoviviparity in Drosophila species[END_REF], which could explain the variation observed in our experiments with D. pachea. Therefore, we monitored egg retention in this species by examining dechorionated eggs from a 1 h egglaying period. We found that all observed embryos (n=52) were early embryos at the syncytial blastoderm stage [START_REF] Wieschaus | Looking at embryos[END_REF]) and egg retention was not observed. Out of 28 embryos monitored, 12 (43%) pursued their development until hatching while the others did not develop at all (Movie S1, Dataset S1). Such mortality has been reported previously [START_REF] Jefferson | BREEDING BIOLOGY OF DROSOPHILA PACHEA AND ITS RELATIVES[END_REF][START_REF] Pitnick | Operational sex ratios and sperm limitation in populations of Drosophila pachea[END_REF] but potentially also dependent on the above-mentioned bleach treatment. The embryos that died during the experiment were excluded from analysis. Furthermore, the duration of hatching, which is the last stage of embryonic development, has been shown to be more variable in comparison to the other embryonic stages in various Drosophila species [START_REF] Chong | Temporal development of Drosophila embryos is highly robust across a wide temperature range[END_REF][START_REF] Kuntz | Drosophila Embryogenesis Scales Uniformly across Temperature in Developmentally Diverse Species[END_REF]. We thus measured both the total embryonic duration, from collection up to larva hatching and the embryonic duration up to the trachea gas filling stage, which precedes the hatching stage (Dataset S1).

Cohort synchronisation of larvae, dissection and imaging of larval mouth hooks

In order to collect cohorts of larvae at a synchronous developmental stage, we first collected embryos from a 2 h egg laying interval (see above) that were placed on a food plate together with fresh yeast. Freshly hatched larvae were retrieved from the yeast paste with fine forceps (Dumont #5, Fine Science Tool) or by filtering the yeast through a nylon mesh (see above). Larvae were transferred into vials containing standard Drosophila pachea food and were examined once a day until all larvae had turned into pupae (Dataset S2).

For imaging of the larval teeth, entire larvae were mounted in 20 μL dimethylhydantoin formaldehyde (DMHF) medium (Entomopraxis) beneath a cover slip (0.17 mm ± 0.01 mm thick, ThermoScientific), which was gently pressed against the microscope slide (ThermoScientific) to orient larval teeth in a flat, lateral orientation to the microscope objective. Larval teeth were imaged at 100 or 400 fold magnification in bright field illumination [START_REF] Strasburger | Drosophila melanogaster meig[END_REF] using the microscope IX83 (Olympus). The instar stage of each dissected individual was determined based on tooth morphology [START_REF] Strasburger | Drosophila melanogaster meig[END_REF] (Figure S1).

Measurement of the duration of puparium formation in D. pachea

The precise duration of puparium formation was characterized by monitoring nine D. pachea pupariating larvae by time-lapse imaging. Larvae at the third instar stage and third instar wandering stage were collected from the D. pachea stock and were transferred into fresh D. pachea standard medium, inside a 5 cm diameter petri-dish and a piece of 1 cm x 4 cm paper sheet (BenchGuard). The dish was then placed into the temperature and humidity controlled chamber at 25°C ± 0.1°C and 80% ± 1% humidity, as previously described. Timelapse acquisition was performed for about 72 h as previously described for embryonic timing characterization. The duration of the white puparium stage was measured from the moment when the larva had everted the anterior spiracles and had stopped moving until the moment when the pupal case had turned brown.

Characterization of developmental timing in pupae

The developmental duration of D. pachea, D. nannoptera and D. acanthoptera was examined by observation of pupae at different time points after puparium formation (APF).

Synchronised pupae were obtained from each species by collecting so-called "white pupae" that had just formed the puparium (Dataset S3). Specimens were collected with a wet brush directly from stock vials. Individuals of the same cohort were placed onto moist Kimtech tissue (Kimberly-Clark) inside a 5 cm diameter petri dish. Petri dishes with pupae were kept at 22°C, 25°C, 29°C, or 32°C inside plastic boxes, which also contained wet tissue paper. Specimens analyzed at 22°C and 32°C were taken from a stock at 25°C at the stage of puparium formation and subsequently incubated at the desired temperature. A group of pupae resulting from a single collection event was considered as a synchronised cohort.

Developmental progress of synchronised cohorts (Table 1, Figure S2) was examined at various time points by visual examination of the pupae using a stereomicroscope VisiScope SZB 200 (VWR) (Dataset S4). Developmental stages were assigned according to morphological markers defined for D. melanogaster by [START_REF] Bainbridge | Staging the metamorphosis of Drosophila melanogaster[END_REF] (Table 2). The markers used to characterize stages 8 to 12 (eye, wing or body pigmentation, Table 2) were not convenient for the characterization by direct observation of D. acanthoptera pupae as these flies develop black eyes, as opposed to most other Drosophila species that have red eyes. In addition, D. acanthoptera is generally less pigmented compared to D. pachea and D. nannoptera [START_REF] Pitnick | New species of cactus-breeding Drosophila (Diptera: Drosophilidae) in the nannoptera species group[END_REF]) and pigmentation changes were not easily detectable through the pupal case. Therefore, we additionally carried out time-lapse imaging of one cohort with five D. acanthoptera pupae to investigate the developmental durations of stages 8-12. The anterior part of the pupal case was removed, letting the head and the anterior part of the thorax visible. Image acquisition was done at 25°C ± 0.1°C and 80% ± 1% humidity, as previously described. Time-lapse acquisition was performed as previously described and recorded with the VLC media player, version 3.0 at an acquisition rate of 1 picture every 13:02 min. Two pupae died during acquisition and were excluded from the analysis (Movie S2).

Data analysis

Data was manually entered into spreadsheets (Datasets S1, S2, S3 and S4) and analysis was performed in R version 3.6 (R Core [START_REF] Team | R: A language and environment for statistical computing[END_REF]. Ages expressed in hours after pupa formation were automatically calculated with respect to the time point of white pupa collection.

Results

D. pachea embryonic and larval development at 25°C last for about 33 h and 216 h, respectively

We roughly examined the duration of embryonic and larval development in D. pachea at 25°C. The average duration of the total embryonic development in D. pachea at 25°C, until hatching of the larva was 32 h 48 min ± 1 h 13 min (mean ± standard deviation ; n = 12) (Figure 1, Movie S1). Embryonic development up to the trachea gas filling stage (see Material and Methods for details) was estimated to be 26 h 48 min ± 1 h 13 min (mean ± standard deviation ; n = 12) (Movie S1). These durations appeared to be longer in D. pachea compared to those reported for various other Drosophila species, such as Drosophila melanogaster, Drosophila simulans, Drosophila sechellia, Drosophila yakuba, Drosophila pseudoobscura, Drosophila mojavensis (Figure 2) [START_REF] David | Essai de définition d'une température optimale pour le développement de la Drosophile[END_REF][START_REF] Kuntz | Drosophila Embryogenesis Scales Uniformly across Temperature in Developmentally Diverse Species[END_REF][START_REF] Powsner | The effects of temperature on the durations of the developmental stages of Drosophila melanogaster[END_REF].

The total duration of D. pachea larval development on standard D. pachea food at 25°C was approximately 9 days (~216 h). The duration of the first and second instar larva were about 2 days each and the third instar stage lasted for about 5 days (Figure 1). In D.

melanogaster, the total duration of the larval stage was about 5 days for larvae reared on optimal food at 25°C, the first and second instars lasting for 1 day each, and the third instar for three days, according to [START_REF] Strasburger | Drosophila melanogaster meig[END_REF]. The larval development of D. pachea appeared thus to be longer compared to those of D. melanogaster at 25°C.

Similar durations of pupal development in D. pachea at 25°C, 29°C and 32°C

The duration of larval development appears to be sensitive to various environmental factors, such as diet [START_REF] Matzkin | Dietary Protein and Sugar Differentially Affect Development and Metabolic Pools in Ecologically Diverse Drosophila[END_REF], crowding, or access to food [START_REF] Vijendravarma | Predatory cannibalism in Drosophila melanogaster larvae[END_REF]. Since pupal development is apparently less affected by such factors, we focussed on the pupal stage to investigate the effect of the rearing temperature on timing of development in D. pachea. We evaluated pupal developmental progress at four temperatures: 22°C, 25, 29°C, and 32°C. Preliminary tests revealed that rearing of D. pachea at temperatures lower than 25°C is prolonged which favors the accumulation of bacterial infections in the food and decreased survival of the flies. At 34°C, D. pachea individuals died within a few days and at 32°C flies survived but did not reproduce. Since we could not cultivate D. pachea at the extreme temperatures of 22°C and 32°C, individuals were selected at the stage of puparium formation in a stock at 25°C and incubated at either temperature. At 25°C -32°C, D. pachea pupae reached the pharate adult stage in less than 55 h but timing was prolonged at 22°C (Figure 3A-E). However, pupal development was accelerated at 29°C and 32°C between stages 8 and 13 (beginning of eye pigmentation until the end of body and wing pigmentation) compared to development at 25°C (Figure 3A-E). In addition, development was consistently slower at 22°C compared to 25°C. However, stages 14 and 15 required more time at 29°C and 32°C with respect to developmental progress at 25°C and resulted in a similar overall duration of about 100 -145 h. Only at 22°C, development was globally slower and adults emerged later, between 150 -190 h. Thus, in D. pachea the rearing temperature influences the relative progress of pupal development at particular stages. While pupal development is slowed-down at temperatures below 25°C, the overall duration appears to be similar at higher temperatures.

The timing of the pupal development is conserved up to the pharate adult stage between D. pachea and various Drosophila species at 25°C

The white pupa stage (see Material and Methods for details) in D. pachea was estimated to last for 102 min ± 41 min (mean ± standard deviation) (n=9) at 25°C. This duration has to be considered as the remaining variation of developmental progress between examined individual pupae in later timing analyses (see Materials and Methods). This duration was similar to previously reported durations for D. melanogaster white pupae of 80-120 min, at 25°C [START_REF] Bainbridge | Staging the metamorphosis of Drosophila melanogaster[END_REF].

At 25°C, the pharate adult stage (stage 7, Table 2) was observed about 55 h after puparium formation and emergence of adults between 115 -145 h after puparium formation (Figures 2A). This timing was similar to those of D. acanthoptera and D. nannoptera (Figure 3B). The developmental duration from puparium formation to pharate adult (stages 1 to 7, from 0 h APF to about 55 h APF) was also similar to those reported for D. melanogaster and D. guttifera (Figure 3B) [START_REF] Bainbridge | Staging the metamorphosis of Drosophila melanogaster[END_REF][START_REF] Fukutomi | Pupal development and pigmentation process of a polka-dotted fruit fly, Drosophila guttifera (Insecta, Diptera)[END_REF]. However, at later pupal development durations of stages were prolonged in D. pachea, D. nannoptera and D. acanthoptera compared to D. melanogaster and D. guttifera.

The emergence of the adult fly from the pupal case (stage 15) is highly variable within D. pachea, adults between 102 h -142 h APF. The variance of this stage was significantly different between the three species (Levene's test: F = 3.4414, Df = 2, p = 0.03847), the stage 15 being longer in D. acanthoptera compared to D. pachea and D. nannoptera (Figure 3B).

Discussion

A possible temperature-buffering mechanism during pupal development

The trend of a decrease of developmental duration when rearing temperature increases was not observed in D. pachea at high temperatures. Overall, pupal development duration were similar at 25°C and 32°C, while it was prolonged at 22°C. On the contrary, the duration of the overall pupal development decreases with increasing rearing temperature in D. melanogaster [START_REF] Ashburner | Laboratory culture of Drosophila[END_REF][START_REF] Powsner | The effects of temperature on the durations of the developmental stages of Drosophila melanogaster[END_REF]. In addition, temperature fluctuations during pupal development of D. melanogaster are known to either increase or decrease developmental speed [START_REF] Ludwig | The effect of alternating temperatures on the pupal development of Drosophila melanogaster Meigen[END_REF][START_REF] Petavy | Viability and rate of development at different temperatures in Drosophila: a comparison of constant and alternating thermal regimes[END_REF].

In this species, the first 24 h of pupal development are more sensitive to temperature changes compared to the rest of the pupal stage [START_REF] Ludwig | The effect of alternating temperatures on the pupal development of Drosophila melanogaster Meigen[END_REF][START_REF] Petavy | Viability and rate of development at different temperatures in Drosophila: a comparison of constant and alternating thermal regimes[END_REF]. While D. melanogaster is a cosmopolitan species that lives in a wide climate range [START_REF] David | Genetic variation of Drosophila melanogaster natural populations[END_REF], D. pachea is a desert species endemic of the Sonora [START_REF] Heed | Unique sterol in the ecology and nutrition of Drosophila pachea[END_REF][START_REF] Markow | Drosophila: a guide to species identification and use[END_REF]. The mean daily variations of temperature of this habitat are 18°C -42°C in spring/summer and 6°C -32°C in fall/winter [START_REF] Gibbs | No place to hide: microclimates of Sonoran Desert Drosophila[END_REF]. D. pachea is found in the wild throughout the year but undergoes a strong population decline during August, when the seasonal temperatures are highest [START_REF] Breitmeyer | Resource availability and population size in cactophilic Drosophila[END_REF].

However, adult D. pachea are particularly resistant to high-temperatures and survive up to 44°C in the wild, while most other Drosophila species revealed a decreasing survival already at 38°C [START_REF] Stratman | Resistance to thermal stress in desert Drosophila[END_REF]. Thus, this species may have developed some heat resistance mechanisms, physiological and/or behavioral, that results in a certain tolerance to temperature variations and would buffer temperature changes on the developmental progress. This buffering effect could potentially be important for proper development since heat stress has been reported to increase developmental instability in various species [START_REF] Kristensen | The use of fluctuating asymmetry and phenotypic variability as indicators of developmental instability: a test of a new method employing clonal organisms and high temperature stress[END_REF][START_REF] Nishizaki | Thermal stress increases fluctuating asymmetry in marine mussels: environmental variation and developmental instability[END_REF][START_REF] Polak | Developmental selection against developmental instability: a direct demonstration[END_REF]. However, the specific mechanism by which temperature affects developmental stability is not well understood [START_REF] Abrieux | EYES ABSENT and TIMELESS integrate photoperiodic and temperature cues to regulate seasonal physiology in Drosophila[END_REF][START_REF] Breuker | Heat shock in the developmentally sensitive period of butterfly eyespots fails to increase fluctuating asymmetry[END_REF][START_REF] Carvalho | The 4E-BP growth pathway regulates the effect of ambient temperature on Drosophila metabolism and lifespan[END_REF][START_REF] Enriquez | Cold Acclimation Favors Metabolic Stability in Drosophila suzukii[END_REF]. Rearing at a lower temperature (< 25°C) revealed slower developmental progress, indicating that a potential buffering for colder temperatures does not exist in D. pachea.

Alternatively, the observed buffering phenotype may be temperature independent and could perhaps ensure the emergence of the adult fly at a particular moment of the day, such as dawn or dusk, when the environmental temperature might be most suitable for the freshly emerged individual. In the last pupal stage that corresponds to the adult emergence, we observed timing variation between individuals in D. pachea (up to 75 h between individuals). This variation could potentially depend on individual differences or on environmental factors that we could not control, such as the light/dark illumination cycle at the moment of adult emergence. Such circadian regulation of adult emergence has been observed in various Drosophila species [START_REF] Ashburner | Drosophila: a laboratory handbook[END_REF][START_REF] Mark | The circadian clock gates Drosophila adult emergence by controlling the timecourse of metamorphosis[END_REF][START_REF] Powsner | The effects of temperature on the durations of the developmental stages of Drosophila melanogaster[END_REF][START_REF] Soto | Pupal emergence pattern in cactophilic Drosophila and the effect of host plants[END_REF]. However, the important variation in the last pupal stage is also found among individuals of the same cohort (Dataset S3 and S4). Future monitoring of the emergence of adults from various cohorts collected at different moments of the day will be necessary to test this hypothesis. Future investigations will be needed to further characterize the potential temperature buffering effect during D. pachea development and to test the influence of the circadian rhythm in this species. In addition, we must further assess temperature dependent pupal development in a wider range of species that live in distinct climate habitats.

Conservation of the overall developmental progress during early pupal stages

The detailed analysis of the timing of pupal stages revealed that the first stages 1 to 7 appear to be rather synchronous among D. melanogaster [START_REF] Bainbridge | Staging the metamorphosis of Drosophila melanogaster[END_REF], D. guttifera [START_REF] Fukutomi | Pupal development and pigmentation process of a polka-dotted fruit fly, Drosophila guttifera (Insecta, Diptera)[END_REF], and the three closely related species D. pachea, D. acanthoptera and D. nannoptera. Later on, pupal development appears to be more variable between species. This may indicate the existence of some developmental constraints, which are limitations of phenotypic variability due to inherent properties of the developmental system [START_REF] Smith | Developmental constraints and evolution: a perspective from the Mountain Lake conference on development and evolution[END_REF][START_REF] Wagner | Homology, genes, and evolutionary innovation[END_REF]. Such constraints probably act on outgrowth of adult organs from primordial structures, so-called imaginal discs, that develop throughout larval stages but undergo extensive tissue growth during pupal development up to the pharate adult stage. Thereafter, the timing of development seems to be less constrained and interspecific variations were observed. At least a part of the variation in the pupal developmental timing could be attributed to the developmental marker used. As the coloration markers are qualitative, it is hard to define precise limits of each stage (ie. eyes turn progressively from yellow to red). A solution might be to identify a combination of multiple markers for each stage or to establish gene expression markers that are known to account for specific developmental processes, as it has been recently done for eye development [START_REF] Escobedo | Characterizing a gene expression toolkit for eye-and photoreceptor-specific expression in Drosophila[END_REF] or male genitalia development [START_REF] Vincent | An atlas of transcription factors expressed in male pupal terminalia of Drosophila melanogaster[END_REF].

D. pachea embryonic and larval developmental durations appear to be longer compared to other Drosophila species

The embryonic developmental duration at 25°C has been investigated in 11 drosophila species other than D. pachea [START_REF] Chong | Temporal development of Drosophila embryos is highly robust across a wide temperature range[END_REF][START_REF] Crapse | Evaluating the Arrhenius equation for developmental processes[END_REF][START_REF] David | Essai de définition d'une température optimale pour le développement de la Drosophile[END_REF][START_REF] Kuntz | Drosophila Embryogenesis Scales Uniformly across Temperature in Developmentally Diverse Species[END_REF][START_REF] Powsner | The effects of temperature on the durations of the developmental stages of Drosophila melanogaster[END_REF] (Figure 2) and ranged from 16 h in D. sechellia to 25 h in D. virilis [START_REF] Chong | Temporal development of Drosophila embryos is highly robust across a wide temperature range[END_REF][START_REF] Kuntz | Drosophila Embryogenesis Scales Uniformly across Temperature in Developmentally Diverse Species[END_REF]) (Figure 2), which appear to be shorter compared to embryonic development of D. pachea at the same temperature. Interspecific variation in the duration of embryonic development might rely on genetic factors, as closely related species tend to have similar embryonic developmental durations compared to those of distantly related ones (Figure 2). Overall, sample size was rather low in our experiments and only present a rough approximation of the time range of Drosohila larval and embryonic development at a single temperature. A detailed examination would be necessary to adequately refine the duration of each developmental stage.

Larval development appeared to be longer in D. pachea compared to those in D. melanogaster [START_REF] Bakker | Feeding period, growth, and pupation in larvae ofDrosophila melanogaster[END_REF][START_REF] Strasburger | Drosophila melanogaster meig[END_REF]. However, the duration of this developmental stage has been shown to be highly variable compared to the other life stages. In particular it has been shown that larvae are very sensitive to food composition and to crowding that affect food quality and food access [START_REF] Matzkin | Dietary Protein and Sugar Differentially Affect Development and Metabolic Pools in Ecologically Diverse Drosophila[END_REF][START_REF] Vijendravarma | Predatory cannibalism in Drosophila melanogaster larvae[END_REF]. Food quality and food access in turn prolong the larval developmental duration [START_REF] Matzkin | Dietary Protein and Sugar Differentially Affect Development and Metabolic Pools in Ecologically Diverse Drosophila[END_REF]. This effect of food on developmental duration might also probably affect embryonic and pupal stages indirectly due to nutrient contribution from the adult and larval stages. A slower development observed in D. pachea raised in the lab might also be due to variations in the ecdysone metabolism. In insects, ecdysone is first provided to the embryo as maternal contribution and then directly produced by the individual [START_REF] Lafont | 4 -Ecdysteroid Chemistry and Biochemistry[END_REF]. However, in D. pachea the first metabolic step of the ecdysone biosynthesis is different compared to other insect species, the conversion of cholesterol into 7dehydrocholesterol being abolished (Lang et al., 2012). Instead, D. pachea metabolizes sterols produced by the Senita cactus on which they feed, such as lathosterol, and potentially campestenol and schottenol [START_REF] Heed | Unique sterol in the ecology and nutrition of Drosophila pachea[END_REF], into steroid hormones differing in their side residues (Lang et al., 2012). Therefore, in the wild, D. pachea may produce different variants of ecdysone that may also differently affect developmental timing compared to the lab conditions that only provide the single ecdysone precursor 7dehydrocholesterol. Thus, it would be interesting to compare developmental durations of D. pachea fed with standard D. pachea food used in the lab or with their natural host plant, the Senita cactus. In addition, further investigations would be needed to elucidate how temperature modulates these mechanisms.

Conclusion

We investigated the effect of temperature on developmental speed in D. pachea, a desert species. We characterized the timing of the life-cycle in this species and observed prolonged developmental durations compared to other Drosophila species. The global developmental duration during metamorphosis is similar at rearing temperatures between 25°C and 32°C although stage specific timing differences were observed. These observations indicate that D. pachea might potentially have evolved mechanisms to buffer the effect of temperature on developmental speed. Such mechanisms might be of importance to preserve the fitness of individuals exposed to extreme temperatures and important temperature variations during their development.

Tables

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Durations of the embryonic development in various Drosophila species at 25°C. The duration of total embryonic development of D. pachea (grey) was established based on time-lapse imaging. The data for the species other than D. pachea were extracted from: blue: Kuntz and Eisen, 2014 (duration up to the trachea filling stage, at 25°C), yellow: David and Clavel, 1966 (total embryonic development, at 25°C) and green: Powsner, 1935 (total embryonic development, at 25°C). The data used to establish the cladogram was extracted from Yassin (2013) and Lang et al. (2014).

Figure 3 :

 3 Figure 3: Progress of pupal development. A-D: Durations of developmental stages in D. pachea pupae at 22°C (A), 25°C (B), 29°C (C) and 32°C (D), observed at various time points. Temperatures are highlighted in blue colour tones according to the legend. Black dots indicate single observations (Dataset S3). E: Overlay of durations from panels A -D. F: Comparison of pupal development at 25°C in D. pachea (blue), D. acanthoptera (green) and D. nannoptera (yellow) based on observations of synchronized cohorts. The stages 8 to 12 were determined in D. acanthoptera by timelapse imaging of developing pupae, after removal of the anterior part of the pupal case (dotted lines). Data of D. melanogaster (pink) and D. guttifera (purple) were retrieved from Bainbridge and Bownes (1981) and Fukutomi et al. (2017), respectively. These species were indicated by stars in the legend.

  Drosophila stocks were retrieved from the San Diego Drosophila Species Stock Center (now The National Drosophila Species Stock Center, College of Agriculture and Life Science, Cornell University, USA). The D. pachea stock 15090-1698.01 was established in 1997 from individuals caught in Arizona, USA. The D. nannoptera stocks 15090-1692.00 and 15090-1693.12 were established in 1992 from individuals caught in Oaxaca, Mexico. The D.

	acanthoptera stock 15090-1693.00 was established in 1976 from individuals caught in
	Oaxaca, Mexico (UCSC Drosophila species stock center San Diego, now The National
	Drosophila Species stock center, Cornell University). These stocks have been kept in good
	conditions at 25°C in our laboratory since 2012.

Table 1 : Total numbers of pupae and synchronised cohorts used in for pupal timing characterization

 1

	Species	Temperature (°C)	Total number of	Total number of
			pupae	synchronised cohorts
	D. pachea	22	134	21
		25	76	11
		29	42	5
		32	141	21
	D. nannoptera	25	40	13
	D. acanthoptera	25	61	7

Table 2 : Summary of morphological markers used to stage pupae, according to Bainbridges and Bownes (1981)

 2

	Pupal	Description
	stage
	1	Pupariation: extremity of trachea are everted, pupa does not move anymore.
	2	Clear, white pupa.
	3	Light pigmentation, dorsal trachea still visible.
	4	Bubbles appear, dorsal trachea is not visible anymore, light pigmentation of the body.
	5	Cranial extremity is retracted, distal extremity of wings appeared.
	6	Yellow body is visible.
	7	Pharate adult, eyes are not yet pigmented.
	8	Eye discs become a bit yellower compared to the rest of the body.
	9	Orange eyes, transparent wings are visible.
	10	Deep red eyes.
	11	Bristles are visible on the thorax.
	12	Wings are clear grey, clear pigmentation of the body.

13

Wings are completely black, grey pigmentation appears on the body.

Acknowledgements

We thank all team members of the Evolution and Genetics team, including Jean David, for stimulating and constructive discussions. We thank Virginie Courtier for comments on the manuscript and for covering a part of the experimental costs. Funding sources BL was supported by a pre-doctoral fellowship Sorbonne Paris Cité of the Université Paris 7 Denis Diderot and by a fellowship from the Labex ''Who am I?'' ["Initiatives d'excellence", Idex ANR-18-IDEX-0001, ANR-11-LABX-0071]. This work was further supported by the CNRS, by a grant of the European Research Council under the European Community's Seventh Framework Program [FP7/2007-2013 Grant Agreement no. 337579] given to Virginie Courtier-Orgogozo and by a grant of the Agence Nationale pour la recherche [ANR-20-CE13-0006] given to ML.

Availability of data and material

The movies S1 and S2 supporting the results of this article are available in the DRYAD repository, https://datadryad.org/stash/share/dfhCAtgopC4JY6qmkjK6Q_UEMmf2WSfc1gdETPPI7gk.

Conflict of interest

The authors have no conflict of interest to declare. 2). Pupae of each stage are presented in dorsal (D) and ventral (V) view. The stage (Table 2) is indicated by a number. Arrows point to relevant morphological markers: C: stage 3, dorsal trachea still visible; D: stage 4, bubbles appear and trachea not visible; E: stage 5, distal margins of wings appear; F: stage 6, yellow body visible; G: stage 7, non-pigmented eyes visible; H: stage 8, yellow eyes appear; I: stage 9, orange eyes; J: stage 10, red eyes; K: stage 11, thorax bristles visible; L: stage 12, grey wings; M: stage 13, black wings; N: stage 14, meconium visible. The scale is 100 µm.

Supplementary data

Movie S1: Time-lapse of embryonic development of D. pachea at 25 °C.

Out of the 28 embryos, 12 completed their development up to the larva hatching. The 16 embryos that did not complete their embryonic development were excluded from the analysis.

Movie S2: Time-lapse of pupal development of D. acanthoptera from 52 h APF up to the emergence of the adult at 25°C

Out of 5 pupae, 3 completed their development up to adult emergence. The two that died during the time-lapse were excluded from analysis.