Jean-François Laslier

François Maniquet

Classical electoral competition under approval voting

We study a Downsian model of electoral competition with an arbitrary number of parties. The voting rule is approval voting. We assume that voters are strategic in the sense of the Leader Rule of Laslier (2009, Jnl. Th. Pol.). We show that if a Condorcet winner policy exists, then there exists an electoral competition equilibrium supporting this policy. Moreover, if the set of policies is one-dimensional and voters have single-peaked preferences, then it is the only electoral competition equilibrium.

Introduction

In large societies, collective decisions cannot be taken directly but have to be delegated to professional decision makers. In a democracy, these delegates are typically elected through a competitive mechanism. The simplest expression of such a mechanism is the now standard Downsian model of Politics [START_REF] Downs | An Economic Theory of Democracy[END_REF] in which a relatively small number of candidates face a relatively large number of voters, the candidates are purely office-motivated and the voters 1 policy-motivated. For the purpose of winning the election, each candidate freely and independently proposes a policy from a fixed and common set of possible policies. Voters are only interested in policies and not in candidates per se. They trust that the elected candidate will implement the policy she is proposing.

The usual case in the literature considers only two candidates under plurality voting. Then voters only face a binary choice, so that each voter simply votes for the candidate whose policy she prefers. In that case, competition for office drives the candidates to propose popular policies. In particular, if there exists a policy preferred to any other by a majority of voters -a Condorcet alternative -then both candidates propose this same policy. This statement is even an if and only if statement since, as soon as no Condorcet alternative exists, the two-party Downsian game has no pure-strategy equilibrium. Formal political science has studied this question in great details, and the literature on two-party competition under plurality rule is very large; see for instance the books of Ordeshook (1992), [START_REF] Roemer | Political Competition: Theory and Applications[END_REF], [START_REF] Dennis | Public Choice III[END_REF], or [START_REF] Austen | Positive Political Theory II: Strategy and Structure[END_REF]. This chapter is devoted to the study of multiparty electoral competition under approval voting in the Downsian political context where collective choice is delegated to office-motivated candidates. To recall, approval voting is the electoral rule under which voters are given the right to approve of as many candidates as they wish, and each approval gives one point to the approved candidate. The winner of the election is the candidate having received the largest number of approvals.

Rational behavior of the voters rests on their beliefs about two things. On one hand, they have to wonder which candidate is most likely to win the election and which candidates can challenge this front-runner. On the other hand, they have to make up their mind as to the policies that each candidate would implement, if elected. Rational behavior of the candidates choosing platform campaigns, in turn, rests on their knowledge about the choices of the voters and of the other candidates. We study electoral competition in a framework where the candidates choose rationally (and simultaneously) their political platforms, and the voters react to these platforms. With more than two candidates, not only the voting rule matters but the behavior of voters is no longer as straightforward as it is with two candidates. Studying approval voting, we make the assumption that voters follow the Leader Rule, a behavioral rule which has a rational foundation [START_REF] Laslier | The Leader rule : a model of strategic approval voting in a large electorate[END_REF]) and satisfies the criterion of sincerity of Brams and Fishburn (1986) and admissibility of [START_REF] Dellis | Policy Moderation and Endogenous Candidacy in Approval Voting Elections[END_REF].

We prove that when voters follow that rule, the outcome of the electoral competition among candidates converges towards the Condorcet winner policy in the following sense: if a Condorcet winner policy exists, then there exists an equilibrium that supports it; and if, moreover, the set of policies is one-dimensional and voters' preferences are single-peaked, then this equilibrium is the only one. The prediction of the model is thus that the approval voting electoral rule drives office-motivated candidates to policy moderation.

In Section 2, we present the model and we recall the definition of the Leader Rule. In Section 3, we present the results. In Section 4, we discuss some possible extensions.

The model 2.1 Candidates, voters, and preferences

There is a set X of possible policies. We consider two models below. In the first one, X is a finite set with no particular structure. In the second one, X is the real line.

In both models we make the following assumptions on voters and candidates. There is a set {1, ..., N} of N voters. Voters have preferences over X. There is a set {1, ..., K} of K candidates. Each candidate k ∈ {1, ..., K} has to choose a policy

x k ∈ X.

We assume that policy x k is implemented if candidate k is elected. Consequently, a voter prefers candidate k to candidate k 0 if and only if she prefers x k to x k 0 , k, k 0 ∈ {1, ..., K}, and we can equally well speak in terms of preference over candidates or preference over policies. The way voters vote among policies {x 1 , ..., x K } is described below. Let us begin by describing the preferences of the candidates. As a result of the election, a fraction of the voters, which we denote by s(k), approve of policy x k , k ∈ {1, ..., K}. The number of approvals of k is thus Ns(k). This number is called the score of k. The winning candidate is the one with highest score. If several candidates obtain the same, highest, score, the winner is decided by a fair lottery. We assume that the objective of a candidate is to maximize the probability of winning the election.

Let x, y ∈ X. Voters may prefer x to y, y to x, or be indifferent. We assume that the profile of voters' preferences over X is fixed. Given this preference profile of the population {1, ..., N}, we can compute g(x, y) ∈ [0, 1], the fraction of the voters who strictly prefer x to y and i(x, y) ∈ [0, 1], the fraction of the voters who are indifferent between x and y. Note that g(x, y)g(y, x) measures the relative plurality in favor of policy x against policy y. By definition, g(x, y) + g(y, x) + i(x, y) = 1.

We suppose that the number of voters is large.

Individual voting behavior

Let us now describe how voters choose their vote. Here, we follow the behavioral rule developed in [START_REF] Laslier | The Leader rule : a model of strategic approval voting in a large electorate[END_REF] and we adapt it to the current model. A rational voter responds to the number of approval votes granted by the other voters to the various candidates (their scores). Let us assume that s(k) represents the fraction of voters approving of k when we do not take account of a given voter's vote. First, the voter deduces from (s(1), . . . , s(K)) a strict ranking c 1 , c 2 , ..., c K of the candidates. Candidate c 1 is the leader, according to that voter. This ranking needs to be compatible with the scores in the following sense.

Definition 1

The ranking c 1 , c 2 , ..., c K of the candidates is compatible with a score vector s = (s(1), . . . , s(K)) if for all k, k0 ∈ {1, . . . , K},

s(c k ) > s(c k 0 ) ⇒ k < k 0 .
If the score vector is such that the K candidates have distinct scores then there is a unique compatible ranking. That is the case analysed in [START_REF] Laslier | The Leader rule : a model of strategic approval voting in a large electorate[END_REF]. Otherwise, the candidates with identical scores can be ranked in any way, providing multiple compatible rankings.

Recall that each voter has fixed preferences over X. For any list of policy positions x = {x 1 , . . . , x K }, there is an induced preference relation of this voter over candidates. When the voter has strict preferences over the candidates, the Leader Rule stipulates that she approves of all the candidates she strictly prefers to c 1 and of no candidate she finds strictly worse than c 1 , and she votes for c 1 if and only if she prefers c 1 to c 2 . When the voter is likely to be indifferent to several candidates, the rule can be generalized as follows.

Assumption 1 Leader Rule: Given a strict ranking c 1 , c 2 , ..., c K of the candidates, a voter behaves as follows:

• If she is indifferent among all candidates, then she approves each of them with probability 1/2, independently.

In all other cases, for any candidate d: If the score vector is such that the K candidates have distinct scores, then this postulated behavior defines a unique ballot for any voter, except in the case where all candidates are indifferent. If the score vector contains ties, several rankings of the candidates are compatible. That may lead to different responses for some voters. For instance, let us assume that s(1) = s(2) > s(3) and the preferences of the voter are: Candidate 1 is preferred to candidate 3, preferred to candidate 2. If the strict ranking of the candidates compatible with scores is 1, 2, 3, then the voter approves only of 1. If the ranking is 2, 1, 3, then she approves of 1 and 2.

Let us briefly present the rationale for the Leader Rule. Assume that the scores represent how voters plan to vote, but for each voter and for each candidate she plans to approve, there is a small chance ² that the vote is not recorded, or that she forgets to cast that vote, etc. Then, the actual number of approvals for a candidate k ∈ K becomes a random variable of mean (1 -²)Ns(k). As a consequence, in spite of the fact that the expected scores of two candidates differ, there is always a positive probability that they tie, so that the vote of this voter is pivotal. Reasoning on these pairwise ties and neglecting three-way ties, a voter votes for a candidate if and only if the most likely serious tie event involving that candidate is one where the former is strictly preferred to the latter (a tie is serious if the voter is not indifferent between the two candidates). [START_REF] Laslier | The Leader rule : a model of strategic approval voting in a large electorate[END_REF] proves that it gives the above voting behavioral rule, and Nunez (2010) presents this rule and other related models for large electorates.

Electorate voting

To define the electorate response to a score vector s = {s(1), . . . , s(K)}, suppose that s has exactly M compatible rankings. We make the assumption that a proportion 1/M of the population of voters adopts each of these rankings, independently of the types. For instance, in the above example s(1) = s(2) > s(3), among all voters sharing the same preferences, fifty per cent will behave according to the ranking 1, 2, 3, and fifty per cent according to 2, 1, 3. This assumption only makes sense in sufficiently large populations. This is precisely our definition of a large population.

Assumption 2 Uniform tie-breaking: Given a score vector s, each voter chooses a ranking of the candidates compatible with s, and responds to this ranking. The choice of the ranking is uniform among the rankings compatible with s, and it is independent of the voter's preferences and of the other voters' choices.

The above assumptions form a simple and natural extension of the Leader Rule defined by [START_REF] Laslier | The Leader rule : a model of strategic approval voting in a large electorate[END_REF] to handle the possibility of ties, although it is only justified by some kind of law of insufficient reason, as is often the case for uniform rules. Notice that, in any case, since each ballot is defined by the Leader Rule applied to the appropriate ranking, all ballots are sincere and admissible.

Equilibrium

We define an electoral competition game as one with K + N players, the K candidates and the N voters. In the first stage of the game, each candidate k chooses a policy x k ∈ X. In the second stage of the game, voters vote, using approval voting. Each vote has a given probability of not being recorded, as explained above. Depending on which votes are recorded, candidates receive numbers of approvals. The candidate with the largest number of approvals is the winner of the election. Ties are broken by a fair lottery. Voters derive utility from the (lottery over) policies that were chosen by the winning candidates. Candidates derive utility from the probability of being elected.

If we restrict our attention to the second stage of the game, then we are back to the game studied by [START_REF] Laslier | The Leader rule : a model of strategic approval voting in a large electorate[END_REF] except that candidates' score vector can now contain ties. The Leader Rule tells us how voters react to the expected vector of scores. Now, the expected vector of scores itself is completely determined by the voters' expected votes. The equilibrium notion we consider, which we call uniform consistency, is the fixed point of that relation. Definition 2 A score vector s = (s 1 , s 2 , ..., s 3 ) is uniformly consistent with policy positions x = (x 1 , x 2 , ..., x K ) if s is the score vector that is obtained when voters react, according to assumptions 1 and 2, to s itself.

We are interested in the subgame perfect equilibria of the electoral competition game. Using the above definition, such an equilibrium is a pair (x, s) of positions and scores such that s is uniformly consistent with x and for no candidate k there exists a unilateral deviation x 0 = (x 0 k , x -k ) and a score vector s 0 uniformly consistent with x 0 such that the probability of k winning the election is higher in s 0 than in s.

For the sake of completeness, we prove the following result, which consists of adapting Laslier's result to the current framework. That results concerns cases where candidates choose policies in such a way that no voter is indifferent between any two policies. In those cases, if a strict Condorcet winner policy exists, then there is a unique score vector uniformly consistent with it. That vector is easily built by using the g function.

Recall that a Condorcet winner policy is one that is preferred to any other by a majority of voters.

Definition 3 A list of policy positions x = (x 1 , x 2 , ..., x K ) admits a Condorcet winner policy x `if and only if for all k 6 = `,

g(x `, x k ) ≥ 1/2.
It admits a strict Condorcet winner if the above inequality is strict for all k 6 = `.

A policy x `may be a Condorcet winner in a list of policy positions x = (x 1 , x 2 , ..., x K ) even if there exists a policy y ∈ X that is preferred to x `by a majority of voters, provided y is not in the list of policy positions. We will come back on that key issue in the next section.

Proposition 1 Let us assume that the list of policy positions x = (x 1 , x 2 , ..., x K ) is such that no voter is indifferent between any pair of policies. If x admits a strict Condorcet winner policy x `, then there exists a unique score vector s that is uniformly consistent with x. This score vector is defined by: For all k 6 = `, s k = g(x k , x `) < 1/2, and

s `= min k6 =`g (x `, x k ).
Proof. 1) The score vector s is consistent with x: Let L (2) denote the set of candidates obtaining the second largest score. By construction, for all k, k 0 ∈ L (2) : g(x k , x `) = g(x k 0 , x `). For all k ∈ L (2) , a fraction 1 |L (2) | of voters has ranking `, k, . . .. Among those voters, a fraction g(x `, x k ) vote for `, a fraction g(x k , x `) vote for k, and for all h 6 = `, k, a fraction g(x h , x `) vote for h. Consequently,

s(`) = 1 |L (2) | X k∈L g(x `, x k ) = g(x `, x k 0 ) ∀ k 0 ∈ L (2) ,
and for all k 6 = `: s(k) = g(x k , x `). Finally, as for all k ∈ L (2)

, k 0 / ∈ L (2) ∪ {`} : s(k) > s(k 0 ), by construction, g(x k , x `) > g(x k 0 , x `). This implies g(x `, x k ) < g(x `, x k 0 ) so that s `= min k6 =`g (x `, x k ).
2) Unicity: Let L (1) denote the set of candidates obtaining the largest score. Assume `/ ∈ L (1) . Then,

s(`) = 1 |L (1) | X k∈L (1) g(x `, x k ) > 1 2 .
First case: L (1) contains more than one candidate. Then, for all k ∈ L (1) ,

s(k) = 1 |L (1) | X ⎛ ⎝ k 0 ∈ L (1) k 0 6 = k ⎞ ⎠ g(x k , x k 0 ) + 1 |L (1) | 1 |L (1) | -1 X ⎛ ⎝ k 0 ∈ L (1) k 0 6 = k ⎞ ⎠ g(x k , x k 0 ) = 1 |L (1) | -1 X ⎛ ⎝ k 0 ∈ L (1) k 0 6 = k ⎞ ⎠ g(x k , x k 0 ).
Summing up these scores, and recalling that g(x k , x k 0 ) + g(x k 0 , x k ) = 1 (no voter is indifferent between x k and x k 0 ), we get X k∈L (1) s

(k) = 1 |L (1) | -1 X k,k 0 ∈L (1) g(x k , x k 0 ) + g(x k 0 , x k ) = 1 |L (1) | -1 (|L (1) |)(|L (1) | -1) 2 = |L (1) | 2 ,
so that for all k ∈ L (1) :

s(k) = 1 2 .
To summarize, we have s(k) < s(`), a contradiction. Second case: L contains one candidate, say 1. We must have s(1) > 1 2 and for all k in the set of candidates ranked second, s(k) = g(x k , x 1 ) < 1 2 , which is inconsistent with s(`) > 1 2 . There are two important directions in which the above result does not extend. First, even if the profile has a Condorcet winner, if the Condorcet winner is not strict, then it is possible that no uniformly consistent scores exist.

Example 1 Consider a set of three candidates {1, 2, 3} such that the pairwise comparisons among candidates are: g(x 1 , x 2 ) = .5, g(x 1 , x 3 ) = .6, g(x 2 , x 3 ) = .1. No uniformly consistent scores exist for this profile. To see that one can check the impossibility for each ordering, strict or not, of the candidates according to s. For instance if 1 is alone at the first place in s, 2 at the second place, and 3 at the third, then the scores should be s 1 = g(x 1 , x 2 ) = .5 and s 2 = g(x 2 , x 1 ) = .5 also, a contradiction. If 1 and 2 tie at the first place and 3 comes third, then s 1 = g(x 1 , x 2 ) = .5, s 2 = g(x 2 , x 1 ) = .5, and s 3 = (g(x 3 , x 1 ) + g(x 3 , x 2 ))/2 = .65 > .5, a contradiction. The reader will easily complete this proof.

Second, if (a non-negligeable fractions of) voters have indifferences, then there may be several score vectors uniformly consistent with the policy positions and even a strict Condorcet winner may fail to be ranked first in such a score vector.

Example 2 Consider a set of three candidates {1, 2, 3} and their policy positions x = (x 1 , x 2 , x 3 ) inducing the preferences described in the following table (which reads: 4 voters are indifferent between 1 and 2 and strictly prefer any of these two to 3, etc.):

4 3 2 1, 2 3 3 3 1 2 2 1
Observe that 3 is a strict Condorcet winner. Nevertheless consider the following score vector: s(1) = 7, s(2) = 6, s(3) = 5, in which candidate 3 is ranked last. One can easily check that, applying the Leader Rule, all voters vote for two candidates, and that this score vector is consistent with x.

Of course, if the profile of candidates has no Condorcet winner, it is allthe-most possible that no uniformly consistent score exists. The fact that uniformly consistent scores may fail to exist is a difficulty for the study of electoral competition under AV in the general case, under the uniform tiebreaking assumption. The results (in the next section) will thus be limited to some observations, in the case of existence of a Condorcet winner. The result conveyed in example 2 above involves preferences that are not single-peaked: the only rankings compatible with the existence of preferences 312 and 321 would have 3 in the middle, which excludes preferences (12)3. The last result in the next section shows that example 2 cannot hold if preferences are single-peaked.

Results

We are now equipped to prove our two results. Both results hold even if agents have indifferences among some pairs of policy positions. They confirm the close relationship between approval voting and the Condorcet winner. Proposition 3 essentially states that a Condorcet winner policy, if it exists, can always result from electoral competition under approval voting, and Proposition 3 essentially states that, in single-peaked domains, this is the only possible outcome. The section is completed by showing that these results do not hold if approval voting is replaced with plurality voting.

Condorcet-consistency

In the previous section, we defined a Condorcet winner by reference to a list x = (x 1 , . . . , x K ) of policy positions. If we look at the entire set X of possible policies, we can define a Condorcet winner policy as one that is preferred to any other policy in X by a majority of voters. Let us note that there is no logical relation between the existence of a Condorcet winner in X and the existence of a Condorcet winner relative to a list of K policy positions in X.

The first result states that if a strict Condorcet winner exists in X, then, independently of the structure of X, all candidates choosing that policy position is an equilibrium.

Proposition 2 If x C ∈ X is a Condorcet winner policy then the strategy profile in which all candidates choose x C is an equilibrium of the electoral competition game. If x C is a strict Condorcet winner then the equilibrium is strict.

Proof. Let x C ∈ X be a Condorcet winner policy. Let x = (x 1 , . . . , x K ) be defined by: For all k ∈ {1, . . . , K} : x k = x C . Then, by the Leader Rule, each candidate is elected with probability 1/K. Suppose candidate 1 (for instance) deviates to x 1 6 = x C . There are now two different policy positions to choose from. Independently of how they are ranked, we have s(1) = g(x 1 , x C ) ≤ .5 and for all k ∈ {2, . . . , K} : s(k) = g(x C , x 1 ) ≥ .5. Candidate 1's probability of being elected is now either 1/K or 0. In any case, the deviation is not profitable. If x C is a strict Condorcet winner policy, then s(1) = g(x 1 , x C ) < .5 and for all k ∈ {2, . . . , K} : s(k) = g(x C , x 1 ) > .5. Consequently, 1's probability of being elected decreases to 0.

Median convergence

We consider in this section the standard, one-dimensional, single-peaked model. The set of possible policies is the real line X = R. Each voter j ∈ N has a preferred policy p j . Moreover, for two policies x, y ∈ X on the left of p j (resp., on the right of p j ), x is strictly preferred to y if and only if x is closer to p j than y: y < x < p j (resp., p j < x < y). Let x m ∈ X be the median of the voters' preferred policies -as many voters have their preferred policy at the left as at the right of x m . We suppose that this point exists and is unique. Then, as is well-known, this policy-moderated, centrist outcome, x m is a strict Condorcet winner: for any y 6 = x m , g(x m , y) > .5, a strict majority of the population strictly prefers x m to y. The previous result applies and all candidate policy positions being concentrated at the median point is a strict equilibrium of the electoral competition. The following proposition, our main result, also proves it is the only equilibrium. That shows that, under approval voting, electoral competition drives candidates to propose the Condorcet policy platform.

Proposition 3 In the single-peaked model: (i) The strategy profile in which all candidates choose the median policy position is a strict equilibrium of the electoral competition game. (ii) It is the only equilibrium.

Proof. Point (i) follows from Proposition 2. (ii) We first note three facts related to the single-peaked profile structure. Let x, y, z ∈ X be such that x < y < z.

• Fact 1: no voter is indifferent between the three positions.

• Fact 2: the voters (if any) who are indifferent between x and z strictly prefer y to both x and z.

• Fact 3: g(y, x) ≥ g(z, x) and g(y, z) ≥ g(x, z).

Next we observe that, from the definition of the Leader Rule (Assumption 1), if two candidates k, k 0 , propose the same policy x k = x k 0 , they obtain the same number of votes and any other candidate l obtains the same number of votes as l would obtain if there was only one candidate at position x k . Now, let x = (x 1 , . . . , x K ) be some list of policy positions chosen by the candidates. For the ease of reading, and when no confusion in the course of the proof can arise, we can neglect the possibility of several candidates located at the same position and we will speak of "a set of candidates" rather than "a set different policy positions chosen by candidates."

There is at least one candidate, say 1, with a probability of winning the election less than or equal to 1/K. We will prove that deviating to x 0 1 = x m is profitable. 1 Let s = (s(1), . . . , s(K)) be a score vector associated to x 0 = (x 0 1 = x m , x 2 , . . . , x K ). Notice that s(1) is equal to or larger than some average of g(x m , x k ), for k ∈ {2, ..., K}, and because x m is a strict Condorcet winner, s(1) > .5. Let L (1) denote the set of candidates obtaining the largest score.

If all the candidates except candidate 1 are located at x m then the probability of winning goes from 0 to 1/K when candidate 1 deviates to x m . We can thus suppose that some candidates are not located at x m and we will prove that when deviating to x m , 1 ∈ L (1) and L (1) contains at most K -1 candidates, which makes the deviation profitable for candidate 1. Assume 1 / ∈ L (1) . We distinguish three cases. Case 1: L (1) contains one candidate, say 2. Let L (2) denote the set of candidates obtaining the second largest score. Assume that L (2) contains more than one candidate. By Fact 1, all the scores of 2 and the candidates of L (2) are determined by the preferences over these candidates. To simplify, let L (2) = {k, k 0 } (the argument extends if there are more than two candidates). If

x k < x 2 < x k 0 , then s(2) = g(2, k) + g(2, k 0 ) 2 + i(2, k) + i(2, k 0 ), s(k) = g(k, 2) + i(2, k), s(k 0 ) = g(k 0 , 2) + i(2, k 0 ). We compute s(k) + s(k 0 ) = g(k, 2) + g(k 0 , 2) + i(2, k) + i(2, k 0 ) < 1, so that s(k) = s(k 0 ) < .5, in contradiction to the fact that s(1) > .5. If x 2 / ∈ (x k , x k 0 ), then (assuming, w.l.o.g., x k ∈ (x 2 , x k 0 )) s(2) = g(2, k) + g(2, k 0 ) 2 + i(2, k), s(k) = g(k, 2) + i(2, k), s(k 0 ) = g(k 0 , 2).
Given that g(k, 2) ≥ g(k 0 , 2), this implies g(k, 2) = g(k 0 , 2) and i(2, k) = 0. Therefore, s(2) > s(k) implies s(k) < .5, in contradiction to the fact that s(1) > .5. That proves that L (2) contains one and only one candidate. We cannot have L (2) = {1}, as this would imply s(2) < s(1), a contradiction. Let L (2) = {k}. Therefore, we must have s(2) > s(k) > . . . ≥ s(1) ≥ . . . . That implies s(1) ≥ g(1, 2) > .5. To have s(k) > s(1), it must be the case that s(2

) = g(2, k) + i(2, k) and s(k) = g(k, 2) + i(k, 2). By Fact 2, x 0 1 / ∈ (x 2 , x k ). Then, either x 0 1 < x 2 < x 3 or x 3 < x 2 < x 0 1 . As a result, g(1, k) > g(k, 2) + i(2, k), a contradiction.
Case 2: L (1) contains two candidates, say 2 and 3, with x 2 ≤ x 3 . We have s(1) ≥ g(x 0

1 ,x 2 )+g(x 0 1 ,x 3 ) 2 > .5. Note that all voters who are indifferent between x 2 and x 3 vote exactly in the same way, as they all strictly prefer any policy in (x 2 , x 3 ) to either x 2 or x 3 , and they prefer x 2 or x 3 to any position out of (x 2 , x 3 ). Consequently, either

s(2) = g(x 2 , x 3 ) + i(x 2 , x 3 ) s(3) = g(x 3 , x 2 ) + i(x 2 , x 3 ), or s(2) = g(x 2 , x 3 ) s(3) = g(x 3 , x 2 ).
In either case, g(x 2 , x 3 ) = g(x 3 , x 2 ) which implies that x 2 < x 0 1 = x m < x 3 and s(1) = g(x 0

1 , x 3 ) + i(x 2 , x 3 ). By Fact 3, s(1) ≥ s(2) = s(3), contradicting the assumption on the score vector.

Case 3: L (1) contains three or more candidates, say 2, 3 and 4, with x 2 ≤ x 3 ≤ x 4 (a similar argument goes true if L (1) contains more than three agents). By Assumption 2, s(2), s(3), s(4) are determined as the average between the scores that are compatible to any of the six possible strict rankings of 1, 2, 3. The scores, for each ranking, are as follows. Using g(2, 3) ≤ g(2, 4) and g(4, 3) ≤ g(4, 2), we obtain

s(2) s(3) s(4) 234 g(2, 3) + i(2, 3) g(3, 2) + i(2, 3) g(4,
s(2) ≤ g(2, 4) + i(2, 3) 3 s(3) = g(3, 2) + g(3, 4) 2 + 1 3 (i(2, 3) + i(3, 4)) s(4) ≤ g(4, 2) + i(3, 4) 3
Given that g(2, 4) + g(4, 2) ≤ 1 and g(3, 2) + g(3, 4) ≥ 1, we can only have

s(2) = s(3) = s(4) if i(2, 3) = i(2, 4) = i(3, 4) = 0 and g(3, 2) + g(3, 4) = 1.
Consequently, s(2) = s(3) = s(4) = .5 whereas s(1) > .5, a contradiction.

That proves that 1 ∈ L (1) . We want to prove that all candidates in L (1) are located at x m . Assume two different locations are represented in L (1) : L (1) = {1, k} with x k 6 = x m . Then because x m is a strict Condorcet winner, s(1)s(k) = g(1, k)g(k, 1) > 0, a contradiction. Moreover, we know from Case 3 above that a three candidate tie is possible only if all the scores are .5, which is impossible if candidate 1 is one of them. It follows that all winning candidates are at x m . That completes the proof.

Comparison with Plurality voting

The above result should be contrasted with what happens under other voting rules. Consider Plurality rule and take K > 2 (at least 3 candidates). The models of rational voting which are similar to the one used here, such as those of [START_REF] Myerson | A theory of voting equilibrium[END_REF], [START_REF] Roger | Comparison of scoring rules in Poisson voting games[END_REF], or Laslier (2009), provide, as can be easily seen, the following behavior.

Rational behavior for the voter, under the plurality rule is basically to vote for the one she prefers among the two first-ranked candidates. Consider the simple case of the single-peaked model on the real line. Suppose that all candidates are at the median, each one receiving 1/K of the votes and having thus the probability 1/K of being elected. Then suppose that one candidate, say k = 1, moves slightly away from this position to some new position, say x 0 1 = x m + ε on the right of x m . This produces a situation in which the electorate is essentially split in two: the left-wing prefers x 2 = x 3 = ... = x 4 = x m and the right-wing prefers x 1 = x 0 1 = x m + ε. This potentially gives to the mover almost 1/2 of the votes while the remaining (K -1) candidates have to share the remaining votes. Under most reasonable assumption as to voters' behavior, the strength of the splitmajority phenomena will be such that candidate 1 will be elected with probability 1. Therefore the situation in which all candidates propose the median is not an equilibrium, exept if voters' beliefs are such that all votes gather on two candidates only. An equilibrium is obtained when only two candidates are located at the median and receive half of the votes while the other candidates receive none. This point (only two parties can survive under Plurality voting), which has been emphasized by Cox (1997) after [START_REF] Duverger | Les partis politiques[END_REF] is not valid for approval voting. Further studies on this subject also endogenise the number of candidates running for office: see [START_REF] Dellis | Approval voting with endogeneous candidates Games and[END_REF] and Dellis (2010).

Extensions

We need to discuss two extensions of the above model. First, we have assumed that candidates maximize their probability of winning the election. Alternatively, we could have assumed that they try to maximize their victory margin (or minimize their defeat margin), that is, for all k ∈ {1, . . . , K}, candidate k maximizes s(k) max k 0 ∈{1,...,K}\{k} s(k 0 ) .

This assumption is more difficult to justify in the case of approval voting than in the standard two-party plurality case because it is not clear whether one should consider the abolute or relative number of approval votes. Anyway, we conjecture that our three results remain true under this assumption.

The second extension is about the source of uncertainty facing voters. We have assumed that each vote of each voter had a fixed probability of not being recorded. We might have assumed, instead, that each voter had a fixed probability of not going to vote. Under this assumption, there is some correlation between the probability that votes are not recorded. Indeed, if a voter planning to vote for k and k 0 does not vote, none of her two votes are recorded. Unfortunately, our results do not extend to that case. [START_REF] Nunez | Approval Voting and the Poisson-Myerson Environment[END_REF] has shown that Laslier (2009)'s result does not hold under this alternative assumption. The same kind of example as the one developed in [START_REF] Nunez | Approval Voting and the Poisson-Myerson Environment[END_REF] applies in the model we have studied in this chapter.

In conclusion, we have found a new kind of elections in which approval voting leads to electing a Condorcet winner. Compared to [START_REF] Laslier | The Leader rule : a model of strategic approval voting in a large electorate[END_REF], our results show that if policy positions are endogeneous and follow from candidate competition, then strategic voting based on vote uncertainty leads to the election of the Condorcet winner when it exists. A consequence of electoral competition is that voters may be indifferent between pairs of candidates, a case which was excluded from Laslier's analysis. We showed that indifferences could prevent the general result from holding. Nonetheless, when voters have single-peaked preferences, in spite of possible indifferences, electoral competition leads all candidates to propose the median policy platform.

•

  If the voter is not indifferent between d and c 1 , she approves of d if and only if she prefers d to c 1 . • If she is indifferent between d and c 1 (for instance in the case d = c 1 ), she approves of d if and only if she prefers d to c i , where c i is the first candidate, according to the ranking c 1 , c 2 , ..., c K who is not indifferent with c 1 .

We will thus prove that the equilibrium is in dominant strategy.


Ariana Gutiérrez
email: m.agutierrezo@correo.itlalaguna.edu.mx

Héctor Ríos

Manuel Mera
email: mmerah@ipn.mx

Denis Efimov
email: denis.efimov@inria.fr

Rosane Ushirobira
email: rosane.ushirobira@inria.fr.

An Interval Predictor-based Robust Control for a Class of Constrained Nonlinear Systems

Keywords: Constrained Nonlinear Systems, Interval Predictor, Model Predictive Control

This paper proposes the design of a robust sampled-time controller to stabilize continuous-time nonlinear systems, taking into account state and input constraints. The proposed controller comprises the design of a robust control law, which is based on an interval predictor-based state-feedback controller and a Model Predictive Control (MPC) approach, which deals with the state and input constraints. The interval predictor-based state-feedback controller is designed based on a Lyapunov function approach that provides a safe set, where the state constraints are not transgressed. Out this set, the MPC is activated guaranteeing the fulfillment of the state and input constraints. The proposed switched control strategy guarantees the practical Uniform Asymptotic Stability of the considered nonlinear systems. A constructive method, based on linear matrix inequalities (LMIs), is proposed to compute the controller gains and the state of the system is not required. Some simulation results illustrate the feasibility of the proposed scheme.

I. INTRODUCTION

R EAL dynamical systems, which in general can be represented by nonlinear models, are constrained due to the structural limitations of the system, i.e., the system dynamics have state and input constraints, (see e.g. [START_REF] Allenspach | Nonlinear model predictive control and guidance for a propeller-tilting hybrid unmanned air vehicle[END_REF], [START_REF] Li | Robust distributed model predictive control of constrained continuous-time nonlinear systems: A robustness constraint approach[END_REF], and [START_REF] Weiss | Model predictive control for spacecraft rendezvous and docking: Strategies for handling constraints and case studies[END_REF]). In addition, the systems can be affected by parameter uncertainties, that can modify their behaviour. In order to design a controller that can counteract the effects of parameter uncertainties, taking into account state and input constraints, robust control techniques are required. For instance, the MPC is a technique in which the current control action is obtained by solving on-line, at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state. The solution produces an optimal control sequence and the first control in that sequence is applied to the system [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF].

Regarding the control design for constrained systems, in [START_REF] Leurent | Robust-adaptive control of linear systems: beyond quadratic costs[END_REF], an adaptive MPC for linear systems with unknown parameters and disturbances is proposed. Also, in [START_REF] Bujarbaruah | Robust MPC for LPV systems via a novel optimization-based constraint tightening[END_REF], an MPC for constrained uncertain linear systems is proposed. In this work, the robustness of the system constraints and Input-to-State Stability (ISS) of the closed-loop system is guaranteed. Meanwhile, for linear parameter varying (LPV) systems, the MPC approach is also well-used. For instance, in [START_REF] Ding | On dynamic output feedback robust MPC for constrained quasi-LPV systems[END_REF], an output feedback robust MPC for quasi-LPV systems with bounded noise is proposed. This work utilizes both, an ellipsoidal bound and a polyhedral bound in the main optimization problem, and asymptotic convergence of the state trajectories to a neighborhood of the origin is guaranteed. In [START_REF] Ding | Output feedback robust MPC for LPV system with polytopic model parametric uncertainty and bounded disturbance[END_REF], an output feedback robust MPC is proposed for disturbed LPV systems. In this algorithm single-step dynamic output feedback robust MPC, where the infinite-horizon control moves, are parameterized as a dynamic output feedback law. However, this proposal demands heavy computational burden to achieve the control task. The MPC can also be applied to discrete-time systems, in [START_REF] Mayne | Robust output feedback model predictive control of constrained linear systems: Time varying case[END_REF], an output feedback MPC for discrete-time linear systems affected by bounded state and output disturbances is considered. The controller is based on a stable state estimator and an MPC. The proposed robust output feedback controller requires the online solution of a standard quadratic program. The closed-loop system renders a specified invariant set robustly exponentially stable. It is worth mentioning that most of the MPC results given in the literature are devoted to linear systems, and this could be restrictive.

Concerning the application of the MPC approach to nonlinear systems, in [START_REF] Manzano | Robust learning-based MPC for nonlinear constrained systems[END_REF], a robust learning-based MPC is proposed for nonlinear systems subject to input and output constraints. A nonparametric machine learning technique is used to estimate the prediction model; and thus, an MPC, without terminal constraint, is obtained ensuring the asymptotic stability of the closed-loop system. In [START_REF] Sun | Robust MPC for tracking constrained unicycle robots with additive disturbances[END_REF], a robust MPC is proposed for tracking unicycle robots with input constraint and bounded disturbances. In this work ISS properties are guaranteed and the constraints are ensured by tightening the input domain and the terminal region. The MPC can also be applied to nonlinear discrete-time systems, in [START_REF] Xie | Robust MPC for disturbed nonlinear discrete-time systems via a composite self-triggered scheme[END_REF], a periodic formulation of MPC with a composite self-triggered mechanism is proposed to reduce the communication and computational burden while maintaining the desired control performance for disturbed nonlinear discrete-time systems. However, the algorithm has a relatively low tolerance to disturbances due to the state error bounds are derived by means of Lipschitz properties. In the previous works, a heavy computational burden is needed, due to the MPC being active during the control task.

The MPC can be also combined with other techniques, e.g., sliding-mode control (SMC), and interval observers techniques, in order to improve its behavior. For instance, in [START_REF] Rubagotti | Robust model predictive control with integral sliding mode in continuous-time sampled-data nonlinear systems[END_REF], a combination of robust MPC with SMC is proposed for nonlinear constrained uncertain systems. The proposed approach provides Input-to-State practically Stable properties. Meanwhile, in [START_REF] Ji | Stabilizing terminal constraint-free nonlinear MPC via sliding mode-based terminal cost[END_REF], an MPC is proposed for nonlinear systems based on a terminal cost function characterized by an implicit SMC law. However, in order to guarantee the asymptotic stability of the closed-loop system, a linearized model is required. In [START_REF] Reis De Souza | Robust output feedback model predictive control for constrained linear systems via interval observers[END_REF], an interval observer and predictor are incorporated into the classic MPC scheme for discrete-time constrained linear systems with bounded state disturbances. Nevertheless, most of the above-mentioned works can only guarantee ISS properties, require the system trajectories and do not take into account sampled control.

Motivated by these issues, this paper proposes the design of a robust sampled controller to stabilize continuoustime nonlinear systems, taking into account state and input constraints. The robust controller is based on a discretetime interval predictor-based state-feedback controller and an MPC approach. The proposed control approach possesses the following features: 1) Characterization of a safe set, where the state constraints are not transgressed. 2) Low computational cost since the MPC is only activated out of the safe set. 3) Practical Uniform Asymptotic Stability of the constrained nonlinear systems is proved. 4) The synthesis of the controller is constructive since is based on LMIs. 5) The system states are not required 1 .

The organization of this work is as follows. In Section II the problem statement is described. Section III presents some preliminary concepts. The robust control design is introduced in Section IV. Simulation results are depicted in Section V. Finally, the conclusions are presented in Section VI.

Notation: The sets of real and natural numbers are defined by R and N, respectively; R + = {s ∈ R : s ≥ 0}, and N + = {s ∈ N : s ≥ 0}. For a matrix A, we define A + = max{0, A} and A -= A + -A (similarly for vectors). For two vectors x 1 , x 2 ∈ R n or matrices A 1 , A 2 ∈ R n×n , the relations x 1 ≤ x 2 and A 1 ≤ A 2 are understood in an element-wise sense. The relation P ≺ 0 (P 0) means that a symmetric matrix P ∈ R n×n is negative (positive semi-) definite. For a Lebesgue measurable function w : R + → R q , define w (t0,t1) := ess sup t∈(t0,t1) w(t) ; then, w ∞ = w (0,+∞) and the set of all the functions w with the property w ∞ < +∞ is denoted as L q ∞ . A matrix is called Metzler when all its non-diagonal coefficients are non-negative.

II. PROBLEM STATEMENT

Consider the following class of nonlinear system

ẋ(t) = f (x(t)) + Bu(t), t ≥ 0, (1)
where

x(t) ∈ R n is the state, f : R n → R n is a locally Lipschitz function, u(t) ∈ R p is the control input. The matrix B is known of suitable dimension. It is considered that u ∈ U := {u ∈ R p | -u max ≤ u ≤ u max }, with 0 < u max ∈ R p
as the vector containing the maximum value that each control signal can take. Additionally, the control is sampled,

i.e., u(t) = u(t k ), for t ∈ [t k , t k+1 ) and k ∈ N + .
Moreover, the solutions of the system (1) must be constrained inside the polytope

X := {x ∈ R n |bx ≤ 1 kx }, (2)
where b = (b 1 , . . . , b kx ) ∈ R kx×n , with b i ∈ R n being some given vectors that characterize the state constraints, k x ∈ N is the number of the polytope faces, and

1 kx = (1, . . . , 1) ∈ R kx .
Before proceeding, the following assumptions are imposed on the system (1). Assumption 1. There exist a known Metzler matrix A 0 ∈ R n×n and known matrices A j ∈ R n×n , j = 1, N for some N ∈ N, such that the following relations are satisfied

f (x) =   A 0 + N j=1 α j (x)A j   x, N j=1 α j (x) = 1, α j (x) ∈ (0, 1), j = 1, N . Assumption 2. There exist known vectors x 0 , x 0 ∈ R n such that x 0 ≤ x(0) ≤ x 0 .
Assumption 1 characterizes the class of nonlinearities of the system (1), i.e., those that can be expressed by a sum of a linear term, A 0 x, and a convex nonlinear function, N j=1 α j (x)A j x. On the other hand, Assumption 2 and the condition that A 0 must be Metzler, are required to design an interval predictor (IP) for the system [START_REF] Allenspach | Nonlinear model predictive control and guidance for a propeller-tilting hybrid unmanned air vehicle[END_REF].

The aim of this manuscript is to design a sampled control law in order to ensure the stabilization of the system (1), taking into account the input and state constraints, i.e., x ∈ X and u ∈ U , under Assumptions 1 and 2.

III. PRELIMINARIES

Lemma 1. [START_REF] Efimov | Interval estimation for LPV systems applying high order sliding mode techniques[END_REF] Let x ∈ R n be a vector variable, x ≤ x ≤ x for some x, x ∈ R n , then

• If A ∈ R m×n is a constant matrix then A + x -A -x ≤ Ax ≤ A + x -A -x.
(3)

• If A ∈ R m×n is a variable matrix and A ≤ A ≤ A for some A, A ∈ R m×n , then A m ≤ Ax ≤ A M . (4)
with

A m = A + x + -A + x --A -x + +A -x -and A M = A + x + -A + x --A -x + + A -x -. Lemma 2. [17] Let λ : [a, b] → R

IV. CONTROL DESIGN

The proposed controller comprises the design of a robust control law, which is composed by an interval predictorbased state-feedback controller and the MPC, which deals with state and input constraints. We will show that there exists a safe set for the interval-predictor based state-feedback controller, where the state constraints are not transgressed, that characterizes the regions where each controller is activated: the MPC is activated outside the safe set while the statefeedback controller is applied inside of it. Since the control is sampled, for the continuous-time system (1), a discretetime interval predictor is designed, which takes into account the error of discretization and guarantees a careful treatment of the constraints.

The following section addresses the design of the IP for the nonlinear system (5) satisfying Assumptions 1 and 2.

A. Interval Predictor

According to Assumption 1, the system (1) has the following representation

ẋ(t) =   A 0 + N j=1 α j (x)A j   x(t) + Bu(t). (5)
Based on [START_REF] Reis De Souza | Robust output feedback MPC for LPV systems using interval observers[END_REF], applying (4) to α j (x)x, the relation -x -≤ α j (x)x ≤ x + holds. Thus, according to (3), the following inequalities can be obtained

∆A m ≤ N j=1 α j (x)A j x ≤ ∆A M ,
where

∆A m = -∆A + x --∆A -x + and ∆A M = ∆A + x + + ∆A -x -, with ∆A + = N i=1 A + i and ∆A -= N i=1 A - i .
Then, a continuous-time IP for system (5) can be stated as follows [START_REF] Leurent | Robust-adaptive interval predictive control for linear uncertain systems[END_REF]

ż(t) = A 0 z(t) + A 1 z + (t) + A 2 z -(t) + Bu(t), (6)
where z = [x , x ] ∈ R 2n is the extended state vector of the predictor, the extended system matrices

A 0 ∈ R 2n×2n , A 1 ∈ R 2n×2n , A 2 ∈ R 2n×2n
, and B ∈ R 2n×p are given as

A 0 = A 0 0 0 A 0 , A 1 = 0 -∆A - 0 ∆A + , A 2 = -∆A + 0 ∆A -0 , B = B B .
Being properly initialized by x(0) = x 0 and x(0) = x 0 , this IP ensures the desired interval inclusion property [START_REF] Leurent | Robust-adaptive interval predictive control for linear uncertain systems[END_REF]:

x(t) ≤ x(t) ≤ x(t), ∀t ≥ 0.
Then, for any t ≥ t 0 ≥ 0 the solution of system ( 6) is given by

z(t) = e A0(t-t0) z(t 0 ) + t t0 e A0(t-τ ) Bu(τ )dτ + t t0
e A0(t-τ ) g(z(τ ))dτ, [START_REF] Ding | On dynamic output feedback robust MPC for constrained quasi-LPV systems[END_REF] where g(z(t)) = A 1 z + (t) + A 2 z -(t). The expression [START_REF] Ding | On dynamic output feedback robust MPC for constrained quasi-LPV systems[END_REF], with a sampled control u(t) = u(t k ), for t ∈ [t k , t k+1 ), and k ∈ N + , and t 0 = 0, generates a discrete-time sequence:

z(t k+1 ) = e A0h z(t k ) + h 0 e A0(h-s) Bds • u(t k ) + t k+1 t k e A0(t k -s) g(z(s))ds,
where h = t k+1 -t k is the sampling interval, which we select to be constant for brevity, and t k is the sampling time instant. Then, adding and subtracting the term 8) can be rewritten as

h 0 e A0(h-s) ds • g(z k ), it follows that z(t k+1 ) = e A0h z(t k ) + h 0 e A0(h-s) Bds • u(t k ) + h 0 e A0(h-s) ds • g(z(t k )) + ϕ k , (8) with ϕ k = t k+1 t k e A0(t k+1 -s) [g(z(s)) -g(z(t k ))]ds. Let us consider the notation z(t k ) = z k , u(t k ) = u k , and G = h 0 e A0(h-s) ds ∈ R 2n×2n . Thus, (
z k+1 = A 0 z k + Bu k + Gg(z k ) + ϕ k , (9)
with A 0 = e A0h ∈ R 2n×2n and B = GB ∈ R 2n×p . However, note that the interval predictor (9) cannot be implemented since ϕ k , which characterizes the discretization accuracy and that is dependent on z(t) with t ∈ [t k , t k+1 ], cannot be computed. Thus, consider the following predictor

ẑk+1 = A 0 ẑk + Bu k + Gg(ẑ k ). (10)
Then, in order to stabilize the nonlinear dynamics (1), we need to design u k to take the trajectories of the system [START_REF] Manzano | Robust learning-based MPC for nonlinear constrained systems[END_REF] to zero dealing with the state and input constraints. Let us propose the following control law for the system ( 9)

u k = ūk if ẑk ∈ R, ûk if ẑk ∈ R, ( 11
)
where ūk is the control signal computed by the MPC algorithm and ûk is the state-feedback control law. Be aware that ûk is applied only inside the safe set R, which is defined further on, where the states constraints are not transgressed.

In the following sections we describe the design of each part of the controller.

B. State-Feedback Control

Let us propose the following state-feedback control

ûk = K 0 ẑk + K 1 ẑ+ k + K 2 ẑ- k . ( 12
)
Be aware that ( 12) is a nonlinear state-feedback control law due to the terms ẑk and ẑ+ k . The next theorem provides a way to compute the controller gains K 0 , K 1 , K 2 ∈ R p×2n . Lemma 3. Let Assumptions 1 and 2 be satisfied, and the control law (12) be applied to the system [START_REF] Mayne | Robust output feedback model predictive control of constrained linear systems: Time varying case[END_REF]. Suppose that there exist some symmetric matrices

Q 0 , Q 1 , Q 2 ∈ R 2n×2n , a diagonal matrix R 3 ∈ R 2n×2n , some matrices 0 ≺ X 1 = X 1 ∈ R 2n×2n , Y 1 , Y 2 ∈ R 2n×2n X 2 ∈ R 2n×p , K 1 ∈ R p×2n , and K 2 ∈ R p×2n , such that the following set of LMIs          -X1 Y2 Y3 0 Σ16 Σ17 X1 Σ22 Σ23 0 Σ26 0 0 Σ33 0 Σ36 0 0 -γI I 0 0 -X1 0 0 -γI 0 -X2          0, (13)
Σ16 = X1A 0 + Y1B , Σ17 = 2γkaX1, Σ22 = 4γk 2 e 2 A 1 A1 + Q1, Σ23 = 4γk 2 e 2 A 1 A2 + R3, Σ26 = A 1 G + K 1 B , Σ33 = 4γk 2 e 2 A 2 A2 + Q2, Σ36 = A 2 G + K 2 B ,
is feasible for some fixed constants

k e1 = 1 + e k ξ k ξ , k e2 = k e1 k ξ , k ξ = ξν -1 (1 -e -νh ), k a = k e1 ξ -||A 0 || -L||G||, with γ, ξ, α > 0, L = 2 max{||A 1 ||, ||A 2 ||}, and ν = |Re{λ max {A 0 }}|. If the following inequality X + min{Q 1 , Q 2 } + 2 min{R 1 , R 2 } 0, (14)
holds with X = -αX -1

1 + X -1 2 , R 1 = X -1 1 Y 1 , and R 2 = X -1
1 Y 2 ; the IP (10) is initialized as ẑ0 = (x 0 ,x 0 ) , and the controller gains are selected as K 0 = X 2 X -1 1 , K 1 , K 2 , and S 1 ; then, the origin of the system dynamics (9) is practically Uniformly Asymptotically Stable (practically UAS) 2 . Moreover, the safe set is given as

R = {z k ∈ R 2n : ẑ k X -1 1 ẑk ≤ ε}, (15)
with ε = α -1 q 1 and

q 1 = 8(k e1 -||A 0 || -L||G||) 2 z2 + 8k 2 e2 ||B|| 2 ū2 max + (2ẽz + ẽ2 )λ max {P } + 2[zc 1 + ẽc 2 + c 4 φ]ẽ,
2 For more details of this type of convergence rate, please see [START_REF] Moreau | Practical stability and stabilization[END_REF].

z = 2k x λ -1 max {b b}, u max = ||u max ||, ẽ = [k e1 (ξ + 1) + k ξ (||A 1 || + ||A 2 ||)]z + k ξ [||B||u max + λ -1/2 max {R 0 }], φ = ẽ -[2||A 0 || + 2L||G||]z.
for a given b, u max , and

c 1 = (|| Ã0 || + || Ã1 || + || Ã2 ||)||P B||K, (17a)
c 2 = (0.5||K 0 || + ||K 1 || + ||K 2 ||)||P BK 0 || (17b)
+ (||K 0 || + 0.5||K 1 || + ||K 2 ||)||P BK 1 || + (||K 0 || + ||K 1 || + 0.5||K 2 ||)||P BK 2 ||, c 3 = || S 1 P B||K, (17c)
c 4 = ||P B||K, (17d)
where then, the trajectories of the system (1) fulfill the state constraints, i.e., x ∈ X .

Ã0 = A 0 + BK 0 , Ã1 = GA 1 + BK 1 , Ã2 = GA 2 + BK 2 , and K = ||K 0 || + ||K 1 || + ||K 2 ||. Additionally, for P = αP/q 1 ∈ R 2n×2n , with P = X -1 1 , if the following inequalities b i P -1 x b i ≤ 1, b i P -1 x b i ≤ 1, (18)
Due to space limitation, the proofs of all the results are omitted.

C. Design of the Model Predictive Control

For the design of the MPC the following assumption is considered.

Assumption 3. There exist some gains K 0 , K 1 , K 2 ∈ R p×2n , satisfying the conditions of Lemma 3,and

ûk (t) = K 0 ẑk (t)+ K 1 ẑ+ k (t) + K 2 ẑ- k (t) ∈ U
, for all z k ∈ R and t ≥ 0. Assumption 3 implies that the invariant set R, which is the safe set of system (1), is completely contained inside the polytope (2), i.e., the constrained region X .

Let us consider that the prediction can be performed in a receding horizon fashion with a window length N > 1.

The interval predictor [START_REF] Manzano | Robust learning-based MPC for nonlinear constrained systems[END_REF] has a sequence of inputs S N = {s 0 , . . . , s N -1 }, with s m ∈ U for all m = 0, N -1, the values ẑk,m+1 will be calculated for m = 0, N -1 under substitution u k+m = s m . Let us define ẑk+j = ẑk,j , for j = 0, N . Thus, the optimal control problem to be solved by the MPC is given as follows:

Problem 1. For given matrices 0 ≺ W = W ∈ R 2n×2n , 0 ≺ H = H ∈ R 2n×2n , and M = M ∈ R p×p , to find the control signals

S k N = argminV N (ẑ k,0 , . . . , ẑk,N , S N ), (19)
with the cost function

V N V N (ẑ k,0 , . . . , ẑk,N , S N ) = V f (ẑ k,N ) + N -1 m=0 (ẑ k,m , s m ),
where V f (ẑ) = ẑ W ẑ, (ẑ, s) = ẑ H ẑ + s M s, such that the following constraints are satisfied: a) ẑk+j , for j = 0, N are computed by [START_REF] Manzano | Robust learning-based MPC for nonlinear constrained systems[END_REF]. b)

u k ∈ U . c) ẑk,N ∈ R.
Finally, considering the results of Lemma 3 and the solution of the Problem 1 the following theorem is presented.

Theorem 1. Let the Assumptions 1-3 be satisfied; then, the closed-loop system composed by (1), ( 10) and (11) has the following features: i) Recursive feasibility to reach R in N steps; ii) The origin of the system dynamics (9) is practically UAS, in the safe set R; iii) The state and input constraints are not transgressed.

Note that the proposed MPC is based on the conventional MPC scheme; thus, the proof of Theorem 1 is based on classic results for MPC schemes [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF].

V. SIMULATION RESULTS

Consider the following nonlinear system

ẋ1 ẋ2 = -1 6 + 2 sin(x1) 2 sin(x2) -1 x1 x2 + 0 1 u.
It is possible to show that this system satisfies Assumption 1. The matrices A 0 and A j can be obtained by a convex polythopic method, e.g., that one given in [START_REF] Nagy | Systematic multimodeling methodology applied to an activated sludge reactor model[END_REF]. Let us consider that u max = 5, x 1 ∈ (-12, 12) and x 2 ∈ (-5, 5), and hence the state constraints are b 1 = (1/12, 0) , b 2 = -b 1 , b 3 = (0, 1/5) , and b 4 = -b 3 . The initial conditions for the system are x(0) = (2, -1) and for the IP are ẑk (0) = (-2.5, -1.6, 2.5, 1.6) .

All the simulations are done in MATLAB with Euler discretization method, integration step equal to 0.0001 [s] and sampling time h = 0.1 [s]. The solution for the LMI's are found by means of SDPT3 solver among YALMIP while the MPC is implemented using the nlmpc MATLAB toolbox.

For the matrix inequalities of Lemma 3, selecting γ = 0.1, and fixing Q 1 = Q 2 = -0.05I, the following feasible solution is obtained: Then, with α = 0.1 it is possible to verify that condition ( 14) is fulfilled. For the MPC, let us consider W = I , H = 1000I, M = 0.0001, and N = 10.

P =    0.
The system trajectories, the IP [START_REF] Manzano | Robust learning-based MPC for nonlinear constrained systems[END_REF] trajectories and the corresponding state constraints are depicted in Fig. 1. Note that both the IP and the system trajectories never transgress the system constraints. On the other hand, in Fig. 2 it is noticed that the initial conditions of the system are outside the safe set R, but inside of the state constraints. However, the MPC takes the trajectories of the system towards the safe set R and they remain inside of it. Additionally, in Fig. 3 it can be seen that the control signal do not transgress the input constraints.

VI. CONCLUSIONS

This paper deals with the design of a robust sampled-time controller for the stabilization of continuous-time nonlinear systems, taking into account state and input constraints. The proposed controller comprises the design of a robust control law, which is based on an interval predictor-based state-feedback controller and an MPC approach, that deals with the state and input constraints. A safe set is provided where the state constraints are not violated. The safe set characterizes the region where the MPC is activated, i.e., out of this set, guaranteeing the fulfillment of the state and input constraints. The proposed switched control strategy guarantees the practical Uniform Asymptotic Stability of the considered nonlinear systems without using the system states.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. System trajectories and state constraints

  11 P 21 , with P 11 ∈ R n×n , P 21 ∈ R n×n , P 22 ∈ R n×n , and

	P =	P 11 P 21 P 21 P 22	,

hold for P x = P 11 -P 21 P -1 22 P 21 and P x = P 22 -P 21 P -1

Note that its noisy measurements may be used to refine the available information on the set of admissible values.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support from TecNM projects, from SEP-CONACYT-ANUIES-ECOS NORD Project 315597 and ECOS NORD Project M20M04, from CONAHCYT CVU 932774, from CONAH-CYT CVU 270504 Project 922, and from SIP Project 20230170.


Thomas Abegg

Janine Cossy

Christophe Meyer

Cascade Cope/Winstein Rearrangements: Synthesis of Azido-Cycloheptadienes from Dialkenylcyclopropanes Possessing a Vinyl Azide

cis-1,2-Dialkenylcyclopropanes incorporating a vinyl azide, generated by Knoevenagel condensations between the corresponding cyclopropanecarbaldehydes and -azido ketones, undergo cascade Cope and Winstein [3,3]-sigmatropic rearrangements, under mild conditions. The sequence allows access to diversely substituted 1,4-cycloheptadienes armed with a secondary allylic azide with up to three stereocenters.

Azides have found countless applications in organic chemistry encompassing their use as amines precursors to the development of bioorthogonal ligation reactions. 1,[START_REF] Rostovtsev | A Stepwise Huisgen Cycloaddition Process: Copper(I)-Catalyzed Regioselective "Ligation" of Azides and Terminal Alkynes[END_REF] Recent years have seen a renewed interest in the chemistry of aliphatic azides with a particular focus on allylic and vinylic azides. [START_REF] Carlson | Allylic Azides: Synthesis, Reactivity, and the Winstein Rearrangement[END_REF][4][START_REF] Sivaguru | New Strategies for the Synthesis of Aliphatic Azides[END_REF] The reactivity of allylic azides is dominated by their ability to undergo a facile allylic transposition, known as the Winstein rearrangement (Scheme 1A). [START_REF] Carlson | Allylic Azides: Synthesis, Reactivity, and the Winstein Rearrangement[END_REF] Steric and electronic factors influencing the position of the equilibrium between interconverting allylic azides have been investigated [START_REF] Carlson | Allylic Azides: Synthesis, Reactivity, and the Winstein Rearrangement[END_REF]6 and the Winstein rearrangement has been coupled with other transformations that can exclusively or preferentially (kinetic dynamic transformations) occur on one regio-or stereoisomer. [START_REF] Carlson | Allylic Azides: Synthesis, Reactivity, and the Winstein Rearrangement[END_REF][START_REF] Craig | Claisen rearrangements of equilibrating allylic azides[END_REF][START_REF] Ott | Dynamic Kinetic Resolution of Allylic Azides via Asymmetric Dihydroxylation[END_REF] The synthesis of allyl azides is usually accomplished by substitution reactions involving allylic halides or alcohol derivatives, or by ring-opening of vinyl oxiranes or bromonium ions, with a nucleophilic azide source, [START_REF] Carlson | Allylic Azides: Synthesis, Reactivity, and the Winstein Rearrangement[END_REF] sometimes in the presence of a transition metal catalyst (Scheme 1B, path a). [START_REF] Carlson | Allylic Azides: Synthesis, Reactivity, and the Winstein Rearrangement[END_REF]9 Allylic C-H bond azidation (Scheme 1B, path b), 10 allene hydroazidation (Scheme 1B, path c) 11 or 1,3-dienes azido-functionalizations (Scheme 1B, path d) 12 have also emerged as alternative strategies. Because of their versatile reactivity, vinyl azides have been identified as an "emerging class" of hetero-substituted alkenes. 4 Whilst denitrogenative transformations are the most documented, these latter substrates can serve as precursors of functionalized aliphatic azides. [START_REF] Sivaguru | New Strategies for the Synthesis of Aliphatic Azides[END_REF] However, to our knowledge, the synthesis of allyl azides from vinyl azides A seems to be restricted to a copper-catalyzed formal (3+2) cycloaddition with vinyl diazoacetates which is followed by a Winstein rearrangement of the initially formed allylic azides B, the driving force of which is the formation of the more stable tetrasubstituted cyclopentenoates C (Scheme 1C). [START_REF] López | Synthesis of Functionalized Cyclopentene Derivatives from Vinyldiazo Compounds and VinylAzides through Sequential Copper-Pomoted [3+2] Cycloaddition/Azide Rearrangement[END_REF] Interestingly, sigmatropic rearrangements constitute a class of reactions in which vinyl azides have not been involved so far, despite their potential interest to access other classes of unsaturated aliphatic azides. Perhaps the required conditions and reagents may have deterred chemists from considering these latter hazardous, reactive and thermally labile substrates, which can generate 2H-azirines and/or vinyl nitrenes upon heating. 14 Scheme 1. Allylic azides: Synthesis, Winstein Rearrangement and Vinyl Azides as Potential Precursors.

Herein, we report the Cope rearrangement of cis-dialkenyl cyclopropanes D incorporating a vinyl azide, 15 as a route to cycloheptadienes E possessing a tertiary allylic azide and their subsequent isomerization into azido-cycloheptadienes F by a Winstein rearrangement (Scheme 1D). The preparation of cis- (2-alkenylcyclopropyl)alkenyl azides D required for this study was envisioned by a Knoevenagel condensation between the corresponding cyclopropanecarbaldehydes and -azido carbonyl compounds. [START_REF] Knittel | Mitt.: Synthese von 1,3-Diaryl-3-azido-propenonen (-Azidochalkonen)[END_REF][START_REF]Attemps to use ethyl azidoacetate as partner in the Knoevenagel condensation (EtONa/EtOH) with cyclopropyl aldehydes led to very low yields of adducts, despite two previously reported examples (without indication of the yields)[END_REF] Aldehyde 1a was involved in a Knoevenagel condensation with -azido acetophenone 2 in the presence of piperidinium acetate (MeOH, rt, 16 h). [START_REF] Knittel | Mitt.: Synthese von 1,3-Diaryl-3-azido-propenonen (-Azidochalkonen)[END_REF] Under these conditions, the 1,4-cycloheptadiene 3a, possessing a secondary azide, was identified as the new product and was isolated in high yield (92%). Hence, the cis-1,2-dialkenylcyclopropane 4a, arising from the Knoevenagel condensation underwent an efficient Cope rearrangement and the resulting 1,4-cycloheptadiene 5a, incorporating a tertiary allylic azide, further evolved by a Winstein rearrangement which is completely shifted toward the more stable secondary allylic azide 3a, possessing an ,-unsaturated ketone (Scheme 2). Scheme 2. Cascade Knoevenagel condensation-Cope and Winstein sigmatropic rearrangements from aldehyde 1a.

The scope of the cascade Knoevenagel condensation-Cope and Winstein sigmatropic rearrangements was then explored with diversely substituted cis-(2-alkenylcyclopropane)-carbaldehydes 1b-1l and -azido acetophenone 2 (Table 1). The reaction accommodates a substituent at the  position of the alkene on the cyclopropyl aldehyde 1b, as illustrated with the formation of cycloheptadiene 3b (76%) (Table 1, entry 1). Aldehyde 1c possessing an ,-disubstituted olefin was a viable partner in the Knovenagel condensation with 2 but the Cope/Winstein cascade rearrangements leading to cycloheptadiene 3c (66%) incorporating an enol phosphate only occurred after heating the crude vinyl azide intermediate (EtOH, reflux, 16 h) (Table 1, entry 2). The reactivity of aldehydes 1d-1g possessing an ,-disubstituted olefin was next explored. Aldehydes 1d and 1e, substituted by a phenyl or a (tert-butyldiphenylsilyloxy)methyl group on the alkene (E/Z > 96:4), respectively led to the azido-cycloheptadienes 3d (73%) and 3e (78%) as single detectable diastereomers (dr > 96:4, by 1 H NMR spectroscopy) (Table 1, entries 3 and4). Interestingly, in the case of aldehydes 1f and 1g in which the ,-disubstituted alkene was not fully controlled (E/Z = 92:8), cycloheptadienes 3f (72%) and 3g (87%) were respectively obtained as single diastereomers (Table 1, entries 5 and 6). As confirmed later in the case of substrate 1i (vide infra), both geometric isomers of 1f and 1g probably underwent the Knoevenagel condensation with 2 but only the (E) isomers subsequently evolve by a Cope rearrangement. 2-Vinylcyclopropanecarbaldehydes possessing a 1,2,3-trisubstitued three-membered ring could also be considered as substrates but the Cope rearrangement proceeds only when the additional substituent (R 4 ) is trans to the alkenyl chains.

3 b 1d, R 3 = Ph (E/Z = 96:4) 3d, R 3 = Ph 73% 4 1e, R 3 = CH2OTBDPS (E/Z = 96:4) 3e, R 3 = CH2OTBDPS 78% 5 1f, R 3 = n-Pent (E/Z = 92:8) 3f, R 3 = n-Pent 72% 6 1g, R 3 = (CH2)2CO2Et (E/Z = 92:8) 3g, R 3 = (CH2)2CO2Et 87% 7 c 66% 1h 3h (dr > 96:4) 8 d 68%
1i, R 3 = Ph (E/Z = 65:35) 3i (dr > 96:4)

9 e 1j 3j (0%) 6 (54%) 1k-1l 3k-3l 10 1k, R 3 = H 3k, R 3 = H 90% 11 1l, R 3 = Ph (E/Z > 96:4) 3l, R 3 = Ph 90%
a The crude product was further heated in refluxing EtOH for 16 h. b Experiments conducted at a scale ≥ 1 mmol. c The crude product was further heated in CDCl3 at 40 °C for 24 h. d The crude product was further heated in refluxing CH2Cl2 for 24 h. e Reaction run in EtOH (rt, 16 h then 70 °C, 24 h).

In the case of aldehyde 1h substituted by a phenyl group, the Knoevenagel condensation with 2 leading to azide 4h was complete after the usual reaction time (16 h) but cycloheptadiene 3h was only a minor product (20%) at this stage. Heating was required to trigger the Cope rearrangement, which was immediately followed by the Winstein transposition leading to cycloheptadiene 3h as a single diastereomer (66%), as evidenced by monitoring using 1 H NMR spectroscopy (CDCl3, 40 °C, 24 h) (Table 1, entry 7). The reactivity of aldehyde 1i (E/Z = 65:35) paralleled that of 1f, 1g and 1h since both geometric isomers underwent the Knoevenagel condensation with 2 but subsequent heating (CH2Cl2, reflux, 24 h) triggered the cascade Cope-Winstein rearrangements of the (E) geometric isomer exclusively which afforded cycloheptadiene 3i (68%, based on (E)-1i) (Table 1, entry 8), whereas the vinyl azide generated from (Z)-1i did not react further. The presence of an ester in substrate 1j retarded the Cope rearrangement so the crude product arising from the Knoevenagel condensation was heated in refluxing EtOH. However, under these conditions, the azidocycloheptadiene 3j generated by the cascade sigmatropic rearrangements underwent elimination of HN3 to produce the conjugated cycloheptatriene 6 (Table 1,entry 9). By contrast with 1h and 1j, a silyloxymethyl substituent on the cyclopropane ring in substrates 1k and 1l did not have any adverse effect on the rate of the Cope rearrangement which proceeded at rt and after Winstein rearrangement, the desired tri-and tetrasubstituted cycloheptadienes, 3k and 3l respectively, were isolated in 90% yield (Table 1, entries 10 and 11). The relative stereochemistry of the azido-cycloheptadiene 3l was established by X-ray diffraction analysis 19 and that of the other products 3d-3k was assigned by analogy. The stereochemical outcome can be understood considering that the Knoevenagel condensation between aldehydes and -azido ketones is known to afford -azido trisubstituted enones D of (Z) configuration [START_REF] Knittel | Mitt.: Synthese von 1,3-Diaryl-3-azido-propenonen (-Azidochalkonen)[END_REF] and that the Cope rearrangement of divinylcyclopropanes occurs through a boat transition state TS1. 15 Additionally, the subsequent rapid Winstein rearrangement of the initially formed tertiary allylic cycloheptadienyl azides E, which have never been detected as intermediate products, would proceed in a suprafacial manner and with complete chirality conversion. [START_REF] Carlson | Allylic Azides: Synthesis, Reactivity, and the Winstein Rearrangement[END_REF] It is worth noting that despite the presence of a trisubstituted alkene, the low steric demand of the azido group enables dialkenylcyclopropanes D to undergo the Cope rearrangement under mild conditions (at rt, in most cases, or 40-80 °C) which do not jeopardize the thermally labile vinyl azide moiety. Limitation of the scope to substrates D devoid of substituent on the three-membered ring cis to the alkenyl groups (R 4' = H) or to (E)-,-disubstituted alkenes (R 3' = H) can also be understood from the transition state model TS1. Indeed, for substrates D in which R 3' or R 4' would differ from hydrogen atoms, the transition state of the Cope rearrangement would be significantly destabilized by steric interactions (A 1,3 strain or eclipsing interaction between the R 3' and N3 substituents) 15,[START_REF] Osler | The Cope Rearrangement of gem-dimethyl substituted divinylcyclopropanes[END_REF] and hence the reaction would require higher temperatures, incompatible with the vinyl azide (Scheme 3). [START_REF]For additional experiments dealing with attempted Cope rearrangement of substrates of type D incorporating a 1,2,3-cistrisubstituted or a 1,2-trans-disubstituted cyclopropane, see the Supporting Information[END_REF]

Scheme 3. Transition state of the Cope rearrangement of cis-dialkenyl cyclopropanes D.

The scope of the Knoevenagel condensation-Cope/Winstein rearrangements cascade was further explored with different -azido ketones (Table 2). The reaction of aldehyde 1d with -azido p-methoxy-and p-chloroacetophenone, 7 and 8, as well as with o-fluoro-acetophenone 9, under usual conditions (piperidinium acetate, MeOH, rt, 16 h) delivered the azido-cycloheptadienes 12 (72%), 13 (87%) and 14 (96%), respectively (Table 2, entries 1-3). Besides acetophenones, -azido chalcone 10 and -az-ido'-(benzoyloxy)acetone 11 could be successfully used as partners with aldehyde 1d and delivered the corresponding cycloheptadienes 15 (89%) and 16 (60%) (Table 2, entries 4 and 5). It is worth pointing out that -azido ketone 11 can act as a surrogate for -azido acetaldehyde in the Knoevenagel condensation since chemoselective reduction of the ketone and the ester in compound 16 with DIBAL-H, followed by oxidative cleavage of the resulting 1,2-diol with NaIO4, delivered the ,-unsaturated aldehyde 17 (59%) (Table 2, entry 5). The reaction between 1l and -azido ketone 8 also afforded the tetrasubstituted azido-cycloheptadiene 18 in excellent yield (96%) (Table 2, entry 6).

Seven-membered carbocycles are less represented than cyclic systems of smaller size in bioactive compounds but are currently eliciting a growing interest in medicinal chemistry. [START_REF] Ouvry | Recent applications of seven-membered rings in drug design[END_REF] Hence, with several azido-cycloheptadienes of type F in hands, some post-functionalization reactions were carried out to highlight the synthetic interest of these latter functionalized sevenmembered carbocycles (Scheme 4). Scheme 4. Post-functionalization reactions.

The electron-rich olefin in compound 3l could be chemoselectively dihydroxylated under standard conditions to afford the corresponding 1,2-diol 19 (96%) with high diastereoselectivity (dr > 90:10), presumably arising from a steric control (dihydroxylation occurring trans to both adjacent allylic substituents) (Scheme 4, eq. 1). Transformations involving the azido group were next investigated. The azido-cycloheptadiene 3d was involved in a copper-catalyzed (3+2)-cycloaddition with propargyl alcohol to deliver the corresponding 1,4-triazole 20 (85%) (Scheme 4, eq. 2). Reduction of the azido group in compounds 3d, 3g and 3k was efficiently accomplished using SnCl2 (MeOH, rt) [START_REF] Maiti | Facile Conversion of Azides into Amines[END_REF] and the crude amine hydrochlorides were converted to N-Boc carbamates 21 (93%), 22 (83%) and 23 (49%), respectively (Scheme 4, eq. 3). The presence of an enone moiety in these latter products led us to explore the feasibility of the conjugate addition of organocuprates. The N-Boc carbamate 21 underwent 1,4-addition of Me2CuLi in the presence of Me3SiCl. 23 After subsequent protodesilylation of the silyl enol ether with n-Bu4NF, the tetrasusbtitued cycloheptene 24 was obtained as a single diastereomer (79%) and the relative stereochemistry of this latter crystalline compound was unambiguously established by X-ray diffraction analysis. Since the NHBoc group did not exert a directing effect in previously reported examples of cuprates addition to -amino ,-unsaturated compounds, 23 the stereochemical outcome may be due to a faster silylation of the d-* cuprate-enone complex when the copper is trans to the bulky phenyl group of 21. [START_REF] Frantz | Isotope Effects and the Mechanism of Chlorotrimethylsilane-Mediated Addition of Cuprates to Enones[END_REF] This could explain the trans relationship between the phenyl and the methyl groups after reductive elimination of the copper(III) -adduct, whilst protodesilylation of the silyl enol ether would result in proton transfer on the face opposite to the phenyl group (Scheme 4, eq. 4).

In conclusion, we have shown that 1,2-dialkenyl cyclopropanes, generated by a Knoevenagel condensation between cis- (2-alkenyl)cyclopropanecarbaldehydes and -azido ketones, readily undergo cascade Cope and Winstein rearrangements leading to azido-cycloheptadienes incorporating up to three stereocenters. Hence, an access to allylic azides, complementary to the existing strategies, has been devised and the products are seven-membered carbocycles with new substitution patterns, armed with an azido group potentially useful for further derivatization. These results highlight the synthetic interest of vinyl azides in sigmatropic rearrangement as a route to different classes of aliphatic azides.

  [START_REF]For additional experiments dealing with attempted Cope rearrangement of substrates of type D incorporating a 1,2,3-cistrisubstituted or a 1,2-trans-disubstituted cyclopropane, see the Supporting Information[END_REF]

Table 1 . Cascade Knoevenagel condensation-Cope and Winstein sigmatropic rearrangements from cyclopropane- carbaldehydes 1b-1l and

 1 -azido ketone 2.

	entry	substrate	product	yield
	1			76%
		1b	3b
	2 a			66%
		1c	3c
		1d-1g	3d-3g (dr > 96:4)

Table 2 . Cascade Knoevenagel condensation-Cope and Winstein sigmatropic rearrangements from cyclopropane- carbaldehydes 1d-1l and -azido ketones 7-11.

 2

	entry	aldehyde	azido ketone	product	yield
	1	1d	7, R = OMe	12, R = OMe	72%
	2	1d	8, R = Cl	13, R = Cl	87%
	3	1d			96%
			9	14
	4	1d			89%
			10	15
	5	1d
			11
				16	60%
				17	59%
	6	1l			96%
			8	18

ACKNOWLEDGMENT

One of us (T. A.) thanks the MESRI and ESPCI Paris-PSL for a PhD thesis grant. The authors acknowledge MS3U of Sorbonne Université for HRMS analysis and Geoffrey Gontard (IPCM) for X-ray crystallographic analysis.

ASSOCIATED CONTENT

Supporting Information

Experimental procedures, 1 H, [START_REF] López | Synthesis of Functionalized Cyclopentene Derivatives from Vinyldiazo Compounds and VinylAzides through Sequential Copper-Pomoted [3+2] Cycloaddition/Azide Rearrangement[END_REF] C and 19 F NMR spectra, and characterization data of new compounds (PDF)

Crystallographic data for compounds 3l and 24 (cif)

AUTHOR INFORMATION

Corresponding Author *christophe.meyer@espci.fr.

Author Contributions

All authors have given approval to the final version of the manuscript.