Anh-Lam Do
email: anh-lam.do@gipsa-lab.grenoble-inp.fr

Olivier Sename
email: olivier.sename@gipsa-lab.grenoble-inp.fr

Luc Dugard
email: luc.dugard@gipsa-lab.grenoble-inp.fr

Optimisation par Algorithme Génétique d'une Commande LPV de Suspension Semi-active

Keywords: suspension semi-active, système LPV, algorithmes génétiques

.

I. Introduction

L'approche de commande H ∞ est un outil efficace pour améliorer les performances d'un système en boucle fermée dans les gammes de fréquence prédéfinies. L'étape clé de la conception du contrôleur H ∞ repose sur la sélection de fonctions de pondération qui dépend de la compétence et de l'expérience des ingénieurs. Cependant, ce n'est pas toujours une tâche facile même pour de bons ingénieurs. Dans beaucoup d'applications, la difficulté dans le choix des fonctions de pondération augmente encore parce que la spécification de performance n'est pas définie avec précision et il est tout simplement impossible d'atteindre les meilleures performances (conception optimale) ou d'améliorer en même temps plus d'un objectif (conception multiobjectif). Ainsi, l'optimisation des fonctions de pondération pour satisfaire les performances souhaitées est encore un problème ouvert.

Dans la littérature, quelques travaux orientés vers cet objectif ( [START_REF] Beaven | Weighting function selection in the H∞ design process[END_REF], [START_REF] Hu | Systematic H∞ weighting function selection and its application to the real-time control of a vertical take-off aircraft[END_REF]...) proposent de considérer chaque système, si complexe soit-il, comme une combinaison de sous-systèmes du premier et second ordre, pour lesquels il est facile de trouver les fonctions de pondération. Cependant, il n'y a aucune méthode explicite pour retrouver ces fonctions dans le cas général. Une manière de faire est de procéder par essai/erreur. Une solution au problème est de faire appel à un outil d'optimisation, c'est ce qui est proposé pour le système LTI dans [START_REF] Alfaro-Cid | Optimisation of the weighting functions of an H∞ controller ternational[END_REF]. Nous allons voir ici que le choix des Algorithmes Génétiques (GAs) semble naturel, leur formulation étant bien adaptée pour ce type de problématique.

Dans cette communication, nous cherchons les meilleures fonctions de pondération pour obtenir un contrôleur stabilisant un système de type LPV et garantissant un performance prédéfinie du système en boucle fermée. Il est important de noter que ce problème est plus général que le problème H ∞ où le contrôleur ne garantit que la stabilité et la performance H ∞ . Pour illustrer cette idée, le cas particulier de commande de suspensions semi-actives est étudié. Tout d'abord, à partir de critères de performance industriels des systèmes de suspension tels que ceux définis dans [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions : some practical aspects[END_REF], une fonction objectif appropriée est formulée. Cette fonction se compose de deux objectifs antagonistes : le confort et la tenue de route. Deuxièmement, les fonctions de pondération sont ensuite réglées par l'algorithme génétique SPEA2 [START_REF] Zitzler | SPEA2 : Improving the strength pareto evolutionary algorithm[END_REF] de façon à minimiser la fonction objectif. Enfin, les meilleures fonctions de pondération sont choisies pour la synthèse de contrôleur LP V . Il est important de noter que cette méthode fournit un nouvel outil efficace et générique pour la conception de commande de suspension, où les objectifs de performance industrielle sont inclus directement dans la procédure de conception, et non pas dans l'objectif de performance H ∞ comme dans les études précédentes [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions : some practical aspects[END_REF], [START_REF] Poussot-Vassal | New semi-active suspension control strategy through LPV technique[END_REF]. La section 2 est consacrée à l'introduction de l'algorithme génétique et le SPEA2. Dans la 3 eme section, on présente la formulation du modèle LPV pour l'étude de la dynamique verticale de véhicule. Dans la section 4, la nouvelle méthode de synthèse du contrôleur LPV avec l'aide de SPEA2 est détaillée. Les simulations et les résultats obtenus sont donnés dans la section 5. Quelques conclusions et perspectives terminent cette communication.

II. Algorithmes génétiques

A. Introduction

Les algorithmes génétiques ont connu un fort développement depuis la première étude de Holland [START_REF] Holland | Adaptation in natural and artificial systems, an introductory analysis with application to biology, control and artificial intelligence[END_REF], l'ouvrage orienté pour la théorie de Goldberg [START_REF] Goldberg | Genetic Algorithms in Searching Optimisation and Machine Learning[END_REF] et l'ouvrage orienté pour l'application de Davis [START_REF] Davis | Handbook of genetic algorithms[END_REF]. Les algorithmes sont basés sur le mécanisme de la sélection naturelle et se sont révélés très efficaces dans l'optimisation avec beaucoup d'applications réelles comme dans les stratégies de financement et d'investissement, la robotique, l'ingénierie de conception, les télécommunications... Ils peuvent conduire à des techniques d'optimisation globale (malgré le coût élevé de calcul) (voir [START_REF] Marler | Survey of multi-objective optimization mehtods for engineering[END_REF]) en utilisant l'approche probabiliste, la re-cherche de multi-points, la combinaison aléatoire (croisement, mutation) et de l'information de l'itération précédente pour évaluer et améliorer la population. Un grand avantage des GAs par rapport à d'autres méthodes de recherche (les méthodes de gradient par exemple), est qu'ils cherchent sans considérer la nature des fonctions d'objectif et des contraintes. Les GAs sont initialisés avec une population aléatoire. Par les opérations génétiques : la sélection, le croisement et la mutation, une nouvelle population est obtenue. En utilisant le processus de sélection, les individus les plus "convenables" qui sont basés sur la valeur de la fonction d'adaptation sont choisis ; le croisement et la mutation s'appliquent alors pour créer la nouvelle population. L'opération génétique sur les individus de population continue jusqu'à ce que le critère d'optimisation soit satisfait ou qu'un certain nombre de générations soit atteint.

B. Optimisation multi-objectif

Une application bien connue des GAs est de trouver la solution optimale pour le problème d'optimisation multiobjectif comportant des objectifs multiples et conflictuels. C'est un problème très populaire dans la pratique et peut être décrit comme suit :

min x∈C F (x) =      f 1 (x) f 2 (x)
. . .

f n obj (x)      , n obj 2, (1)
∀i ∈ {1, 2, ..., n obj } : f i (a) f i (b) ∃j ∈ {1, 2, ..., n obj } : f j (a) < f j (b) (2)
Tous les vecteurs de décision qui ne sont pas dominés par un vecteur de décision sont appelés non-dominés ou Paretooptimal. La famille de vecteurs non-dominés est notée par Ensemble-Pareto. Dans l'Ensemble-Pareto, on ne peut pas améliorer l'un des objectifs sans dégrader au moins un autre.

C. SPEA2 -Strength Pareto Evolutionary Algorithm 2

Comme mentionné dans [START_REF] Zitzler | Comparison of multiobjective evolutionary algorithms : Empirical results[END_REF], deux problèmes majeurs doivent être abordés quand un algorithme génétique est appliqué à l'optimisation multi-objectif :

-Comment faire pour choisir la "meilleure" fonction d'adaptation et de sélection, respectivement, afin d'orienter la recherche vers l'Ensemble-Pareto ? -Comment maintenir une population diversifiée, afin d'éviter la convergence prématurée et réaliser un Ensemble-Pareto bien réparti ?

Il est prouvé que les algorithmes élitistes (par exemple PESA [START_REF] Corne | The pareto enveloppe-based selection algorithm for multiobjective optimization[END_REF], SPEA2 [START_REF] Zitzler | SPEA2 : Improving the strength pareto evolutionary algorithm[END_REF], NSGA-II [START_REF] Deb | A fast elitist multi-objective genetic algorithm : NSGA-II[END_REF]) contribuent à améliorer la convergence du problème MOEAs-Multi-Objective Evolutionary Algorithms [START_REF] Zitzler | Comparison of multiobjective evolutionary algorithms : Empirical results[END_REF]. Les points communs de ces MOEAs élitistes sont les suivants : (1) la valeur de la fonction d'adaptation de chacun de membres de la population est calculée grâce au tri non-dominé et (2) la diversité de la solution est conservée (parce que la distance des individus est maximisée dans le processus d'optimisation).

Dans cet article, le SPEA2 est utilisé parce qu'il est très efficace pour l'optimisation multi-objectif [START_REF] Bleuler | Multiobjective genetic programming : Reducing bloat using SPEA2[END_REF], [START_REF] Li | Multiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-II[END_REF]. Les idées principales du SPEA2 sont :

-Parallèlement à l'ensemble de population

m s zs = -F spring -F mr m us zus = F spring + F mr -k t (z us -z r ) (3)
Les efforts (F spring , F mr ) dans les équations dynamiques ci-dessus sont les parties dynamiques des efforts réels. F spring = k s z def et F mr est donnée par

F mr = a 2 ( żdef + v 0 x 0 z def ) + a 1 (I) tanh(a 3 ( żdef + v 0 x 0 z def )) (4)
où z def = z sz us et żdef = żs -żus et où a 2 , a 3 , v 0 et x 0 sont des paramètres constants et a 1 dépend du courant électrique (0 < a 1min ≤ a 1 ≤ a 1max ). Le quart de véhicule étudié dans cet article correspond à la "Renault Mégane Coupé" (voir [START_REF] Zin | A nonlinear vehicle bicycle model for suspension and handling control studies[END_REF]). Les paramètres sont m s = 315 kg, m us = 37.5 kg, k s = 29500 N/m, k t = 210000 N/m.

Les paramètres du modèle de l'amortisseur ont été choisis comme ceux de [START_REF] Nino-Juarez | Minimizing the frequency in a black box model of a MR damper[END_REF] : a 2 = 1500 Ns/m, a 3 = 129 s/m, v 0 = 0.788x10 -3 m/s, x 0 = 1.195x10 -3 m, a 1min = 0 N, a 1max = 400 N. Deux paramètres pour la conception du contrôleur (voir [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF]) sont F 0 = (a 1max + a 1min )/2=200 N,

F 1 = (a 1max -a 1min )/2=200 N.
En définissant ρ 1 = tanh(a 3 ( żdef + v0 x0 z def )), la représentation d'état du modèle 1/4 de véhicule peut être écrite comme suit :

ẋs = A s x s + B s ρ 1 I + B s1 w y = C s x s (5)
    , B s1 =     0 0 0 kt mus     , C s =     1 0 -1 0     T B.

Formulation de modèle LPV

La formulation LPV du modèle de suspension semiactive est présentée en détail dans [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF]. Ici, on ne donne que le résultat final. Le modèle non-linéaire (5) se ré-écrit dans le cadre de l'approche LPV

ẋ = A (ρ 1 * , ρ 2 * ) x + Bu c + B 1 w y = Cx (6) où x = x s T x f T T A (ρ * 1 , ρ * 2 ) = A s + ρ * 2 B s2 C s2 ρ * 1 B s C f 0 A f , B = 0 B f , B 1 = B s1 0 , C = C s 0 T B s2 = 0; -F0 ms ; 0; F0 mus T , C s2 = a3v0 x0 ; a 3 ; -a3v0 x0 ; -a 3 T ,
x f est le vecteur d'état d'un filtre strictement propre F f défini par

F f : ẋf u = A f B f C f 0 x f u c (7)
où A f , B f , C f sont les matrices constantes. Deux paramètres variants sont définis par

ρ 1 * = tanh(C s2 x s ) tanh( C f x f F 1 ) F 1 C f x f (8) ρ 2 * = tanh(C s2 x s ) C s2 x s (9)
On note également que ρ * 1 et ρ * 2 ne sont pas indépendants (voir Fig. 2).

C. Objectifs de performance pour la suspension

Les objectifs de performance pour le confort et la tenue de route sont issus des idées de [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions : some practical aspects[END_REF].

-Confort dans les hautes fréquences ([3 -10] Hz) : min

IV. Optimisation de contrôleur LPV utilisant SPEA2

La structure de commande proposée dans [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF] se retrouve dans la Fig. 3.

Les filtres utilisés pour la synthèse sont définis comme suit :

F f = Ω f s + Ω f ( 10
)
W zs = k zs s 2 +
W zus z r u z def żdef ρ 1 * , ρ 2 * K(ρ 1 * , ρ 2 * ) P F f z 1 z 2 - - - W zr - w 1 u f u c
Fig. 3. Schéma bloc pour la commande de suspension semi-active .

Le système augmenté du système en boucle ouverte (5) avec les filtres ( 10)-( 12) est représenté par

ξ = A(ρ * , ν)ξ + B 1 (ρ * , ν)w + B 2 u c z = C 1 (ρ * , ν)ξ + D 11 (ρ * , ν)w + D 12 u c (14) y = Cξ
où ξ se compose des états des systèmes ( 5),( 10)-( 12),

z = z 1 z 2 T et ρ * = ρ * 1 ρ * 2 T .
Objectif -Chercher un contrôleur LP V qui stabilise le système ( 6) et qui satisfait les objectifs de performance de suspension spécifiés dans III-C.

A. Synthèse du contrôleur stabilisant

Considérons un contrôleur LP V associé au système [START_REF] Deb | A fast elitist multi-objective genetic algorithm : NSGA-II[END_REF] et décrit par les équations d'état

K c (ρ * ) : ẋc u c = A c (ρ * ) B c (ρ * ) C c (ρ * ) D c (ρ * )
x c y (15) où x c , y et u sont l'état, l'entrée et la sortie du contrôleur, respectivement.

Selon [START_REF] Scherer | Multiobjective outputfeedback control via LMI optimization[END_REF] et [START_REF] Apkarian | A convex characterization of gain scheduled H∞ controllers[END_REF], s'il existe Â, B, Ĉ D, X, Y satisfaisant les LMIs suivantes

M 11 (ρ * i , ν) M 12 (ρ * i , ν) M T 12 (ρ * i , ν) M 22 (ρ * i , ν) ≺ 0 ( 16
) pour i = 1 : 4 où M 11 (ρ * i , ν) = A(ρ * i , ν)X + XA(ρ * i , ν) T + B 2 Ĉ(ρ * i , ν) + Ĉ(ρ * i , ν) T B T 2 M 12 (ρ * i , ν) = Â(ρ * i , ν) + A(ρ * i , ν) T + C T D(ρ * i , ν) T B T 2 M 22 (ρ * i , ν) = Y A(ρ * i , ν) + A(ρ * i , ν) T Y + B(ρ * i , ν)C + C T B(ρ * i , ν) T
alors le contrôleur LP V (15) qui stabilise le système (14) est donné par :

K c (ρ * ) = Co{ A c (ρ * i ) B c (ρ * i ) C c (ρ * i ) D c (ρ * i ) } (17)
avec

Dc(ρ * i ) = D(ρ * i ) Cc(ρ * i ) = Ĉ(ρ * i ) -Dc(ρ * i )C 2 X M -T Bc(ρ * i ) = N -1 B(ρ * i ) -Y B 2 Dc(ρ * i ) Ac(ρ * i ) = N -1 Â(ρ * i ) -Y A(ρ * i )X -Y B 2 Dc(ρ * i )C 2 X M -T -Bc(ρ * i )C 2 XM -T -N -1 Y B 2 Cc(ρ * i ) (18)
i = 1 : 4 correspond aux 4 sommets du polytope (P 1 , P 2 , P 3 , P 4 ) (voir fig. 2). M et N sont définies telles que M N T = I n -XY soit résolue par une décomposition en valeurs singulières et une factorisation de Cholesky.

Jusqu'ici, une question peut être posée : comme le but original est de synthétiser un contrôleur qui stabilise le système [START_REF] Zitzler | SPEA2 : Improving the strength pareto evolutionary algorithm[END_REF], quelle est l'utilité des filtres (10)-( 12) ? En fait, chaque valeur possible des filtres pourra donner un contrôleur qui stabilise le système [START_REF] Zitzler | SPEA2 : Improving the strength pareto evolutionary algorithm[END_REF]. L'algorithme génétique SPEA2 va choisir les meilleurs parmi ces contrôleurs. Alors, les filtres sont des outils utilisés par l'algorithme génétique SPEA2 pour la recherche de meilleurs contrôleurs, satisfaisant les objectifs de performance décrit dans III-C.

B. Optimisation du contrôleur

On n'utilise pas la méthode conventionnelle H ∞ /LP V proposée dans [START_REF] Scherer | Multiobjective outputfeedback control via LMI optimization[END_REF] et [START_REF] Apkarian | A convex characterization of gain scheduled H∞ controllers[END_REF] où le niveau d'atténuation de perturbation γ est l'objectif à minimiser ; ici l'objectif de performance est celui de la suspension, défini par

J = J Confort J Tenue de route (19)
où J Confort et J Tenue de route sont des indices de performance correspondant au sommet P 3 (Remarquons que les solutions à P 3 et P 4 sont identiques en raison de la propriété symétrique du polytope (P 1 , P 2 , P 3 , P 4 )). Il est important de noter qu'en basses fréquences l'amélioration (réduction de gain) de zs /z r implique une amélioration de z s /z r . Avec cette remarque et la définition de critère de performance dans la section III-C, J Confort et J Tenue de route sont définis par

J Confort = 10 1 zs /z r (f )df (20)
J Tenue de route = σ

f ∈ [START_REF] Davis | Handbook of genetic algorithms[END_REF][START_REF] Marler | Survey of multi-objective optimization mehtods for engineering[END_REF][START_REF] Zitzler | Comparison of multiobjective evolutionary algorithms : Empirical results[END_REF][START_REF] Corne | The pareto enveloppe-based selection algorithm for multiobjective optimization[END_REF][START_REF] Deb | A fast elitist multi-objective genetic algorithm : NSGA-II[END_REF][START_REF] Bleuler | Multiobjective genetic programming : Reducing bloat using SPEA2[END_REF][START_REF] Li | Multiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-II[END_REF][START_REF] Zin | A nonlinear vehicle bicycle model for suspension and handling control studies[END_REF][START_REF] Nino-Juarez | Minimizing the frequency in a black box model of a MR damper[END_REF][START_REF] Scherer | Multiobjective outputfeedback control via LMI optimization[END_REF][START_REF] Apkarian | A convex characterization of gain scheduled H∞ controllers[END_REF] (z us /z r (f ))

Remarque 1 : Pour le système de suspension semi-active étudiée, il y a des parties non-linéaires (saturation d'entrée a 1 , la partie non-linéaire hyperbolique tangente de la suspension semi-active), et les contrôleurs LP V stables sont préférés. La solution théorique de l'existence d'un contrôleur LP V stable n'est pas donnée ici. Pour garantir un contrôleur LP V stable, il suffit que tous les contrôleurs locaux à chaque sommet du polytope soient stables. Par algorithme génétique, on peur obtenir ce type des contrôleurs en éliminant les "solutions instables" correspondant à au moins un contrôleur local instable lors de la synthèse. Ceci peut être obtenu simplement en choisissant J = ∞ pour les "solutions instables". Alors, la fonction objectif est choisie comme suit. Algorithme 1 : Affectation de valeur de fonction d'objectif si (16) est faisable alors tous les contrôleurs locaux K ci sont stables alors Calculer J en utilisant les Eq. ( 19) -( 21)

sinon J = ∞ fin sinon J = ∞ fin.
Méthode proposée pour optimiser la commande LP V de suspension semi-active -Etape 1 : Initialiser la première génération avec des valeurs positives aléatoires des paramètres des filtres (10)-( 12). -Etape 2 : Résoudre les LMI [START_REF] Li | Multiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-II[END_REF]

V. Résultats

Pour la simulation, la force du ressort k s (Fig. 1) est une fonction non-linéaire de z def (voir [START_REF] Zin | A nonlinear vehicle bicycle model for suspension and handling control studies[END_REF]) et la force de l'amortisseur est donné par (4). Dans la suite, les différents cas sont considérés pour l'évaluation de la performance de la méthodologie proposée :

-"Soft MRD" = Renault Mégane Car avec un amortisseur MR passif avec a 1 = a 1min = 0 N . -"Hard MRD" = Renault Mégane Car avec un amortisseur MR passif avec a 1 = a 1max = 400 N . -"H ∞ /LP V (ACC 2010)" = Renault Mégane Car avec un amortisseur MR dont le coefficient d'amortissement est piloté par l'algorithme H ∞ /LP V proposé dans [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF]. -"LP V /SPEA2" = Renault Mégane Car avec un amortisseur contrôlé par la méthodologie proposée. Les paramètres des filtres utilisés pour la synthèse de H ∞ /LP V dans [START_REF] Do | An LPV control approach for semi-active suspension control with actuator constraints[END_REF] et les paramètres obtenus par la méthode proposée LP V /SP EA2 sont donnés dans le Tableau I. Notons que pour les deux méthodes, la même structure de filtre a été choisie.

Dans les Fig. 4-8, globalement, la méthode proposée offre le meilleur compromis : une très bonne performance pour le confort en hautes fréquences et la force de contact (tenue de route) ; une bonne performance pour le confort en basses fréquences et le rebond de la roue (tenue de route).

VI. Conclusion

Nous avons proposé une nouvelle méthode de type LP V avec l'aide de l'algorithme génétique SPEA2 pour la commande de suspensions semi-actives (plus généralement, cette méthode peut être étendue pour la commande de systèmes représentés par des modèles LPV). Les simulations ont été faites avec un modèle vertical, non-linéaire, simplifié. Les résultats fréquentiels montrent l'efficacité de la méthode pour obtenir un contrôleur "optimal". Dans les travaux futurs, la réduction du conservatisme de la solution concernant la synthèse de contrôleur LP V stable ou "stabilisation forte" sera étudiée. using genetic algorithms and structured genetic algorithms. In-

  où x est le vecteur de décision, C l'ensemble des vecteurs de décision (ou l'espace de recherche), et F (x) le vecteur des objectifs. L'existence d'une solution idéale x * qui permet de minimiser simultanément toutes les fonctions objectif f 1 , f 2 ...f n obj est en fait rarement possible. Ainsi, dans ce cas, le concept de Pareto-optimal est habituellement utilisée pour décrire la solution du problème d'optimisation multi-objectif et elle est définie comme suit :Considérons deux vecteurs de décision a, b ∈ C. Le vecteur a domine b si et seulement si :

Fig. 1 .

 1 Fig. 1. Modèle 1/4 de véhicule avec un amortisseur MR semi-actif.

  où x s =(z s , żs , z us , żus )T , w=z r , y=z s -z us .

Fig. 2 .

 2 Fig. 2. Ensemble de (ρ 1 * , ρ 2 * ) (zone grisée).

10 3

 10 zs /z r (f )df . -Confort dans les basses fréquences ([0 -4] Hz) : min σ(z s /z r )(f ). -La tenue de route ([0 -20] Hz) : min σ(z us /z r (f )) (rebond de la roue) et de 20 0 (z usz r )/z r (f )df (Force de contact).

Fig. 4 .

 4 Fig. 4. Accélération zs/zr.

Fig. 5 .

 5 Fig. 5. Déplacement du chassis zs/zr.

Fig. 6 .

 6 Fig. 6. Rebond du pneu zus/zr.

Fig. 7 .

 7 Fig. 7. Force de contact (zszr)/zr .

Fig. 8 .

 8 Fig. 8. Comparaison des performances.

  Considérons un modèle simple 1/4 de véhicule se composant d'une masse suspendue (m s ) et d'une masse nonsuspendue (m us ), reliées par un ressort de raideur k s et un amortisseur semi-actif. La roue est représentée par un ressort de raideur k t . Dans ce modèle, z s (z us , respectivement) est le déplacement vertical de m s (m us , respectivement) autour de sa position statique, z r représente le profil vertical du sol. Le contact roue-route est supposé permanent. Les dynamiques du quart de véhicule sont données par :

	P , à chaque génération, une archive ensemble P de taille fixe est
	utilisée pour stocker des solutions non dominées.
	-La fonction d'adaptation de chaque solution est défi-
	nie par la valeur "strength" (basée sur la relation de
	Pareto-dominance) et l'estimation de la densité (fonc-
	tion de la distance au k-ème point le plus proche). La
	solution non-dominée a une valeur de fonction d'adap-
	tation inférieure à 1.

-La sélection de l'environnement (environmental selection) est appliquée pour les deux ensembles de la population et d'archive P + P . Les solutions non-dominées sont choisies et sont copiées vers l'archive P . Si le nombre de solutions non-dominées s'adapte à la taille de l'archive, l'étape de sélection de l'environnement est terminée. S'il est plus petit, un nombre approprié d'individus les mieux dominés dans P + P est ajouté à P . Sinon, une procédure de troncature de l'archive est utilisée, c.à.d, la solution qui a la distance minimale aux autres solutions est éliminée.

-La sélection par tournoi binaire (binary tournament selection) est appliquée sur P pour créer une génération intermédiaire (matting pool). Ensuite, les opérateurs de recombinaison et de mutation sont appliqués à la génération intermédiaire, afin de créer la nouvelle population de P .

III. Formulation LPV du modèle de suspension semi-active

A. Description du système

  2ξ 11 Ω 11 s + Ω 11

	2 2 s 2 + 2ξ 21 Ω 21 s + Ω 21 s 2 + 2ξ 12 Ω 12 s + Ω 12 W zus = k zus s 2 + 2ξ 22 Ω 22 s + Ω 22 Définissons ν = [ Ω zs z us W zs	2 2	(11) (12)

f Ω 11,12 ξ 11,12 k zs Ω 21,22 ξ 21,22 k zus ] T (13) le vecteur de décision qui est optimisé par le SPEA2.

  Paramètres des filtres pour la synthèse LP V .

	Contrôleurs	Filtre			W zs					Wz us
		Ω f	Ω 11	Ω 12	ξ 11	ξ 12	k zs	Ω 21	Ω 22	ξ 21	ξ 22	kz us
	H∞/LPV (ACC 2010)	50	125.7	125.7	20	1	1	69.1	69.1	20	1	1
	LPV/SPEA2 (proposée)	9.3	50.71	0.012	55.39	78.4	68.96	94.39	28.23	81.52	48.21	21
					TABLE I
									Références


Guillaume Tomasini

INTEGRABILITY OF WEIGHT MODULE OF DEGREE 1

Introduction

Let g denote a simple Lie algebra over the field C of complex numbers. Let h be a Cartan subalgebra of g. A weight module is a g-module, h-diagonalizable, having finite dimensional weight spaces. The set of all weight g-modules is a category containing the BGG category O. This category of weight modules has been much studied in recent years (e.g. [START_REF] Fernando | Lie algebra modules with finite-dimensional weight spaces[END_REF][START_REF] Futorny | The weight representations of semisimple finite dimensional Lie algebras[END_REF][START_REF] Britten | A classification of simple Lie modules having a 1-dimensional weight space[END_REF][START_REF] Mathieu | Classification of irreducible weight modules[END_REF][START_REF] Britten | Complete reducibility of torsion free Cnmodules of finite degree[END_REF][START_REF] Britten | Tensor product realizations of simple torsion free modules[END_REF][START_REF] Grantcharov | Category of sp(2n)-modules with bounded weight multiplicities[END_REF][START_REF] Grantcharov | Cuspidal representations of sl(n + 1)[END_REF][START_REF] Mazorchuk | Cuspidal sln-modules and deformations of certain brauer tree algebras[END_REF][START_REF] Mazorchuk | Blocks of the category of cuspidal sp 2n -modules[END_REF]). It is then a natural question to find those weight modules that integrate to continuous (resp. unitary) representations of simply-connected real Lie groups. They should form a small but interesting class of representations, with small Gelfand-Kirillov dimension. In this paper, we treat the case of weight modules with weight multiplicities equal to 1.

Let us explain our strategy. Let G be a simply-connected real Lie group. Assume the g-module V integrates to a continuous representation π of G in some Hilbert space H. Let K be a compact subgroup of G and denote by (π |K , H |K ) the representation (π, H) restricted to K. Then it is well known that the representation (π |K , H |K ) is unitarizable. Therefore, we can express H |K as a direct sum of simple finite dimensional unitary representation of K. A consequence is that the (complexified) Lie algebra of K should act nicely on V : the module V should be a k-finite g-module. If K is big enough, this condition is strong enough to imply that V should be a highest (or lowest) weight module. We then use a classification result due to Benkart, Britten and Lemire to describe the possible modules. Then it remains to check whether or not these modules can be integrated. Our results also make use of a theorem of Jørgensen and Moore and classical results about discrete series.

2. Some facts about weight modules 2.1. Weight modules of degree 1. Let g denote a simple Lie algebra over the field C of complex numbers. Fix a Cartan subalgebra h. A g-module V is called a weight module if [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF] The module V is finitely generated, [START_REF] Britten | Complete reducibility of torsion free Cnmodules of finite degree[END_REF] We have the following decomposition of V :

V = λ∈h * V λ , V λ := {m ∈ V | ∀ H ∈ h, H • m = λ(H)m}, 1 (3)
The weight spaces V λ are all finite dimensional. We call degree of a weight module the supremum of the dimension of the weight spaces:

deg(V ) = sup λ {dim(V λ )} ∈ Z ≥0 ∪ ∞.
When deg(V ) ∈ Z ≥0 , we call V a bounded module. In particular, if V is a weight gmodule of degree 1, then all the non zero weight spaces are 1-dimensional. The weight modules of degree 1 have been studied by Benkart, Britten and Lemire [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF]. In particular, they constructed weight modules of degree 1, N (a) (with a ∈ C n ) for sl(n + 1, C) and M (b) (with b ∈ C n ) for sp(n, C) (see [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF] for the explicit construction of these modules). Moreover, they proved the following: Theorem 2.1 (Benkart, Britten, Lemire [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF]). Let V be a simple infinite dimensional weight g-module of degree 1. Then [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF] The Lie algebra g is isomorphic to either sl(n + 1, C) or to sp(n, C).

(2) The Gelfand-Kirillov dimension of V equals the rank of g.

(3

) If g = sl(n + 1, C), then there is a ∈ C n such that V ∼ = N (a). (4) If g = sp(n, C), then there is b ∈ C n such that V ∼ = M (b).
For our purpose we shall need another notion. Let l be a subalgebra of g. A g-module V is a (g, l)-module of finite type if as an l-module, V splits into a direct sum of simple finite dimensional l-modules, with finite multiplicities. For instance, a weight module is a (g, h)-module of finite type. The general notion of (g, k)-module has been studied in details by Penkov, Serganova and Zuckerman in [START_REF] Penkov | Generalized Harish-Chandra modules[END_REF][START_REF] Penkov | On the existence of (g, l)-modules of finite type[END_REF][START_REF] Penkov | Generalized Harish-Chandra modules: a new direction in the structure theory of representations[END_REF][START_REF] Penkov | Generalized Harish-Chandra modules with generic minimal k-type[END_REF][START_REF] Penkov | Construction of generalized Harish-Chandra modules with arbitrary minimal t-type[END_REF].

2.2. Classification of (g, l j )-module of finite type and of degree 1. Let n be a positive integer greater than 1. Let g denote the Lie algebra sl(n + 1, C). Let h denote the standard Cartan subalgebra of g, consisting of diagonal matrices. Denote by H 0 , H 1 , . . . , H n-1 its canonical basis. Denote by E j (resp. F j ) the vector in g corresponding to the elementary matrix E j+1,j+2 (resp. E j+2,j+1 ) for 0 ≤ j ≤ n -1. Then g is Lie-generated by the vectors {H j , E j , F j } 0≤j≤n-1 . Denote by l j the maximal standard Levi subalgebra of g Lie-generated by h and the vectors {E k , F k } k =j .

For future use, we shall find those infinite dimensional weight g-modules of degree 1 whose restriction to some l j is a direct sum of finite dimensional l j -modules. To this aim, we need the following general result: Lemma 2.2 (Fernando [5], Benkart-Britten-Lemire [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF]). Let a be a simple Lie algebra over C. Let t be a Cartan subalgebra of a. Let V be a simple weight a-module. Let R denote the root system of (a, t). Then [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF] For any α ∈ R, and any X ∈ a α \ {0}, the action of X on V is either locally finite or injective. (2) Let α, β ∈ R be such that there are X ± ∈ a ±α \ {0} and Y ± ∈ a ±β \ {0} satisfying X ± both act locally finitely on V and Y ± both act injectively on V . Then α + β ∈ R.

Proof. See [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF]Section4].

Corollary 2.3. Let V be a simple weight g-module. Let 0 ≤ j ≤ n -1. Assume that V is a (g, l j )-module of finite type. Then V is a highest weight or a lowest weight module.

Proof. From lemma 2.2, the vectors E k and F k act locally finitely or injectively on V . By hypothesis, they act locally finitely for k = j. Now, lemma 2.2 applied to E j , F j and either E j-1 , F j-1 or E j+1 , F j+1 shows that at least one of the vectors E j and F j acts locally finitely on V . In the first case, the module is a highest weight module. In the second case, it is a lowest weight module.

Denote by {ω i } i=1..n the fundamental weights for g. Recall now the following: In the notation of theorem 2.1, these modules correspond to N (a, 0, . . . , 0), N (-1, . . . , -1 i , -1a, 0, . . . , 0), and N (-1, . . . , -1, -1a), for a ∈ C.

In the sequel we will often work with the sl(n + 1, C)-module N (a, 0, . . . , 0). For the convenience of the reader we write down here the action of the vectors {H j , E j , F j } 0≤j≤n-1 on a basis. The module N (a, 0, . . . , 0) has a basis

x(k) indexed by k ∈ Z n ≥0 . If k = (k 1 , . . . , k n ) ∈ Z n ≥0 , we set |k| := k 1 + • • • + k n .
The action is given by:

H 0 • x(k) = (a -k 1 -|k|)x(k) (1a) H j • x(k) = (k j -k j+1 )x(k) (1b) E 0 • x(k) = k 1 x(k -ǫ 1 ) (1c) F 0 • x(k) = (a -|k|)x(k + ǫ 1 ) (1d) E j • x(k) = k j+1 x(k -ǫ j+1 + ǫ j ) (1e) F j • x(k) = k j x(k + ǫ j+1 -ǫ j ) (1f)
From this classification, we are in position to prove the Proposition 2.5. Let 0 ≤ j ≤ n -1. Let V be a simple infinite dimensional (g, l j )module of finite type and of degree 1. Then [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF] If j = 0, then V or its contragredient is isomorphic to N (a, 0, . . . , 0), for some

a ∈ C \ Z ≥0 or to N (-1, m, 0, . . . , 0) for some m ∈ Z ≥0 . (2) If j = n -1, then V or its contragredient is isomorphic to N (-1, . . . , -1, a), for some a ∈ C \ Z <0 or to N (-1, . . . , -1, -1 -m, 0) for some m ∈ Z ≥0 . (3) If 0 < j < n -1, then V or its contragredient is isomorphic to N (-1, . . . , -1 j+1 , m, 0, . . . , 0) or N (-1, . . . , -1 j , -1 -m, 0, . . . , 0), for some m ∈ Z ≥0 .
Proof. From corollary 2.3, V or its contragredient is a simple highest weight module. Therefore we know that V or its contragredient is given by proposition 2.4. It thus remains to check whether or not the modules in proposition2.4 satisfy the restriction property.

Assume that j = 0. Let 0 < k < n-1 and consider the module V = N (-1, . . . , -1, a, 0, . . . , 0) where a is a complex number in position k +1. Then the highest weight x for this module satisfies:

H k-1 • x = (-1 -a)x, H k • x = ax.
If k-1 = 0, then by our assumption on V , x should generate a finite dimensional module. This imposes that the vectors H 1 , . . . , H n-1 of the Cartan subalgebra acts on x by non negative integers. Therefore a and -1a should be non negative integers, which is impossible. The same argument shows that for k = 1 we must have a ∈ Z ≥0 . Consider now the module N (-1, . . . , -1, a) for a ∈ C. Using once again the same argument, we show that necessarily a should be a negative integer. In this case, the module is finite dimensional.

We need to show now that N (a, 0, . . . , 0) for a ∈ C \ Z ≥0 and N (-1, m, 0, . . . , 0) for m ∈ Z ≥0 do satisfy the restriction property. Let us prove it for N (a, 0 . . . , 0).

We want to find those linear combinations µ k x(k) which are highest weight vectors for the action of l 0 . From the explicit action given by formulae (1), we conclude that the highest weight vectors are the linear combinations of the following linearly independent highest weight vectors:

x(k 1 , 0, . . . , 0), k 1 ∈ Z ≥0 .
We shall prove now that the module U(l 0 )x(k 1 , 0, . . . , 0) is a simple highest weight module. Since it is a highest weight module, it is indecomposable. It is simple if and only if it does not contain a highest weight vector linearly independent of x(k 1 , 0, . . . , 0). But any such vector is a linear combination of x(k ′ , 0, . . . , 0) for some k ′ ∈ Z ≥0 . However the action of

nH 0 +(n-1)H 1 +• • •+H n-1 , vector generating the center of l 0 , on x(k ′ , 0, . . . , 0) is: (nH 0 + (n -1)H 1 + • • • + H n-1 ) • x(k ′ , 0, . . . , 0) = na -(n + 1)k ′ .
Therefore the center acting as a scalar on U(l 0 )x(k 1 , 0, . . . , 0), we conclude that there is no highest weight vector in this module but the multiples of x(k 1 , 0, . . . , 0). Thus proving that the module is simple. Hence it is clear that we have the following branching:

N (a, 0, . . . , 0) |l 0 = k∈Z ≥0 U(l 0 )x(k, 0, . . . , 0).
Therefore we proved that N (a, 0, . . . , 0) is a (g, l 0 )-module of finite type as asserted. The proof for N (-1, m, 0, . . . , 0) is the same.

The case j > 0 is analogous.

Type A case

In this section, we shall find which degree 1 sl(n + 1, C)-module integrate to a continuous representation of some real Lie group whose complexified Lie algebra is sl(n + 1, C).

A natural action of SU

(1, n). Let n be a positive integer. Let SU (1, n) denote the subgroup of GL(n + 1, C) consisting of those matrices g such that t ḡ × -1 I n × g = -1 I n
and whose determinant is 1. We shall label rows and columns from 0 to n. This is a real Lie group. It acts on

S n := {(z j ) ∈ C n | n j=1 |z j | 2 = 1} via g •    z 1 . . . z n    =                      n j=0 g j 1 z j n j=0 g j 0 z j . . . n j=0 g j n z j n j=0 g j 0 z j                     
,

where g = (g j k ) 0≤j,k≤n ∈ SU (1, n) and z 0 = 1. Since SU (1, n) preserves the quadratic form -|Z 0 | 2 + |Z 1 | 2 + • • • + |Z n | 2 ,
the denominator is never 0 and g • z is in S n for any z ∈ S n . Denote by dσ the measure on S n induced from the Lebesgue measure of C n and by Ω n the volume of S n . It is well known that Ω n = 2π n (n-1)! . We also denote by H(C n ) the space of holomorphic functions from C n to C. Then the action of SU (1, n) on S n induces a natural continuous representation on L 2 (S n , dσ Ωn ) ∩ H(C n ). We can further construct a unitary representation ρ on this space by

ρ(g)(ϕ)(z) :=   n j=0 (g -1 ) j 0 z j   -n × ϕ(g -1 • z). Let k = (k j ) ∈ Z n ≥0 . Set P (k)(z) := n j=1 z k j j .
Then the family (P (k)) k∈Z n ≥0 is an orthogonal basis for the Hilbert space L 2 (S n , dσ Ωn )∩H(C n ). Moreover, we have

P (k) 2 = n j=1 k j ! |k| j=1 (j + n -1)
, where |k| :

= n j=1 k j .
Consider the following 1-parameter families:

e itH 1 :=           1 e -it e it 1 . . . 1           , . . . , e itH n-1 :=         1 . . . 1 e -it e it         . e tX 1 :=           1 cos t -sin t sin t cos t 1 . . . 1           , e tY 1 :=           1 cos t -i sin t -i sin t cos t 1 . . . 1           , . . . , e tX n-1 :=         1 . . . 1 cos t -sin t sin t cos t         , e tY n-1 :=         1 . . . 1 cos t -i sin t -i sin t cos t         , e tX 0 :=         cosh t -sinh t -sinh t cosh t 1 . . . 1         , e tY 0 :=         cosh t -i sinh t i sinh t cosh t 1 . . . 1         , e itH 0 :=        e -it e it 1 . . . 1         .
Then the Lie algebra su(1, n) of SU (1, n) is generated (as a Lie algebra) by

iH 0 , . . . , iH n-1 , X 0 , . . . , X n-1 , Y 0 , . . . Y n-1 . Set E 0 := X 0 + iY 0 2 , F 0 := X 0 -iY 0 2 , E j := - X j + iY j 2 , F j := X j -iY j 2 , 1 ≤ j ≤ n -1.
Then (H j , E j , F j ) 0≤j≤n generates a Lie algebra g isomorphic to sl(n + 1, C). The Cartan subalgebra h of g is the subalgebra generated by {H 0 , . . . , H n-1 }. We can compute as usual the action of g on the basis (P (k)) k∈Z n ≥0 . We get:

   H 0 • P (k) = (-n -|k| -k 1 )P (k) E 0 • P (k) = k 1 P (k -ǫ 1 ) F 0 • P (k) = (-n -|k|)P (k + ǫ 1 ) , ( 2a
)    H j • P (k) = (k j -k j+1 )P (k) E j • P (k) = k j+1 P (k -ǫ j+1 + ǫ j ) F j • P (k) = k j P (k + ǫ j+1 -ǫ j ) , ∀ 1 ≤ j ≤ n -1, (2b)
where ǫ j is the vector in Z n ≥0 whose entries are all zero except the jth entry which is 1. In the sequel we shall deform this infinitesimal representation and show the 1-parameter deformation thus constructed integrates to a continuous representation of the universal cover of SU (1, n). We shall further explicit those values of the parameter such that the representation is unitary.

3.2. Deformation of the natural action of sl(n + 1, C). To each k ∈ Z n ≥0 , let us associate a vector e(k). Let

H :=                u := k∈Z n ≥0 u k e(k) k∈Z n ≥0 |u k | 2 n j=1 k j ! |k| j=1 (j + n -1) < ∞                .
We define on H a Hilbert space structure by requiring that the basis (e(k)) is orthogonal and that e(k) 2

= n j=1 k j ! |k| j=1 (j + n -1)
. Denote by G the universal cover of SU (1, n). According to the previous subsection, there is a continuous representation ρ of G (in fact, a unitary representation of SU (1, n)) corresponding to the representation of g given on H by formulae [START_REF] Britten | Complete reducibility of torsion free Cnmodules of finite degree[END_REF].

Let a ∈ C \ Z ≥0 . For any l ∈ Z ≥0 , set µ a (l) := l j=1 j + n -1 |j -a -1| . Note that µ a (l)
is a well defined positive real number such that µ -n (l) = 1. We define now operators (H j (a), E j (a), F j (a)) 0≤j≤n-1 on H by their action on the basis e(k) as follows:

∀ 1 ≤ j ≤ n -1, H j (a) = H j , E j (a) = E j , F j (a) = F j , (H 0 (a) -H 0 ) • e(k) = (n + a)e(k), (3a) (E 0 (a) -E 0 ) • e(k) = k 1 µ(|k| -1) µ(|k|) -1 e(k -ǫ 1 ), (3b) (F 0 (a) -F 0 ) • e(k) = (a -|k|) µ(|k| + 1) µ(|k|) + n + |k| e(k + ǫ 1 ). (3c)
Notice that if a = -n then all the new operators coincide with the corresponding undeformed operator. It is easy to check that the new operators give rise to another representation of sl(n + 1, C). We shall refer to the space of this representation as H a (even though as a Hilbert space it is nothing but H). In the sequel we will use

m - k := k 1 µ(|k| -1) µ(|k|) -1 , m + k := (a -|k|) µ(|k| + 1) µ(|k|) + n + |k|. Remark 3.1. Set x(k) := µ(|k|)e(k). Then x(k) 2 = n j=1 k j ! |k| j=1 |j -a -1| . Moreover, x(k)
is also a basis for H a and the deformed action of sl(n + 1, C) on H a is precisely the one given in [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF]. To apply the theorem to our situation, we set S 0 := {iH j , X j , Y j , 0 ≤ j ≤ n -1}. The Hilbert space is H, the dense subset is

Integrability of the representation

D :=      u = k∈Z n ≥0 u k e(k) ∈ H k∈Z n ≥0 |k| N u k e(k) ∈ H, ∀ N ∈ Z ≥0      .
The function f is given by the formulae [START_REF] Britten | Complete reducibility of torsion free Cnmodules of finite degree[END_REF]. At this point, we need to check that the image of f is in A ∞ (D), i.e. to check that the operators defined on D by formulae [START_REF] Britten | Complete reducibility of torsion free Cnmodules of finite degree[END_REF] are bounded for all the norms • l . As f (iH j ) = f (X j ) = f (Y j ) = 0 for j ≥ 1, we only need to consider the three operators f (iH 0 ), f (X 0 ) and f (Y 0 ). As f (iH 0 ) is a scalar operator, the boundedness is clear. We have to prove it for f (X 0 ) and f (Y 0 ). First we note the following: Lemma 3.3. The operators f (X 0 ) and f (Y 0 ) are bounded for all the norms • l if and only if the operators f (E 0 ) and f (F 0 ) are.

Proof. This is clear since X 0 and Y 0 are linear combination of E 0 and F 0 and viceversa.

Lemma 3.4. The operators f (E 0 ) and f (F 0 ) are bounded for all the norms • l if and only if they are bounded for all the norms constructed using the set S0 := {H j , E j , F j , 0 ≤ j ≤ n -1} instead of S 0 .

Proof. Again, this follows from the fact that we can express the elements of S 0 as linear combinations of those in S0 and vice-versa. Proposition 3.5. The operators f (E 0 ) and f (F 0 ) are bounded for all the norms • l .

Proof. From the lemma 3.4, we can use the norms • l constructing from the set S0 . We prove the lemma by induction on l for f (E 0 ). The proof for f (F 0 ) is analogous. Let

u := u k e(k) ∈ D. Then f (E 0 )u 2 = k |u k | 2 |m - k | 2 e(k -ǫ 1 ) 2 = k|k 1 >0 |u k | 2 |m - k | 2 e(k) 2 |k| + n -2 k 1 = k|k 1 >0 |u k | 2 e(k) 2 × k 1 (|k| + n -2) µ(|k| -1) µ(|k|) -1 2 ≤ sup k|k 1 >0 k 1 (|k| + n -2) µ(|k| -1) µ(|k|) -1 2 × k |u k | 2 e(k) 2 = sup k|k 1 >0 k 1 (|k| + n -2) µ(|k| -1) µ(|k|) -1 2 × u 2 .
Using an asymptotic development of µ(|k|-1) µ(|k|) -1

2
we easily see that the above supremum is finite, thus proving that f (E 0 ) is bounded for

• 0 . Assume now that f (E 0 ) is bounded for the norms • l for 0 ≤ l ≤ M -1.
Let A 1 , . . . , A M be elements in S0 . Let u ba as above and consider the expression

A 1 • • • A M f (E 0 )u 2 . Since A 1 • • • A M is
a weight vector in the enveloping algebra, the vectors A 1 • • • A M e(k) are mutually orthogonal. Therefore, we have

A 1 • • • A M f (E 0 )u 2 = k |u k | 2 |m - k | 2 A 1 • • • A M e(k -ǫ 1 ) 2 . ( 4
)
Now from formulae (2), it is clear that

A 1 • • • A M e(k) = P A 1 •••A M (k)e(k + l(A 1 • • • A M )), where P A 1 •••A M (k) is a polynomial in (k 1 , . . . , k n ), product of M monomials of degree 1.
For brevity, we shall denote it by P (k) in the sequel. Moreover

l(A 1 • • • A M ) ∈ Z n and |l(A 1 • • • A M )| := n j=1 |l(A 1 • • • A M ) j |
is a non negative integer smaller than 2M , since for any A ∈ S0 , the vector Ae(k) is either a multiple of e(k) or a multiple of e(k ± ǫ 1 ) or a multiple of e(k ± ǫ j ± ǫ j+1 ). In the sequel, we denote l(A 1 • • • A M ) simply by l.

Let k be such that P (kǫ 1 ) = 0 and k 1 = 0. If P (k) = 0 then there is 1 ≤ j ≤ M such that A j = H 1 . This is proved by induction on M using the action of S0 given by formulae [START_REF] Britten | Complete reducibility of torsion free Cnmodules of finite degree[END_REF]. In particular, if k 1 = 0, P (k) = 0 and P (k

-ǫ 1 ) = 0, then |P (k-ǫ 1 )| |P (k)|
is well-defined and is bounded by a number depending on M only.

Now we write

A 1 • • • A M f (E 0 )u 2 = k | A 1 • • • A M e(k) = 0, A 1 • • • A M e(k -ǫ 1 ) = 0 |u k | 2 |m - k | 2 A 1 • • • A M e(k -ǫ 1 ) 2 + k |A 1 • • • A M e(k) = 0, A 1 • • • A M e(k -ǫ 1 ) = 0 |u k | 2 |m - k | 2 A 1 • • • A M e(k -ǫ 1 ) 2 . ( 5
)
Let us work with the first sum. We can rewrite it in the following form:

|u k | 2 |m - k | 2 A 1 • • • A M e(k -ǫ 1 ) 2 A 1 • • • A M e(k) 2 A 1 • • • A M e(k) 2 .
This in turn is equal to: 2 . Therefore we have:

|m - k | 2 |P (k -ǫ 1 )| 2 |P (k)| 2 e(k + l -ǫ 1 2 e(k + l) 2 × |u k | 2 A 1 • • • A M e(k)
|m - k | 2 |P (k -ǫ 1 )| 2 |P (k)| 2 e(k + l -ǫ 1 2 e(k + l) 2 × |u k | 2 A 1 • • • A M e(k) 2 ≤ sup |m - k | 2 |P (k -ǫ 1 )| 2 |P (k)| 2 e(k + l -ǫ 1 2 e(k + l) 2 A 1 • • • A M u k e(k) 2 ≤ sup |m - k | 2 |P (k -ǫ 1 )| 2 |P (k)| 2 e(k + l -ǫ 1 2 e(k + l) 2 A 1 • • • A M u 2 ≤ sup |m - k | 2 |P (k -ǫ 1 )| 2 |P (k)| 2 e(k + l -ǫ 1 2 e(k + l) 2 × u 2 M .
We must now prove that sup

|m - k | 2 |P (k-ǫ 1 )| 2 |P (k)| 2 e(k+l-ǫ 1 2 e(k+l) 2
is bounded by a number independent of l (but possibly depending on M ). From previous remarks, it is sufficient to prove that |m - k | 2 e(k+l-ǫ 1 ) 2 e(k+l) 2 is bounded. Since |l| ≤ 2M , this is an easy consequence of the explicit expression for m - k . Let us now investigate the second sum in (5), assuming it is not empty. As we already mentioned, there is an index j such that A j = H 1 . Then using commutation relations in the enveloping algebra, we have:

A 1 • • • A M = A ′ 1 • • • A ′ M -1 H 1 + A ′′ 1 • • • A ′′ M -1
. Therefore, we have:

k |A 1 •••A M e(k)=0, A 1 •••A M e(k-ǫ 1 ) =0 |u k | 2 |m - k | 2 A 1 • • • A M e(k -ǫ 1 ) 2 ≤ |u k | 2 |m - k | 2 A ′ 1 • • • A ′ M -1 H 1 e(k -ǫ 1 ) 2 + A ′′ 1 • • • A ′′ M -1 e(k -ǫ 1 ) 2 ≤ |u k | 2 |m - k | 2 (k 1 -k 2 -1) 2 A ′ 1 • • • A ′ M -1 e(k -ǫ 1 ) 2 + A ′′ 1 • • • A ′′ M -1 f (E 0 )u 2 ≤ sup{(k 1 -k 2 -1) 2 } × A ′ 1 • • • A ′ M -1 f (E 0 )u 2 + A ′′ 1 • • • A ′′ M -1 f (E 0 )u 2
The second sum is by induction smaller than f

(E 0 ) 2 M -1 × u 2 M -1 .
For the first sum, the induction shows that

A ′ 1 • • • A ′ M -1 f (E 0 )u 2 ≤ f (E 0 ) 2 M -1 × u 2 M -1 . Thus, it suffices to prove that (k 1 -k 2 -1) 2 is bounded. Since A 1 • • • A M e(k) = 0,
there is an integer j such that 2 } is bounded by a number depending on M only.

A j = H 1 , H 1 A j+1 • • • A M e(k) = 0 and A j+1 • • • A M e(k) = 0. But then A j+1 • • • A M e(k) = C × e(k + l ′ ) for some non zero constant C. As above |l ′ | ≤ 2M . Thus H 1 e(k + l ′ ) = 0, which means that (k + l ′ ) 1 = (k + l ′ ) 2 or also k 1 -k 2 -1 = l ′ 1 -l ′ 2 -1. And we have |l ′ 1 -l ′ 2 -1| ≤ |l ′ | + 1 ≤ 2M + 1, proving thus that sup{(k 1 -k 2 -1)
Altogether, we have proved that:

A 1 • • • A M f (E 0 )u 2 ≤ C(M ) × u 2
M , for some constant C(M ) depending on M only (Note here that u M -1 ≤ u M ). As a consequence, we get 3.4. Unitarisability. We now know a whole family of continuous representation of G. We should ask then which of these are unitary. If the representation H a is unitary then the infinitesimal action given by the Lie basis {iH j (a), X j (a), Y j (a)} should be given by skew-symmetric operators. In other word, we should have (iH j (a)) * = -iH j (a), X j (a) * = -X j (a) and Y j (a) * = -Y j (a). Using the expression of the H j (a), E j (a), F j (a) in term of this basis, it is equivalent to have H j (a) * = H j (a), E j (a) * = F j (a) for j > 0 and E 0 (a) * = -F 0 (a). As H j (a) is a diagonal operator, it is selfadjoint if and only if its eigenvalues are real. This imposes a ∈ R. Remember that the representation that we started with is unitary. So it only remains to prove f (E j ) * = f (F j ) for j > 0 and f (E 0 ) * = -f (F 0 ). The first condition is trivial since f (E j ) = f (F j ) = 0. To check the second condition, we need to compare f (F 0 )e(k), e(l) ande(k), f (E 0 )e(l) . They are equal for all k and l if and only if a ∈ R <0 .

f (E 0 )u M ≤ C(M ) u M , proving that f (E 0 ) is bounded for the norm • M .
Proposition 3.8. The continuous representation H a of G is unitary if and only if a ∈ R <0 . Remark 3.9. When a ∈ Z ≥0 , the representation H a constructed in the remark 3.1 is not unitary (unless a = 0) since it has finite dimension greater than 1 and G is not a compact group.

3.5. SU (p, q) case. Let 1 ≤ p ≤ n. Set q = n + 1p. Let G p,q denote the universal cover of SU (p, q). The complex Lie algebra g = sl(n + 1, C) is the complexification of the Lie algebra of G p,q . Moreover, G p,q contains a compact subgroup K p,q isomorphic to SU (p) × SU (q), whose complexified Lie algebra is isomorphic to the semisimple part of l p-1 . Let us now give the classification of all simple infinite dimensional degree 1 modules coming from a continuous representation of G p,q on some Hilbert space. Theorem 3.10. Let V be a simple infinite dimensional weight g-module of degree 1. Then V integrates into a continuous representation of G p,q on a Hilbert space if and only if [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF] Either V or its contragredient is isomorphic to N (a, 0, . . . , 0)

(for a ∈ C \ Z ≥0 ) or to N (-1, m, 0, . . . , 0) (for m ∈ Z ≥0 ), in case p = 1. (2) Either V or its contragredient is isomorphic to N (-1, . . . , -1, a) (for a ∈ C\Z <0 ) or to N (-1, . . . , -1, -1 -m, 0) (for m ∈ Z ≥0 ), in case p = n. (3) Either V or its contragredient is isomorphic to N (-1, . . . , -1 p , m, 0, . . . , 0) (for m ∈ Z ≥0 ) or to N (-1, . . . , -1 p-1 , -1-m, 0, . . . 0) (for m ∈ Z ≥0 ), in case 1 < p < n.
Moreover, the corresponding representation of G p,q is unitary if and only if

(1) Either V or its contragredient is isomorphic to N (a, 0, . . . , 0) (for a ∈ R <0 ) or to N (-1, m, 0, . . . , 0) (for m ∈ Z ≥0 ), in case p = 1. (2) Either V or its contragredient is isomorphic to N (-1, . . . , -1, a) (for a ∈ R >0 ) or to N (-1, . . . , -1, -1 -m, 0) (for m ∈ Z ≥0 ), in case p = n. (3) Either V or its contragredient is isomorphic to N (-1, . . . , -1 p , m, 0, . . . , 0) (for m ∈ Z ≥0 ) or to N (-1, . . . , -1 p-1 , -1-m, 0, . . . 0) (for m ∈ Z ≥0 ), in case 1 < p < n.
Proof. First, remark that given any continuous representation π of G p,q on a Hilbert space, its restriction to K p,q splits into a direct sum of finite dimensional representations (possibly with infinite multiplicities). Therefore, the corresponding g-module should also split into a direct sum of finite dimensional l p -modules. In other word, a necessary condition is that the underlying g-module is a (g, l p )-module of finite type. Therefore, by proposition 2.5, V or its contragredient should be isomorphic to the asserted modules.

(1) In case p = 1, we already know from corollary 3.6, that N (a, 0, . . . , 0) does integrate into a continuous representation of G 1 . Moreover from proposition 3.8, we know that this representation is unitary exactly when a ∈ R <0 . Now, the module N (-1, m, 0, . . . , 0) is a highest weight module with highest weight λ m = (-1m, m, 0, . . . , 0) ∈ Z n . This is clearly an analytically integral weight, and dominant with respect to the positive roots of l 0 . It is then straightforward to check that it is the underlying g-module of the holomorphic discrete series of SU (1, n) corresponding to the parameter λ m . (2) The case p = n is of course identical to the previous one up to a relabeling of the simple roots.

(3) The intermediate case 1 < p < n is easy, since the possible underlying gmodules all correspond to holomorphic discrete series, their parameter being (0, . . . , 0, -1m, m, 0, . . . , ) or (0, . . . , 0, m, -1m, 0, . . . , 0).

3.6. SL(n, R) case. Assume that n ≥ 3. Let G n denote the universal cover of SL(n, R).

Its complexified Lie algebra is also g = sl(n, C). The compact Lie group

K n = SO(n) is a subgroup of G n .
Theorem 3.11. Let V be a simple weight g-module. Then V can be integrated into a continuous representation of G n on a Hilbert space if and only if V is finite dimensional.

Proof. Assume V can be integrated into a continuous representation of G n in a Hilbert space. The complexified Lie algebra of K n contains the vectors E j + F j . By lemma 2.2, these vectors should act locally finitely on V . Let V λ be a weight space of V . Denote by α j the weight of E j . Then E j + F j : V λ → V λ+α j ⊕ V λ-α j . Therefore E j + F j is locally finite if and only if both E j and F j are. Then the module is finite dimensional as asserted. The converse is obvious.

Type C case

Let n be a positive integer. Let p and q be positive integers such that p + q = n. In this section, we consider the groups Sp(n, R) and Sp(p, q) and their universal cover G n and G p,q . They contain the compact subgroup K n and K p,q isomorphic to SU (n) and SP (p) × Sp(q) respectively (see for instance [START_REF] Knapp | Lie groups beyond an introduction[END_REF]). Denote by g the Lie algebra sp(n, C). Theorem 4.1. Let V be a simple infinite dimensional weight g-module of degree 1. Then [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF] V cannot integrate into a continuous representation of G p,q on a Hilbert space.

(2) V integrates into a continuous representation of G n on a Hilbert space if and only if V or its contragredient is isomorphic to M (-1, . . . , -1) or M (-1, . . . , -1, -2).

In this case, the corresponding representation of G n is simple and unitary, and isomorphic to the even or odd part of the metaplectic representation or its contragredient.

Proof. The proof is analogous to the proof of theorem 3.10. In the case of G p,q , the complexified Lie algebra of K p,q is the Lie algebra l p whose roots with respect to the standard Cartan subalgebra of g are (see [10, Appendix C])

{±2ǫ l , ±(ǫ j ± ǫ k ) : 1 ≤ l ≤ p, 1 ≤ j = k ≤ p} ⊔{±2ǫ l , ±(ǫ j ± ǫ k ) : p + 1 ≤ l ≤ p + q, p + 1 ≤ j = k ≤ p + q}.
The module V should be a (g, l p )-module of finite type. Using an analogue of proposition 2.5, we see that V or its contragredient should be a highest weight module. The simple infinite dimensional highest weight module of degree 1 have been classified by Benkart, Britten, Lemire [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF]Proposition 3.6]. They are only two: their highest weights are -1 2 ω n and ω n-1 -3 2 ω n respectively. It is then easy to check that the possible modules are not (g, l p )-module of finite type.

The case of G n is analogous. The complexified Lie algebra of K n is the Lie algebra l whose roots are {±(ǫ j ± ǫ k ) : 1 ≤ j = k ≤ n -1}.

Once again, V or its contragredient should be isomorphic to a highest weight module. It is then well-known that these two highest weight modules correspond to the even and odd part of the metaplectic representation (see e.g. [START_REF] Li | Minimal representations & reductive dual pairs[END_REF]).

Proposition 2 . 4 (

 24  Proposition 3.4]). Up to isomorphism, the only highest weight g-module of degree 1 are the modules with highest weight aω 1 , aω i -(1 + a)ω i+1 , and aω n , for a ∈ C.

  H a . To prove the integrability of the representation H a we shall use a criterion of Jørgensen and Moore[START_REF] Jørgensen | Operator commutation relations[END_REF]. Let us recall it. Let H ′ be a Hilbert space. Let D be a dense subspace. Denote by • 0 the Hilbert norm. By A(D) we mean the set of all operators on D. Let A 0 = Id and let A 1 , . . . , A d be a basis for some (finite dimensional) Lie algebra included in A(D). We define inductively a norm • l on D by setting u l+1 := max{ A k u l , 0 ≤ k ≤ d}. Denote by D l the completion of D with respect to • l and by L j the space of continuous operators ofD j . Let L u (D ∞ ) := ∩{L j , j ≥ 0} and A ∞ (D) := A(D) ∩ L u (D ∞). This is the set of operators on D bounded for all the norms • l . Theorem 3.2 (Jørgensen-Moore). Let G be a connected simply-connected Lie group, whose corresponding Lie algebra is denoted g R . Let π 0 be a continuous representation of G on H ′ . Set L 0 := dπ 0 (g R ). Let D := C ∞ (π 0 ). Let S 0 be a set of Lie generator for L 0 . Let f : S 0 → A ∞ (D) be such that S := {A + f (A), A ∈ S 0 } generates a finite dimensional Lie algebra L. Then the representation L can be integrated into a continuous representation π of G such that dπ(g R ) = L.

Corollary 3 . 6 .Remark 3 . 7 .

 3637 Let a ∈ C \ Z ≥0 . Then the representation H a of g integrates into a continuous representation of G on the Hilbert space H. Let a ∈ Z ≥0 . Define a representation H a as above by restricting the index set of k to those k ∈ Z n ≥0 such that |k| ≤ a. Then H a is indeed a representation and is finite dimensional. Therefore it also integrates into a continuous representation of G on some Hilbert space.

Acknowledgment:

We are grateful to Professor Bent Ørsted for some very interesting and stimulating discussions of the subjects. The author also gratefully acknowledges support from Aarhus University, and the Max Planck Institut for Mathematics in Bonn.


The destruction of New York City, in whole or in part, established itself as a frequent theme of American cinema during the Cold War, although it predates that conflict. 1 It goes across genres and has appeared in films considering political questions, the impact of technology, the effect of capitalism, the nature of power, urban living and war but is especially prevalent in science fiction. There are solid reasons for this.

Science fiction, as a genre, is particularly suited to reflecting the political and social anxieties of its time especially as its main thematic focus often bears upon a grim future.

Worries about current problems (as well as hopes for a better life) can be projected forward and exaggerated to show what might happen if things develop (or do not develop) in a particular way. In this manner a warning can be given or, in more optimistic times, people can be encouraged to progress in a certain direction. Second, it has frequently been noted that stories about alien invasion often occur at a time when the nation feels threatened and echo attitudes towards those perceived as outsiders or enemies. Furthermore, one of the main themes of science fiction concerns the relationship between humans and technology and so can be used to express important tensions relative to this bond: for example, worries about the possible misuses of scientific knowledge. Cases of this abound: anxieties about the effects of radiation in many 1950s films; the increasing dependence on and the growing powers of computers in the Terminator films or fears of genetic manipulation in a number of recent movies. For these reasons, science fiction can provide especially illuminating insights into the society of its time. 2 This is particularly true of the Cold War. All of the themes mentioned earlier had a particular resonance at this period. Fears of communist assault from without or within and anxieties about the arms race, including the possibility of nuclear holocaust, all found a natural arena of expression in science fiction. And science fiction found one of its most frequent theatres in which to act out these anxieties in New York. Numerous films of the period showed that city either threatened with destruction, nearly destroyed, emptied of population or in ruins. 3 Numerous other cities in the world have faced cinematic ruin but New York leads in terms of sheer volume. There are a number of reasons for this rather morbid fascination. It is, of course, the largest and most famous city in the United States and its economic capital although not its political capital which clearly differentiates it from cities like Paris and London. Furthermore, the rise of New York to giant metropolis -and more particularly that of the island of Manhattan -occurred in the 19 th century and went against the major trend of American history at that period -the westward expansion. The rise to such dominance of a city that is not a nation's political capital is something of an anomaly. New York succeeded because it was a great port: the nation's products poured out of Manhattan to Europe while Europe's immigrants poured into New York. For this reason, New York became the American city. Furthermore, around that port, all the things necessary for trade grew up: insurance companies, stock exchanges, businesses and banks. New York became the seat of American economic power and may have been the first city in the world in which the temples to Mammon were larger and more striking than the temples to God. 4 The development of the theme of New York"s destruction, which appears at a surprisingly early date, is intimately linked to this fact. In his 1838 novel, Home As Found, James Fenimore Cooper described a major fire around Wall Street. 5 For him, the city was a latter day Sodom and Gomorrah and he took a certain glee in demolishing a predictable part of it:

"That Exchange, which had so lately resembled a bustling temple of Mammon, was already a dark and sheeted ruin, its marble walls being cracked, defaced, tottering or fallen." 6 Note how he focuses on the stock exchange which had already become an emblem of American economic power and, more particularly, of greed. To many at the time, speculation on Wall Street looked a lot like gambling and, was, therefore, rigorously condemned. Here, New York is shown as symbolic of the worst excesses of American capitalism which call forth from the author a desire to punish and allows the reader to enjoy its destruction. Ten years later, Edgar Allen Poe showed a New York ruined by an earthquake in his story "Mellonta Tauta", set in the year 2848. 7 The narrator says of the "aboriginal inhabitants":

It is related of them that they were oddly afflicted with monomania for building what was denominated "churches"a kind of pagoda instituted for the worship of two idols that went by the names of Wealth and Fashion. 8

With the earthquake, nature itself, it would seem, rebelled against this greedy, ostentatious and overly artificial city.

Which brings us to another Biblical parallel: the Tower of Babel. New Yorkers have dared to build too high and, as in the Bible, risk punishment for their arrogance. Henry James, a native of the town, was torn between wonder and horror when he returned to the new New York of skyscrapers which he describes in The American Scene (1907). On the one hand, he cannot help but marvel at "the most extravagant of cities, rejoicing, as with the voice of the morning, in its might, its fortune, its unsurpassable conditions". 9 On the other hand, he finds violence and destructiveness within the very architecture of the city. He compares a skyscraper next to Trinity Church to "the mountain-wall that drops the Alpine avalanche". 10

In another place, he describes the city as a "monstrous organism" and says of it:

One has the sense that the monster grows and grows, flinging abroad its loose limbs even as some unmannered young giant at his "larks", and that the binding stitches must for ever fly further and faster and draw harder; the future complexity of the web, all under the sky and over the sea, becoming thus that of some colossal set of clockworks, some steel-souled machine-room of brandished arms and hammering fists and opening and closing jaws. 11

New York is both monster and machine and one that keeps growing while in continual movement. It has "all the signs of the heaped industrial battlefield"yet another violent image -and shows "the universal will to moveto move, move, move as an end in itself, an appetite at any price". 12 Later, John Dos Passos named New York "the City of Destruction" and saw it as a place of personal and national testing. 13 In 1949, E.B. White wrote that "by rights New York should have destroyed itself long ago, from panic or fire or rioting or failure of some vital supply circuit". 14 Here, once again, we have the idea of destruction coming from the city itself.

There is, of course, a positive side to New York. It is the most famous example of the melting pot with its ethnic neighbourhoods and represents a kind of microcosm of America itself. The presence of the nation's most cherished symbol of freedom, the Statue of Liberty, with its famous inscription, also adds to New York's unique power -a symbol made perhaps even more powerful by the fact that it, and the city in which it lives, are both separated from the every day tumble of national politics. Filmmakers have been attracted for decades to its perpetual movement, its endless variety, modern architecture, economic power and its instant recognition. It is both quintessentially American and yet strangely alien to much of the population of that country. 15 It is not surprising, therefore, to see that many of America"s hopes and anxieties were given expression in and around New York.

The city has become the most popular setting for American films and, as such, many of its landmarks are instantly recognisable to people all over the world. For this reason, filmmakers often use a form of shorthand by showing a familiar landmark destroyed (or threatened with destruction) as a way of commenting on the United States. Themes that existed in literature soon appeared in cinema as well. 16 The easily recognisable skyline of New York quickly became a staple shot. King Kong (Merian C. Cooper and Ernest B. Schoedsack, 1933) pioneered use of the city in American films. It is a tale of two islands, the intensely modern Manhattan (whose skyscrapers resemble, bizarrely enough, a mountain range) and the mysterious unknown island where prehistoric creatures still live, and of two monsters: the city and Kong, the giant ape worshipped by the inhabitants of the primitive island. Humans triumph over the beast for Kong is captured and taken to Manhattan. In a wonderful parody of New York"s love affair with capitalism, he is displayed to the public for money but, of course, escapes and begins to destroy those very same wealthy members of society who paid to see him. Kong, of course, did not survive his encounter with Manhattan although, rather than being defeated by modern weapons, in the end, he simply lets go of the Empire State Building (a recently constructed symbol of high technology) and falls to his death. Significantly, King Kong was remade in 1976 by John Guillermin with a pronounced attack on the oil industry.

This time the giant ape took refuge on the World Trade Center at the end.

Numerous other films followed King Kong and the skyline and skyscrapers of Manhattan became familiar to moviegoers. As a very visible symbol of American capitalism, their part in Cold War cinema was guaranteed, and the threat of communism to the American system could be easily dramatized by showing their destruction. 17 Susan Sontag was one of the first to attract scholarly attention to 1950s science fiction film in her essay "The Imagination of

Disaster" where she talked about "the peculiar beauties to be found in wreaking havoc". 18 Indeed, science fiction cinema often enjoys showing destruction. Partly this is cathartic but partly also it expresses a negative judgment on some aspect of American society. The fact that New York, more than any other place gets repeatedly attacked is a clear illustration of America"s love/hate relationship with that city and what it stands for. After the Soviet Union exploded the atomic bomb, a Protestant preacher took some glee in announcing: "Do you know the area that is marked out for the enemy"s first atomic bomb?... New York!" 19

It has been estimated that between 1948 and 1962 approximately 500 feature films and shorts fell into the broad category of science fiction. Never before or since has one genre developed at such a speed in so short a time. 20 Probably television played a role here as the proportion of younger people, especially adolescent males, among cinema goers increased. 21 However, the subsequent decline in science fiction films in the 1960s (paradoxically at the most exciting moment of the space race) suggests that more is involved here. A quick look at the history of the period bears this out. The twin inventions of the atomic bomb and missilesand particularly their possession by the Soviet Union meant that the continental United States, which had felt fairly protected by its relative geographic isolation before, now experienced a new vulnerability. Furthermore, a number of spy cases helped to stimulate a hysterical search for communists within America by the House Un-American Activities Committee (HUAC),

Joseph McCarthy and othersall of which increased this feeling of vulnerability. Added to this was the continuing spread of communism in Eastern Europe and China, the Korean War and the explosion of an atomic bomb by the USSR. All of these events received a great deal of publicity as did government efforts at civil defence: this was the age of "duck and cover" and atomic shelters.

This situation led to an intensification of the arms race. In the 1950s, nuclear testing was done above ground and often within the U.S., in particular in the desert regions of Nevada and Utah. It became more and more clear that, in spite of government denials, the areas around these testing sitesand thus the people who inhabited themwere being contaminated by fallout. 22 Because HUAC and Sen. McCarthy were searching films for evidence of disloyalty, science fiction cinema became one of the rare places where these anxieties could be expressed and the official discourse questioned. 23 Not infrequently, these films were set inside New York City. 24 Science fiction films of the 1950s set a pattern of destruction that would continue long afterward.

Fears related to atomic power appeared in Captive Women of 1952 (Stuart Gilmore), set in the year 3000, which showed three rival tribes fighting for power in the ruins of New York after the atomic horror. In Five (Arch Oboler, 1951) the hero and new Adam went so far as to tell his Eve that he hated New York and was "glad it"s dead". A lifeless and deserted Manhattan appeared again in The World, the Flesh and the Devil (Ranald MacDougall, 1959) where only three people have survived a nuclear holocaust, two men and a woman, and the two men stalk each other through the empty city as they fight for the woman. 1952), a rather bizarre film shot in seven days. It details the invasion of America by an unidentified enemy whose accents sound suspiciously Russian. The attacks occur in a number of places, notably New York, where skyscrapers get bombed. At the end of the film, however, the invasion is revealed to be an illusion induced by some form of mass hypnosis. It is, therefore, not too late to save the nation and the message here seems to be that all citizens must be prepared and vigilant against communism -very much in keeping with the

McCarthyite discourse of the period. However, perhaps because it was made so quickly, the message frequently gets a bit muddled. The hero, for example, is cynical and always on the look out for his own advantage. 25 The publicity for the film was as bizarre as the film itself.

The trailer gleefully announces "See New York disappear" while publicity shots showed beautiful women in bathing suits posing next to models of burnt out skyscrapers. The whole thing seemed to say that Armageddon would be fun especially for men. 26 The number of American science fiction films declined in the 1960s. To a large extent this was because the genre was failing to renew itself and people were growing tired of formulaic, repetitive movies. But, as many scholars have pointed out, current events also played a role.

After the Cuban missile crisis of 1962, worries about Armageddon declined, as did, after the Test Ban Treaty of 1963, anxieties about radiation. The policy of détente also reassured people and convinced them that nuclear war was unlikely. Much of the nation"s attention was now focused on the Vietnam War and on inner turmoil: the civil rights movements, student protests, women"s liberation, etc. These would all find expression in the science fiction of the time. The space race also became a national priority and inspired many stories. 29 Science fiction films exhibited a new social consciousness, which was very much in keeping with the spirit of the time (at least on the left). Present day fears were still projected into the future with often devastating effects but the nature of these anxieties was more varied. 30 Interestingly enough, while images of the skyscrapers continued, prominent use was made of another, even better known New York landmark, the Statue of Liberty.

Representative as it is of the United States itself, and more particularly, of its system of government and of its identity as a nation of immigrants, the statue could become a potent commentary. Lady Liberty had appeared in films before the Cold War (most notably in Hitchcock"s 1942 film Saboteur where the finale takes place there) but its obvious symbolism made it a favourite backdrop in the Cold War. Going a step further, showing its destruction could serve as a tool to make important observations about America and its values. Note how he refuses the idea of an exterior enemy destroying America for "we finally really did it." Here is a fundamental difference with earlier films which saw destruction coming from outside. Taylor gives us a highly critical viewpoint on the Cold War and the arms race.

In the four sequels to the film we learn more about the reasons for the obliteration of human society (although the origins of the original nuclear disaster are never clearly explained). We even witness the final destruction of Earth itself at the end of the second film. Ostensibly about space exploration, people continually enter space but do not reach any place but Earth.

Frozen in place, all travel is done through time. Central to all these movies is human destructiveness and nuclear apocalypse (which has either happened or is about to happen).

Beneath the Planet of the Apes (Ted Post, 1970) even goes so far as to parody the arms race and government propaganda about nuclear weapons by showing a future humanity that actually worships the bomb as God -in a now underground St Patrick"s cathedral. The films reflect a number of other anxieties in late 1960s and early 1970s America, notably with regard to racial conflict and the Vietnam War, as well as fears of totalitarianism and fascist takeover. 33 The Statue is used to similar effect in John Carpenter"s Escape from New York (1981) which shows a devastated Manhattan where the entire island had been turned into the nation's one maximum-security prison. As the introduction to the film explains:

A 50-foot containment wall is erected along the New Jersey shoreline, across the Harlem river, and down along the Brooklyn shoreline. The political aspect of the film is clear from its first scenes which show two prisoners trying to escape Manhattan in a boat. They are ordered back by the helicopter that patrols the waters around the island and when they do not immediately turn around, they are shot in cold blood.

The scene then switches to the Statue of Liberty where a sign reads "Liberty Island Security

Control". The commentary on a potential loss of liberty from an obsession with repressing crime seems obvious. Too much of a preoccupation with security, Carpenter's film seems to say, can lead to a loss of all freedom. The plot concerns Snake Blissken, who is forced to enter Manhattan to rescue the president whose plane has crashed there after a terrorist attack.

Blissken travels through the ruined streets of the city trying to find the gang leader who has kidnapped the president. No one in the film is really likeable: not the gang leader, not the president and especially not the officials who have forced Blissken to rescue him. At the end, Blissken, in a scene reminiscent of High Noon, walks away in disgust.

Interestingly enough, the very absence of a recognisable landmark can also provide a commentary. Soylent Green (Richard Fleischer, 1973) shows New York in the year 2022, victim of pollution, overpopulation, and corporate greed. The film obviously reflects anxieties related to urban decay but has an anti-capitalist subtext linked to its portrayal of the allpowerful Soylent Corporation that connects it to the Cold War. In this nightmare world, real food is practically non-existent and synthetic rations are provided by the Soylent Corporation, whose latest product is called Soylent green. At the end, the hero discovers that Soylent green is made from human corpsescapitalist exploitation has run so far out of control that it has become cannibalism. New York is clearly mentioned as the setting at the beginning of the film but none of the usual landmarks are visible. The absence of colour, landmarks and, indeed, of anything singular about the cityother than its greyness and overpopulationreinforces the point of the film.

Along with capitalism, the American system of government gets criticised in Soylent Green and in other science fiction films of the period. Pillsbury Doughboy marches through city streets bringing Armageddon with it while, in the second film, the Statue of Liberty comes to life to save the city. 1990"s Gremlins II (Joe Dante) presents the annoying little creatures as a comic threat to Manhattan (they even get to sing "New York, New York") while the tiny, adorable mogwai, Gizmo, plays Rambo.

During the Cold War, the destruction of New York City appeared repeatedly. In the 1950s the devastation of American cities (but especially Manhattan) by alien invaders or by giant atomically mutated animals represented fears of nuclear war or of nuclear testing. These films are often seen as allegories on the Cold War, 35 but a very real tension often exists in them: the military is needed to kill the monster but atomic experiments are usually responsible for the threat in the first place. In the 1960s and 1970s the tone of the films became increasingly ironic and there was often a commentary on social questions. In the 1980s fantasy and humour appeared more frequently although some films did present serious threats to New York or other American cities. The over-populated and ultra-modern island of Manhattan has appeared as a metaphor for humanity"s conquest of natureand its consequences. Its familiar landmarks, economic power and very particular beauty have made it a magnet to film directors. Its iconic status as the American city has also given it a special power and attraction. New York proudly proclaims itself as the capital of the world and in a certain sense this is true. The destruction of New York reflects America"s insecurities both about herself and about her place in the world. But there has also been a certain pleasure taken in this devastation; a judgment that is passed on the city and what it stands for -New York"s destruction is often seen as in some way deserved because of its sins.

Interestingly enough, the end of the Cold War did not bring a reprieve for Manhattan and, indeed, the number of destruction scenes actually increased. 36 By the end of the 1990s, films were showing familiar places exploding all over New York. 1995"s Independence Day the unconscious even of those who most hate everything it stands for. There seems to have been no particular goal to the September 11 th attacks except to hurt and to express hatred. It is certainly difficult to see how they could have forwarded any political goals that al Qaeda may have (and whether they have any is a subject of much debate). Or perhaps they simply wanted to tell America that her anxieties were real.

1 I would like to thank Arnaud Regnauld and Henri Zuber for their help and advice.

2 Numerous studies of science fiction cinema exist. To name but a few: Kuhn, ed, Alien Zone; Kuhn, ed, Alien Zone II; Sobchak, The Limits of Infinity; Seed, American Science Fiction and the Cold War and Brosnan, Future Tense; Bartholomew, "Science Fiction Films"; Peary, ed. Omni's Screen Flights/Screen Fantasies. There are many others that there is not the space to list here.

3 A number of films outside the science fiction genre also played with this theme, such as Fail Safe (Sidney Lumet, 1964). Others showed New York as a crime-ridden wreck, like Mean Streets (Martin Scorsese, 1973), Taxi Driver (Martin Scorsese, 1976) and The Warriors (Walter Hill, 1979). 1981"s Fort Apache, the Bronx (Daniel Petrie, 1981) showed a corrupt and racist police force battling even worse criminals and Nighthawks (Bruce Malmuth, 1981) concerned the arrival of Europe"s most famous terrorist in the city. 4 New York, of course, was also faulted for its seediness: bars, prostitution, etc. For histories of New York, see Lankevich, American Metropolis?; Moorhouse, Imperial City; Kaspi, ed. New York, 1940-1945 Weart,Nuclear Fear,194 24 As Sandars observes in Celluloid Skyline, 387-8: "How better to convey the end of the world than to show the destruction of its best-known place… In the nuclear era, the greatest symbol of civil society was inevitably the greatest possible target, whose annihilation would provide the most wrenching symbol of atomic nightmare."

25 For more information on the film and examples of publicity shots and posters see http://www.conelrad.com/features/invasionusa/index.html. Accessed 15/01/03 26 The trailer can be found at http://www.youtube.com/watch?v=cNZaMm5q_BA. Accessed 20/02/2009 27 See Doherty,Teenagers and Teenpics,152. 28 Ibid. 29 And even some that belong to the domain of fantasy like the television series I Dream of Jeannie. 30 See Dean, "Between 2001 and Star Wars". Fried"s The Russians Are Coming! shows the decline of patriotic displays at the end of the 1950s. The following decade saw increasing cynicism about American institutions and values caused by the revelation of racial inequality through the civil rights movement and disillusionment with the Vietnam War. Watergate, of course, further increased cynicism in the 1970s. 31 The inscription on the Statue, by Emma Lazarus, reads: "Give me your tired, your poor,/ Your huddled masses yearning to breathe free,/The wretched refuse of your teeming shore./Send these, the homeless, tempest-tossed to me./ I lift my lamp beside the golden door." 32 The script is available online at http://pota.goatley.com/scripts/pota_finalshoot2.pdf. Accessed 25/02/2009. Interestingly enough, only the first words "My God!" can be found in the shooting script. The rest seems to have been added during filming. 33 For more on this see Greene, Planet of the Apes. 34 The script can be found at http://www.imsdb.com/scripts/Escape-From-New-York.html Accessed 25/2/2009 35 See, for example, Biskind, Seeing is Believing or Sayre, Running Time. 36 Stephen Keane comments in Disaster Cinema, 101, that "in 1998 watching New York getting destroyed became standard fare."

"

  The Destruction of New York City: A Recurrent Nightmare of American Cold War Cinema" Lori Maguire ABSTRACT. This article examines the repeated appearance of scenes showing the partial or complete destruction of New York City in American cinema of the Cold War. While this theme goes across genres, it has them to the dominant fears and anxieties of each period. It concludes by taking a brief look at the continuation of the theme in the post-Cold War period.

  1979's Meteor has a small meteorite destroying part of the city. In keeping with the post-Watergate spirit of the times, the United States government is dishonest, denying the existence of secret weapons aimed at Russia which they have placed in space (and which are the only thing that can destroy the meteorite and save the earth) -although, to be fair, the Russians are doing the same thing. There is also yet another delusional, Cold War obsessed general. 1984"s C.H.U.D. (Douglas Cheek) shows the U.S. government illegally transporting toxic waste through subway tunnels. Exposure to this has transformed people into monsters who feed on the homeless. But, in general, in the 1980s, with the Cold War heating up after the Soviet invasion of Afghanistan and the election of Ronald Reagan, and the feeling of vulnerability created by the Iran hostage crisis, it became more and more common to show exterior threats to New York. Already in 1980, Superman II (Richard Donner and Richard Lester) showed space aliens destroying Times Square. 1982's Q-The Winged Serpent had the Aztec flying serpent god, Quexcoatl, nesting on the top of the Chrysler Building and nourishing himself on the blood of New Yorkers. As the decade progressed, though, parody appeared in films. The two Ghostbusters films (Ivan Reitman, 1984 and 1989) took a comic view of the apocalypsewhich, of course, threatened to occur in New York. In a spoof of giant monster films, a huge

(

  Roland Emmerich) imagined an alien invasion of the world, and, more particularly, of the United States. Although the Americans triumph in the end, the film shows the annihilation, one by one, of the greatest icons of American power and prestige. The pace of devastation continued over the next few years. In 1997's Deep Impact (Mimi Leder), the world is threatened by a comet which, among other disasters, causes a tidal wave to hit New York, destroying the Brooklyn Bridge and much of Manhattan. In 1998's Armageddon (Michael Bay), meteorites menace the earth and a destructive shower of them falls on Manhattan. In same year Godzilla returned to the big screen -this time attacking New York. The tide did not diminish in the following years. Whether consciously or not, Osama bin Laden and the terrorists of 11 September 2001 gave reality to one of the most frequent representations of American anxiety. If unconscious, the attacks show to what extent Hollywood has penetrated

  After wreaking much havoc, he is finally trapped on the rollercoaster on Coney Island and burnt to death. This may have been the first film about a giant animal mutated by atomic power but it was certainly not the last. The following year saw two much more famous additions: that particularly Japanese nightmare, Godzilla(Ishiro Honda, 1954) and Them! (Gordon Douglas, 1954) which showed giant ants, irradiated at the New Mexico atomic test site, attacking Los Angeles.Fears of invasion surfaced in a number of films, particularly in the early 1950s and burning skyscrapers made frequent appearances. One of the first was Invasion USA (Alfred E. Green,

Atomic fears can also be found in The Beast from 20,000 Fathoms (Eugene Lourie) of 1953. Heat from an atomic bomb test in the Arctic unfreezes a dinosaur which immediately heads for Manhattan where he surfaces around Wall Street.

  Earth vs. Flying Saucers (Fred Sears, 1956) while small town America got quietly invaded in Invasion of the Body Snatchers(Don Siegel, 1956).After this, there was a brief reprieve for New York City until the launching of Sputnik in 1956 provoked a new wave of destruction films. Sputnik may have panicked the American government but Hollywood saw it as an opportunity. It convinced movie makers to invest in movies that had a similar theme. According to some estimates, as many as 43 such projects were announced on the day after Sputnik"s lift-off.27 It was even once mockingly suggested that Sputnik "should be in for 10 per cent".28 1957 saw, among other things, Deadly Mantis, in which a giant killer preying mantis defrosts in the Arctic and heads to Manhattan and The Giant Claw in which a giant bird devastated the Big Apple. In 1958's Colossus of New York, a killer robot went on a rampage. Finally, Robot Attack USA (1960) ends with missile attacks on Manhattan.

Invasion USA launched a whole series of atomic disaster films and New York was far from being the only ravaged city. Byron Haskin"s 1953 version of The War of the Worlds, based on H.G. Wells"s novel, gives a fairly realistic vision of an invasion, achieved by the use of archival images from the Second World War. It takes place, however, in Los Angeles. Many of Washington D.C."s monuments were destroyed in

  The most famous example of this is undoubtedly in Franklin J. Schaffner"s Planet of the Apes (1968) loosely based on a 1963 novel by the French writer Pierre Boulle. The plot concerns four astronauts who, leaving earth in 1972, arrive on a planet where apes are the evolved and dominant species while humans are in a primitive stage of development before speech. Only at the end does the one remaining astronaut, George Taylor, enter the so-called "Forbidden Zone" to find the wreckage of the Statue of Liberty. To his shock and horror he realizes that he has not journeyed through space but through time and that, at some point, humanity destroyed itself. Traditionally, of course, the Statue of Liberty functions as the symbol of an ideal America: land of freedom and of refuge.

31

So the film is making an ironic comment on this. Taylor"s last words hammer home the point: Oh my God! I"m back! I"m home. All the time it was… We finally really did it. You maniacs! You blew it up! God damn you! God damn you all to hell! 32

  It completely surrounds Manhattan island. All bridges and waterways are mined. The United States Police Force, like an army, is encamped around the island. There are no guards inside the prison: only prisoners and the worlds they have made. The rules are simple. Once you go in, you don't come out. 34

  The American Scene, 74 10 Ibid., 83 11 Ibid., 75 12 Ibid., 8413 John Dos Passos, The Best Times, 86. See also O'Connell, Remarkable, Unspeakable New York, 135 14 E.B.White, Here is New York (New York: Harper, 1949) pp. 24-5 15 Mulvey and Simmons in New York: City as Text, 1 make the point that since New York is, in a sense, between Europe and America, facing towards both, it "can represent both a spiritual antithesis to the middle America that would repudiate it and an American epitome to the rest of the world that would embrace or repudiate it." Ric Burns wrote in the companion volume to the PBS history of New York: "To many foreigners its represents America; to many Americans, it represents all that is foreign." Burns and Sandars, New York, xiii.16 As far as I can tell, the first film to destroy New York was The Battle Cry of Peace(Wilfrid North, 1915), now lost. It details the city"s invasion by an unidentified enemy that look German. It was generally seen as Allied propaganda. See The New York Times" review, 7 August 1915. New York was attacked again in a forgotten pacifist film of 1933, Men Must Fight (Edgar Selwyn)17 Wollen, "Delirious Projections", 25 sees some parallels between the skyscraper, and film, which developed at the same time.18 Sontag, "The Imagination of Disaster" 213. See alsoMurphy, "Monster Movies". 19 Quoted in Whitfield, The Culture of the Cold War 77. Barry Keith Grant wrote: "We eagerly await the climactic tidal wave that will sweep over New York and its landmarks of western civilisation." ""Sensuous Elaboration"", 22.20 For more on this see Evans, Celluloid Mushroom Clouds, 75 and Michaels, The American Movies Reference Book.21 Although Barclay, in his study of Viewing Tastes of Adolescents in Cinema and Television, 18, found that the genre was not particularly appreciated by British adolescent boys with 39 per cent listing science fiction as their least favourite type of film.22 For more on this, see Evans. Also Perrine, Film and the Nuclear Age and Boyer, By the Bomb's Early Light.

; Muhlstein.

Manhattan to name but a few 5 Such a fire did, in fact, occur in 1835. Fraser, Wall Street, 43

6

James Fenimore Cooper, Home As Found, 108-9

7

The title means "things of the future" in Greek which already links it to science fiction.

8

Poe "Mellonta Tauta", 381 9 Henry James, 23 See, for example,