Optimisation en dimension infinieNous allons \303\251tendre les id\303\251es vues au pr\303\251c\303\251dent TP \303\240 des probl\303\250mes d\303\251finis en dimension infinie, par exemple \303\240 des probl\303\250mes o\303\271 les inconnues sont des fonctions ou des courbes. L'id\303\251e principale est toujours la m\303\252me : si une fonction f poss\303\250de un extremum en x, alors sa d\303\251riv\303\251e est nulle en x. En dimension infinie, l'outil qui g\303\251n\303\251ralise la d\303\251riv\303\251e est la diff\303\251rentielle. Le r\303\251sultat reste le m\303\252me : si une application F est extremale en X, alors sa diff\303\251rentielle dF(X) est l'application nulle.Plus court chemin entre deux pointsOn consid\303\250re deux points A et B du plan. On cherche \303\240 d\303\251montrer que le plus court chemin allant de A \303\240 B est le segment de droite reliant ces deux points. Il existe des m\303\251thodes assez simples pour d\303\251montrer ce r\303\251sultat. Mais nous allons montrer comment le concept de diff\303\251rentielle permet de le traiter, cette m\303\251thode \303\251tant dans beaucoup de probl\303\250mes compliqu\303\251s la seule possible.On consid\303\250re toutes les courbes reliant A \303\240 B. On associe \303\240 chacune sa longueur. Notre probl\303\250me est alors un probl\303\250me d'optimisation : on cherche la courbe minimisant la longueur.Il faut commencer par d\303\251finir l'application qui \303\240 une courbe associe sa longueur. On \303\251tudie ensutie cette application, on d\303\251termine sa diff\303\251rentielle et on cherche la ou les courbes pour lesquelles cette diff\303\251rentielle est l'application nulle. Nous notons xA, xB, yA et yB les abscisses et ordonn\303\251es des points A et B avec xB>xA. On d\303\251finira une courbe reliant A \303\240 B par une fonction x->y(x) telle que y(xA)=yA et y(xB)=yB.La longueur associ\303\251e \303\240 cette courbe est donn\303\251e par l'int\303\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D\303\251finissons l'application qui \303\240 une fonction y associe la longueur de sa courbe.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Regardons des exemples de courbes reliant le point (0,0) au point (1,1) et leurs longueurs (valeurs num\303\251riques).(Remarque : la derni\303\250re op\303\251ration sert \303\240 effacer les valeurs attribu\303\251es \303\240 xA et xB.) 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2MFEhRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSy9GNlEnbm9ybWFsRic=Cherchons maintenant la diff\303\251rentielle de l'application L. Cette application permet de d\303\251crire (au premier ordre) comment la longueur d'une courbe varie lorsqu'on fait un peu varier la courbe (on parle de calcul variationnel). Elle est d\303\251finie de la mani\303\250re suivante : c'est l'application dL telle qu'au premier ordre, on puisse \303\251crire pour toute fonction y et toute fonction perturbatrice \316\267 :L(y+\316\267)=L(y)+dL(y)(\316\267) + o(||\316\267||)Autrement dit, c'est la partie lin\303\251aire du "d\303\251veloppement limit\303\251" de L au voisinage de y.On notera au passage que ce d\303\251veloppement limit\303\251 n\303\251cessite d'avoir d\303\251fini une norme sur l'espace des fonctions consid\303\251r\303\251es ici. Nous ne le ferons pas ici afin de nous concentrer sur le fonctionnement de la m\303\251thode, mais la d\303\251finition d'une telle norme est en fait cruciale. Il s'agit de choisir une norme pour laquelle tout ce que nous \303\251crirons dans la suite sera valable. En particulier, on verra qu'une telle norme doit permettre de contr\303\264ler le fait que nos perturbations \316\267 sont "petites" mais aussi leurs d\303\251riv\303\251es \316\267'.Afin que y+\316\267 soit encore une fonction reliant A \303\240 B, on impose \316\267(xA)=\316\267(xB)=0. Si la fonction y est notre solution, i.e. si elle minimise la longueur, alors on doit avoir pour toute perturbation \316\267, L(y+\316\267)\342\211\245L(y). Cela impose que dL(y) ne peut pas changer de signe avec \316\267 et cela n'est possible que si dL(y) est l'application nulle : pour toute fonction \316\267, dl(y)(\316\267)=0.Etudions la variation L(y+\316\267)-L(y) afin de d\303\251terminer dL :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JlEiTEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNigtRiw2JlEieUYnRi9GMkY1LUkjbW9HRiQ2LlEiK0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlctRiw2JlEkZXRhRidGLy9GM0ZIRkRGLy8lK2V4ZWN1dGFibGVHRkhGREYvRkQtRkE2LlEoJm1pbnVzO0YnRi9GREZGRklGS0ZNRk9GUUZTRlVGWEYrLUY5NiUtRiM2JkY9Ri9GaG5GREYvRkRGL0ZobkZENous pouvons r\303\251unir ces deux int\303\251grales et utiliser un d\303\251veloppement limit\303\251 \303\240 l'int\303\251rieur de l'int\303\251grale. En consid\303\251rant que y'(x) et \316\267'(x) sont de simples nombres a et \316\265, la diff\303\251rence des racines carr\303\251es sous l'int\303\251grale est donn\303\251e par : 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 rempla\303\247ant a par y'(x) et \316\265 par \316\267'(x), utiliser ce d\303\251veloppement limit\303\251 dans l'int\303\251grale pour obtenir l'expression de L(y+\316\267)-L(y) \303\240 l'ordre 1 :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'est ici que nous voyons que pour \303\251crire cela rigoureusement, il faut que la norme d\303\251finie sur notre espace nous permette de contr\303\264ler \303\240 la fois \316\267 mais aussi \316\267'.L'application qui \303\240 une fonction \316\267 associe cette int\303\251grale est lin\303\251aire ! Il s'agit de la diff\303\251rentielle dL de L en la fonction y.Une remarque avant la suite : il est possible de le faire automatiquement \303\240 l'aide de la commande subs :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Pour d\303\251terminer la courbe y qui minimise la longueur, on cherche les fonctions y pour lesquelles cette diff\303\251rentielle est l'application nulle.A cause de la pr\303\251sence de \316\267', il est difficile de d\303\251terminer avec l'expression ci-dessus pour quelles fonctions y la diff\303\251rentielle est l'application nulle. Pour supprimer \316\267' et faire apparaitre \316\267, on peut simplement faire une int\303\251gration par partie.
On r\303\251\303\251crit notre int\303\251grale sous forme symbolique (avec Int au lieu de int) puis on utilise la commande intparts du package students pour effectuer l'ipp et faire appara\303\256tre \316\267.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Or on rappelle que \316\267(xA)=\316\267(xB)=0. Les termes \303\240 gauche sont donc nuls et on en d\303\251duit que la diff\303\251rentielle de l'application L en y est l'application lin\303\251aire qui \303\240 une fonction \316\267 associe l'int\303\251grale ci-dessus.Il est alors possible de d\303\251terminer une condition simple pour que cette application soit nulle pour toute fonction \316\267.Cette condition \303\251tant une \303\251quation diff\303\251rentielle satisfaite par y, on peut la r\303\251soudre avec dsolve. Le faire et conclure. Ne pas oublier de pr\303\251ciser dans dsolve les conditions en xA et xB.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\303\250me brachistochrone (suite)On consid\303\250re de nouveau le probl\303\250me suivant : une bille roule sur une rampe (c'est-\303\240-dire une courbe) reliant le point A=(0,0) au point B= (1,-1). Le probl\303\250me brachistochrone consiste \303\240 trouver la forme de la rampe pour laquelle le trajet de la bille est le plus rapide.Dans la premi\303\250re partie, nous avions d\303\251compos\303\251 la courbe recherch\303\251e en segments de droites et avions cherch\303\251 les points interm\303\251diaires pour lesquels le temps de parcours \303\251tait minimal.Nous allons maintenant essayer de r\303\251soudre le probl\303\250me g\303\251n\303\251ral en cherchant, parmi toutes les courbes lisses reliant A \303\240 B, celle qui minimise le temps de parcours. Nous nous restreindrons tout de m\303\252me aux courbes d\303\251finies par une fonction de classe C2, de la forme (x,y(x)), x variant de 0 \303\240 1.Les lois de la physique permettent de calculer le temps de parcours de la bille roulant sur une telle courbe avec une vitesse initiale nulle : T=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On d\303\251finit l'application T qui \303\240 une courbe y associe le temps de parcours. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2MFEhRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSy8lK2V4ZWN1dGFibGVHRjwvRjZRJ25vcm1hbEYnD\303\251finir quelques exemples de fonctions d\303\251finissant une courbe reliant A \303\240 B et calculer pour chacune le temps de parcours. Voil\303\240 d\303\251j\303\240 un exemple : le segment de droite.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJURidGL0YyLUY2NiQtRiM2JS1GLDYlUSN5MUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZHRkRGR0ZHRkRGRw==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\303\251terminer la diff\303\251rentielle de T. On commencera par faire varier la courbe y avec une perturbation \316\267 et on \303\251tudiera la variation de T qui en r\303\251sulte. Afin de l'\303\251tudier, on fera un d\303\251veloppement limit\303\251 de la quantit\303\251 int\303\251gr\303\251. Attention, il y a ici 4 fonctions qui interviennent dans l'int\303\251grale : y, y', \316\267 et \316\267'. Il faudra faire un DL en deux variables en \316\267=\316\267'=0.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 de faire dispara\303\256tre les termes \316\267'(x), on effectue une int\303\251gration par partie pour certains termes de l'int\303\251grale.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEhRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEhRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEhRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSw==Une fois la diff\303\251rentielle obtenue, on cherche pour quelles fonctions y cette diff\303\251rentielle est l'application nulle. cela revient \303\240 demander \303\240 Maple de r\303\251soudre une \303\251quation diff\303\251rentielle.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La solution fournie par Maple l'est sous forme implicite. Deux constantes apparaissent. Elles d\303\251pendent des conditions au bord : y(0)=0 et y(1)=-1. On peut essayer de r\303\251soudre ces \303\251quations pour les trouver, mais il y a une difficult\303\251 en 0 o\303\271 la solution y n'est en fait pas bien d\303\251finie. Il faut tricher un peu et r\303\251soudre par exemple y(0)=-0.00001. On obtient approximativement :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Il est maintenant possible de repr\303\251senter la solution obtenue d\303\251finie implicitement par une \303\251quation. On utilise pour cela la commande implicitplot.JSFHJSFHJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEhRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEhRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSw==