Cette thèse est une contribution à l’étude de problèmes de contrôle optimal dont le caractère non linéaire se traduit par la présence, dans les équations d’état, d’un terme bilinéaire relativement à l’état et au contrôle. Malgré les difficultés liées à la non linéarité, nous obtenons des propriétés spécifiques au cas bilinéaire. L’introduction générale constitue la première partie. La seconde partie est consacrée à l’étude des équations d’état ; ce sont des équations aux dérivées partielles d’évolution. Nous établissons des estimations a priori sur les solutions à partir des inégalités de Willett et Wong et nous démontrons que les équations d’états sont bien posées. Dans le cas où les contrôles subissent une contrainte liée aux états, ces estimations permettent de déduire l’existence de solutions dans le cadre des inclusions différentielles. Les troisième et quatrième parties de ce mémoire sont dévolues à la démonstration de l’existence de contrôles optimaux, puis à l’analyse de la sensibilité relative à une perturbation qui intervient de façon additive dans l’équation d’état. Le caractère bilinéaire permet de vérifier des conditions suffisantes d’optimalité du second ordre. Nous fournissons sur des exemples, une formule explicite des dérivées directionnelles de la fonction valeur optimale / This thesis is devoted to the analysis of nonlinear optimal control problems governed by an evolution state equation involving a term which is bilinear in state and control. The difficulties due to nonlinearity remain, but bilinearity adds a lot of structure to the control problem under consideration. In Section 2, by using Willet and Wong inequalities we establish a priori estimates for the solutions of the state equation. These estimates allow us to prove that the state equation is well posed in the sense of Hadamard. In the case of a feedback constraint on the control, the state equation becomes a differential inclusion. Under mild assumptions, such a differential inclusion is solvable. In Section 3, we prove the existence of solutions to the optimal control problem. Section 4 is devoted to the sensitivity analysis of the optimal control problem. We obtain a formula for the directional derivative of the optimal value function. This general formula is worked out in detail for particular examples

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-03-22T20:23:25Z No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-23T11:30:21Z (GMT) No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-23T11:30:21Z (GMT). No. of bitstreams: 2 Tese - Gilson do Nascimento Silva - 2017.pdf: 2015008 bytes, checksum: e0148664ca46221978f71731aeabfa36 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We consider Newton’s method for solving a generalized equation of the form f(x) + F(x) 3 0, where f : Ω → Y is continuously differentiable, X and Y are Banach spaces, Ω ⊆ X is open and F : X ⇒ Y has nonempty closed graph. Assuming strong regularity of the equation and that the starting point satisfies Kantorovich’s conditions, we show that the method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. In addition, a local convergence analysis of this method is presented. Moreover, using convex optimization techniques introduced by S. M. Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), we prove a robust convergence theorem for inexact Newton’s method for solving nonlinear inclusion problems in Banach space, i.e., when F(x) = −C and C is a closed convex set. Our analysis, which is based on Kantorovich’s majorant technique, enables us to obtain convergence results under Lipschitz, Smale’s and Nesterov-Nemirovskii’s self-concordant conditions. / N´os consideraremos o m´etodo de Newton para resolver uma equa¸c˜ao generalizada da forma f(x) + F(x) 3 0, onde f : Ω → Y ´e continuamente diferenci´avel, X e Y s˜ao espa¸cos de Banach, Ω ⊆ X ´e aberto e F : X ⇒ Y tem gr´afico fechado n˜ao-vazio. Supondo regularidade forte da equa¸c˜ao e que o ponto inicial satisfaz as hip´oteses de Kantorovich, mostraremos que o m´etodo ´e quadraticamente convergente para uma solu¸c˜ao, a qual ´e ´unica em uma vizinhan¸ca do ponto inicial. Uma an´alise de convergˆencia local deste m´etodo tamb´em ´e apresentada. Al´em disso, usando t´ecnicas de otimiza¸c˜ao convexa introduzida por S. M. Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), provaremos um robusto teorema de convergˆencia para o m´etodo de Newton inexato para resolver problemas de inclus˜ao n˜ao–linear em espa¸cos de Banach, i.e., quando F(x) = −C e C ´e um conjunto convexo fechado. Nossa an´alise, a qual ´e baseada na t´ecnica majorante de Kantorovich, nos permite obter resultados de convergˆencia sob as condi¸c˜oes Lipschitz, Smale e Nesterov-Nemirovskii auto-concordante.