Analyse-Algèbre 4

DEUGTravaux dirigés d'algèbre linéaire, fonctions de plusieurs variables et intégrales multiples. Reprises et complétées à partir de celles fournies par mon collègue P. Gourdel

Semi-periodic solutions of difference and differential equations

International audienceThe spaces of semi-periodic sequences and functions are examined in the relationship to the closely related notions of almost-periodicity, quasi-periodicity and periodicity. Besides the main theorems, several illustrative examples of this type are supplied. As an application, the existence and uniqueness results are formulated for semi-periodic solutions of quasi-linear difference and differential equations

Existence of Almost Periodic Solutions of Discrete Time Equations

International audienceIn this paper, we study almost periodic (a.p.) solutions of discrete dynamical systems. We first adapt some results on a.p. differential equations to a.p. difference equations, on the link between boundedness of solutions and existence of a.p. solutions. After, we obtain an existence result in the space of the Harmonic Synthesis for an equation $A_t (x_t,...,x_{t+p})=0$ when the dependance of $A$ on $t$ is a.p. and when $A_t$ and $D A_t$ are uniformly Lipschitz and satisfy another condition which is exactly the extension of a simple one for the basic linear system. The main tools for that are Nonlinear Functional Analysis and the Newton method

Notion of Weak Variational Solutions for Almost Periodic or More General Problems

International audienceThe aim of this paper is to introduce of notion of weak variational solution in an abstract setting, although we are mainly interested in almost periodic type solutions. We give two existence and uniqueness theorems. Even if assumptions are strong, we obtain two theorems with explict bounds

Existence of different kind of solutions for discrete time equations

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On some general almost periodic Optimal Control problems: links with periodic problems and necessary conditions

In this paper, we are concerned with periodic, quasi-periodic (q.p.) and almost periodic (a.p.) Optimal Control problems. After defining these problems and setting them in an abstract setting by using Abstract Harmonic Analysis, we give some structure results of the set of solutions, and study the relations between periodic and a.p. problems. We prove for instance that for an autonomous concave problem, the a.p. problem has a solution if and only if all problems (periodic with fixed or variable period, q.p. or a.p.) have a constant solution. After that, we give some first order necessary conditions (weak Pontryagin) in the space of Harmonic Synthesis and we also give in this space an existence result

Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations

International audienceWe build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequat Partial Differential Equation are related to the q.p. solutions of an Ordinary Differential Equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which insures that any essentially bounded weak solution is a strong one

Contrôle optimal et oscillations

Dans cette thèse, nous nous sommes intéressés au problème du Contrôle Optimal des oscillations et à quelques applications économiques. Tout d'abord, on a mis en forme le formalisme de Percival liant une fonction quasi-périodique (=q.p.) à sa génératrice sur le tore, avec un lien se comportant bien au niveau différentiel. Une étude du cas à paramètre, nécessaire pour traiter des équations autonomes, permet d'obtenir un théorème d'isomorphisme entre l'espace des fonctions presque- périodiques ( =p. p. ) à paramètres et un espace de fonctions p. p. à valeurs dans un Banach. Relativement au formalisme de Percival, nous introduisons des espaces du type de Sobolev où nous retrouvons l'absence de compacité, mais où l'étude ne nécessite pas d'hypothèses diophantiennes contrairement aux considérations plus géométriques de ces problèmes. Le formalisme de Percival transforme la recherche des solutions q.p. d'une équation différentielle en la recherche de solutions p.p. d'une E.D.P. Une notion de solution faible est introduite et comparée à l'usuelle, et l'on obtient pour celle-ci un théorème d'existence. L'absence de compacité implique une obligation de développer de nouvelles techniques, y compris dans les méthodes de régularisation, et l'on présente pour cela une nouvelle technique. Le cas discret est également traité (existence et structure), après avoir comparé différentes notions de suites p.p, courantes dans la littérature. On étudie alors les problèmes quasi-périodiques ; pour des problèmes autonomes linéaires-concaves, on démontre l'équivalence des problèmes p.p., périodiques et statiques. Dans le cadre non nécessairement autonome, on obtient des théorèmes d'existence et de conditions nécessaire étendant ceux de Da- Prato et Ichikawa. Enfin, des applications économiques sont fournies.PARIS1-BU Pierre Mendès-France (751132102) / SudocPARIS1-CNRS-Maison des sc. éco (751055202) / SudocSudocFranceF

Existence and Uniqueness of Pseudo Almost Automorphic Solutions to Some Classes of Partial Evolution Equations

International audienceIn this paper, we study pseudo almost automorphic solutions to perturbations to Eq.\;(\ref{1}) consisting of the class of abstract partial evolution equations of the form \begin{eqnarray}\label{2} \displaystyle{\frac{d}{dt}} \left[u(t) + f(t, u(t))\right] = A u(t) \; \; t \in \R, \end{eqnarray} where $A$ is the infinitesimal generator of an exponentially stable $c_0$-semigroup acting on \X, $B, C$ are two densely defined closed linear operators on \X, and $f$ is continuous functions. Under some appropriate assumptions, we establish the existence and uniqueness of an almost automorphic (mild) solution to Eq.\,(\ref{2}) using the Banach fixed-point principle

Pseudo almost automorphic solutions for hyperbolic semilinear evolution equations in intermediate Banach spaces

International audienceWe are concerned in this paper with the pseudo almost automorphy of mild solutions for the semilinear evolution equation $x'(t)=Ax(t)+f(t,x)$ where $A$ is a sectorial operator not necessarily densely defined in $X$ generating an hyperbolic semigroup $(T(t))_{t\geq 0}$ in a Banach space $X$ and $f:\R\times X_\alpha \to X$, where $X_{\alpha}$ is an intermediate space. We prove the existence and uniqueness of a pseudo almost automorphic solution in $X_\alpha$, when the function $f:\R\times X_\alpha\longrightarrow X$ is pseudo almost automorphic