The convergence problem for dissipative autonomous systems: classical methods and recent advances

The initial motivation of this text was to provide an up to date translation of the monograph [45] written in french by the first author, taking account of more recent developments of infinite dimensional dynamics based on the {\L}ojasiewicz gradient inequality. In order to keep the present work within modest size bounds and to make it available to the readers without too much delay, we decided to make a first volume entirely dedicated to the so-called convergence problem for autonomous systems of dissipative type. We hope that this volume will help the interested reader to make the connection between the rather simple background developed in the french monograph and the rather technical specialized literature on the convergence problem which grew up rather fast in the recent years

On damped second-order gradient systems

Using small deformations of the total energy, as introduced in \cite{MR1616968}, we establish that damped second order gradient systems \begin{gather*} u^\pp(t)+\gamma u^\p(t)+\nabla G(u(t))=0, \end{gather*} may be viewed as quasi-gradient systems. In order to study the asymptotic behavior of these systems, we prove that any (nontrivial) desingularizing function appearing in KL inequality satisfies \vphi(s)\ge c\sqrt s whenever the original function is definable and $C^2.$ Variants to this result are given. These facts are used in turn to prove that a desingularizing function of the potential $G$ also desingularizes the total energy and its deformed versions. Our approach brings forward several results interesting for their own sake: we provide an asymptotic alternative for quasi-gradient systems, either a trajectory converges, or its norm tends to infinity. The convergence rates are also analyzed by an original method based on a one-dimensional worst-case gradient system. We conclude by establishing the convergence of solutions of damped second order systems in various cases including the definable case. The real-analytic case is recovered and some results concerning convex functions are also derived

A Liapunov function approach to the stabilization of second order coupled systems

In 2002, Fatiha Alabau, Piermarco Cannarsa and Vilmos Komornik investigated the extent of asymptotic stability of the null solution for weakly coupled partially damped equations of the second order in time. The main point is that the damping operator acts only on the first component and, whenever it is bounded, the coupling is not strong enough to produce an exponential decay in the energy space associated to the conservative part of the system. As a consequence, for initial data in the energy space, the rate of decay is not exponential. Due to the nature of the result it seems at first sight impossible to obtain the asymptotic stability result by the classical Liapunov method. Surprisingly enough, this turns out to be possible and we exhibit, under some compatibility conditions on the operators, an explicit class of Liapunov functions which allows to do 3 different things: 1) When the problem is reduced to a stable finite dimensional space, we recover the exponential decay by a single differential inequality and we estimate the logarithmic decrement of the solutions with worst (slowest) decay. The estimate is optimal at least for some values of the parameters

On the convergence to equilibria of a sequence defined by an implicite scheme

We present numerical implicit scheme based on a geometric approach to the study of the convergence of solutions of gradient-like systems given in [2]. Depending on the globality of the induced metric, we can prove the convergence of these algorithms. Dedicated to the memory of Ezzeddine ZAHROUNI 1. Notation For a riemannian manifold (M, g) of dimension N we denote ·, · g the scalar product dened on each tangent space. The induced norm is denoted · g (or · when there is no risk of confusion) For a local system of coordinates on M , g ij will denote the coecient of the matrix dening the scalar product above. Let us recall that a C 1 curve x : [0, 1] → M is called a geodesic between x(0) and x(1) i it is a critical point of the functional L(γ) = 1 0 ||γ (t)|| g dt restricted to the C 1-curves γ : [0, 1] → M such that γ(0) = x(0) and γ(1) = x(1). For a dierentiable function f : M → R and p ∈ M we denote ∇ g f (p) the unique element of the tangent space T p M to M at p such that ∀u ∈ T p M, ∇ g f (p), u g = df (p).u 2. A implicit numerical scheme and main result of the paper Let us consider (M, g) a complete connected non compact riemaniann manifold and E a smooth real function. Associated to E, it is quite natural to consider the following gradient system (1)Ẋ(t) + ∇ g E(X(t)) = 0. In the paper [11] the authors Merlet & Pierre consider the situation when (M, g) is the standard R N with its natural euclidian structure and prove the convergence of a sequence dened by an implicit scheme associated to (1). It is quite natural to extend the scheme there introduced to the case of more general manifolds. Such insights were initially considered in [12] provided (M, g) is a submanifold of R N. However the specic case of the backward Euler scheme was not considered in this paper under the intrinsic point of view, i.e. the backward scheme is constructed ex post in [12], considering the embedded situation. Here we try to focus on the The rst author wishes to thanks the organizers of ICAAM 2019 in Hammamet, Tunisia, where this work was initiated. The second author wishes to thanks CNAM, France where this work was partially completed

Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

International audienceIn the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p

Méthodes de résolution d'équations algébriques et d'évolution en dimension finie et infinie

Dans la présente thèse, on s intéresse à la résolution de problèmes algébriques et d évolution en dimension finie et infinie. Dans le premier chapitre, on a étudié l existence globale et la régularité maximale d un système gradient abstrait avec des applications à des problèmes de diffusion non-linéaires et à une équation de la chaleur avec des coefficients non-locaux. La méthode utilisée est la méthode d approximation de Galerkin. Dans le deuxième chapitre, on a étudié l existence locale, l unicité et la régularité maximale des solutions de l équation de raccourcissement des courbes en utilisant le théorème d inversion locale. Finalement, dans le dernier chapitre, on a résolu une équation algébrique entre deux espaces de Banach en utilisant la méthode de Newton continue avec une application à une équation différentielle avec des conditions aux limites périodiquesIn this work, we solve algebraic and evolution equations in finite and infinite-dimensional sapces. In the first chapter, we use the Galerkin method to study existence and maximal regularity of solutions of a gradient abstract system with applications to non-linear diffusion equations and to non-degenerate quasilinear parabolic equations with nonlocal coefficients. In the second chapter, we Study local existence, uniqueness and maximal regularity of solutions of the curve shortening flow equation by using the local inverse theorem. Finally, in the third chapter, we solve an algebraic equation between two Banach spaces by using the continuous Newton s method and we apply this result to solve a non-linear ordinary differential equation with periodic boundary conditions.METZ-SCD (574632105) / SudocSudocFranceF