Résumé
This work is intended to introduce and to study new gradient-like dynamical systems.<br /> The dissipative aspects of those kind of dynamics stand at the crossroads<br /> of many field in Analysis : Optimization, Mechanics, PDE.<br /><br />A first part of the work is devoted to the construction of mappings that are able<br /> to control gradient (or subdifferntial vector fields). They are called barrier-operators. <br />One of the main motivation is to derive interior descent methods. The abstract <br />framework that is proposed allow to recover many dynamics : projected<br /> gradient, Riemannian methods, continuous Newton method, Lotka-Volterra based<br /> dynamics ... Within such a setting, we may evoke several results concerning strong<br /> viability, well-posedness, and global convergence.<br />Keeping in mind the fact that "good" trajectories are those that remain <br />in the feasible set : it is natural to pay a particular attention to hessian Riemannian<br /> structure induced by Legendre functions. Those Riemannian manifolds enjoys<br /> many properties, and may be characterized as "the metrics that are the more<br /> appropriate to solve, a certain class of variational inequalities" . Another<br /> interesting aspects, is that those type of structure has a sense in Hilbert Spaces :<br /> it corresponds to some well-known subdifferntial formulation of some<br /> parabolic equations arising in Thermodynamic.<br /><br />The second part oif the thesis is devoted to the study of second-order in time<br /> gradient method. It is shown that the use of Hessian-driven damping yields a <br />nice class of dynamical systems. <br />A first interest of those methods, is to give rise to non-descent methods with<br /> convergent trajectories. Indeed, if one follows some minimization purposes,<br /> it may be interesting to avoid local minima in order to attain a global minimum.