Differential inclusions involving normal cones of nonregular sets in Hilbert spaces

Abstract : This thesis is dedicated to the study of differential inclusions involving normal cones of nonregular sets in Hilbert spaces. In particular, we are interested in the sweeping process and its variants. The sweeping process is a constrained differential inclusion involving normal cones which appears naturally in several applications such as elastoplasticity, electrical circuits, hysteresis, crowd motion, etc. This work is divided conceptually in three parts: Study of positively alpha-far sets, existence results for differential inclusions involving normal cones and characterizations of Lyapunov pairs for the sweeping process. In the first part (Chapter 2), we investigate the class of positively alpha-far sets. This class of nonregular sets is very general and includes convex, uniformly prox-regular and uniformly subsmooth sets, among others. It turns out that this class is the best suited to the study of differential inclusions involving normal cones. In the second part (Chapter 3 to the first part of Chapter 8), we provide several existence results for the sweeping process and its variants. In order to do that, we consider three approaches: The Catching-up algorithm, the Galerkin-like method and the Moreau-Yosida regularization. The first method is the most classic in the study of differential inclusions involving normal cones. We used it in the case where the set considered is fixed. The second method (Galerkin-like) consists in approximating the original problem by projecting the state into a finite-dimensional Hilbert space, but not the velocity. Approximate problems always have a solution and, under some compactness conditions, we prove that they converge strongly pointwisely (up to a subsequence) to a solution of the original differential inclusion. Moreover, it is shown that this method is well adapted to deal with differential inclusions involving normal cones, by providing general existence results for the generalized sweeping process. As a result, existence of solutions for the first order and second order sweeping process is obtained. Furthermore, this method is used to show existence of solutions of the perturbed sweeping process with nonlocal initial conditions. The third method is the Moreau-Yosida regularization technique which consists in approximating a given differential inclusion by a penalized one, depending on a positive parameter and then to pass to the limit when the parameter goes to zero. This method is used to deal with state-dependent sweeping processes governed by uniformly subsmooth sets. Finally, in the third part (Second part of Chapter 8 and Chapter 9), we give some characterizations of weak Lyapunov pairs and weak invariance for the perturbed sweeping process with uniformly subsmooth sets.
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Emilio Vilches. Differential inclusions involving normal cones of nonregular sets in Hilbert spaces. Optimization and Control [math.OC]. Université de Bourgogne - Franche Comté; Universidad de Chile, 2017. English. ⟨tel-01562596⟩

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