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Nonsmooth dynamical systems : Application in optimization and sweeping processes

Abstract : In this dissertation, we study some classes of nonsmooth dynamical systems. Namely, perturbed sweeping processes, sweeping processes with velocity constraints, and vibro-impact problems are investigated. The first topic is on the solution existence and uniqueness of nonconvex perturbed sweeping processes. In the setting adopted by Edmond and Thibault [Mathematical Programming 104 (2005), 347--373], we study a class of perturbed sweeping processes. Under suitable assumptions, we obtain two solution existence theorems for perturbed sweeping processes with the constraint sets being prox-regular sublevel sets. The results are applied to analyzing the behavior of some concrete mechanical sweeping processes. The second topic is on some classes of sweeping processes with velocity in a moving set. In addition to the solution existence and the solution uniqueness for the case of a moving convex constraint set, some results on the solution existence and the solution multiplicity where the constraint set is a finite union of disjoint convex sets are also obtained. Our main tool is a theorem on the solution sensitivity of parametric variational inequalities. Beside the traditional requirement that the constraint set moves continuously in the Hausdorff distance sense, we intensively use a new assumption on the local Lipschitz-likeness of the constraint set-valued mapping. The obtained results are compared with the existing ones and analyzed by several examples. In addition, some properties of solutions of convex sweeping processes with velocity constraints are also studied. Namely, the solution sensitivity with respect to the initial value, the boundedness, the closedness, and the convexity of the solution set are discussed in detail. The third topic is on a vibro-impact problem, which is described in the form of second-order measure differential inclusion. We are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions which is proved to converge to a solution of the problem in consideration.
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Submitted on : Thursday, March 10, 2022 - 12:01:27 PM
Last modification on : Thursday, September 8, 2022 - 4:01:05 AM

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Nang Thieu Nguyen. Nonsmooth dynamical systems : Application in optimization and sweeping processes. Optimization and Control [math.OC]. Université de Limoges, 2021. English. ⟨NNT : 2021LIMO0110⟩. ⟨tel-03604211⟩

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