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Version: publishedVersion × Subject: Sweeping process ×

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Now showing 1 - 3 of 3
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    Hausdorff metric BV discontinuity of sweeping processes
    (Bristol : IOP Publ., 2016) Klein, Olaf; Recupero, Vincenzo
    Sweeping processes are a class of evolution differential inclusions arising in elastoplasticity and were introduced by J.J. Moreau in the early seventies. The solution operator of the sweeping processes represents a relevant example of rate independent operator. As a particular case we get the so called play operator, which is a typical example of a hysteresis operator. The continuity properties of these operators were studied in several works. In this note we address the continuity with respect to the strict metric in the space of functions of bounded variation with values in the metric space of closed convex subsets of a Hilbert space. We provide counterexamples showing that for all BV-formulations of the sweeping process the corresponding solution operator is not continuous when its domain is endowed with the strict topology of BV and its codomain is endowed with the L1-topology. This is at variance with the play operator which has a BV-extension that is continuous in this case.
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    Controlled polyhedral sweeping processes: Existence, stability, and optimality conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Henrion, René; Jourani, Abderrahim; Mordukhovich, Boris S.
    This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.
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    Optimal control of the sweeping process
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Colombo, Giovanni; Henrion, René; Hoang, Nguyen D.; Mordukhovich, Borils S.
    We formulate and study an optimal control problem for the sweeping (Moreau) process, where control functions enter the moving sweeping set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and generalized differentiation. The final results obtained are given in terms of the initial data of the controlled sweeping process and are illustrated by nontrivial examples.