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A continuity method for sweeping processes. (English) Zbl 1237.34116
The author considers the “sweeping process” in a Hilbert space \(H\), specifically, the problem \[ -y'(t)\in N_{C(t)}(y(t)),\quad y(0)= y_0\in C(0),\quad y(t)\in C(t),\quad t\in [0,T], \] where \(C: [0,T]\to 2^H\) and \(N_{C(t)}(x)\) the exterior normal cone to \(C(t)\) at \(y(t)\). The goal of the paper is to extend results in the literature in which the moving convex set \(C\) is Lipschitz continuous [see J. J. Moreau, Rafle par un convexe variable. I. Trav. Semin. d’Anal. convexe, Montpellier, Vol. 1, Expose No. 15, 43 p. (1971; Zbl 0343.49019) and Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations 26, 347–374 (1977; Zbl 0356.34067)]. An existence and uniqueness result is proved in the space of mappings of bounded variation. Also the author proves an extension theorem for a rate independent operator. This result is used for proving the continuous dependence of the sweeping process on initial data.

MSC:
34G25 Evolution inclusions
47J30 Variational methods involving nonlinear operators
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