WIAS Preprint No. 2003, (2014)

Hausdorff metric BV discontinuity of sweeping processes



Authors

  • Klein, Olaf
    ORCID: 0000-0002-4142-3603
  • Recupero, Vincenzo

2010 Mathematics Subject Classification

  • 34A60 34C55 34G25, 47H30, 74C05

Keywords

  • Sweeping process,, discontinuity,, bounded variation,, Hausdorff metric

Abstract

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 emphrate independent operator containing as a particular case the so called emphplay operator which is widely used in hysteresis. 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 a counterexample showing that the solution operator of the sweeping process is not continuous when its domain is endowed with the strict topology of $BV$ and its codomain is endowed with the $L^1$-topology. This is at variance with the case of the play operator which instead is continuous in this sense.

Appeared in

  • 727 of Journal of Physics: Conference Series, 2016, pp. 012006/1--012006/12.

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