Sweeping by a continuous prox-regular set $

Abstract The evolution problem known as sweeping process is considered for a class of nonconvex sets called prox-regular (or ϕ -convex). Assuming, essentially, that such sets contain in the interior a suitable subset and move continuously (w.r.t. the Hausdorff distance), we prove local and global existence as well as uniqueness of solutions, which are continuous functions with bounded variation. Some examples are presented.

[1]  C. Castaing,et al.  Nonconvex Sweeping Process , 1996 .

[2]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[3]  P. Dupuis,et al.  On Lipschitz continuity of the solution mapping to the Skorokhod problem , 1991 .

[4]  A. Marino,et al.  Some variational problems with lack of convexity and some partial differential inequalities , 1990 .

[5]  J. Moreau.  Evolution problem associated with a moving convex set in a Hilbert space , 1977 .

[6]  Annamaria Canino,et al.  On p-convex sets and geodesics , 1988 .

[7]  H. Benabdellah.  Existence of Solutions to the Nonconvex Sweeping Process , 2000 .

[8]  R. Rockafellar,et al.  Local differentiability of distance functions , 2000 .

[9]  L. Thibault.  Sweeping process with regular and nonregular sets , 2003 .

[10]  Pavel Drábek,et al.  Nonlinear differential equations , 1999 .

[11]  C. Castaing,et al.  BV solutions of multivalued differential equations on closed moving sets in Banach spaces , 1995 .

[12]  Local properties of geodesics onp-convex sets , 1991 .

[13]  Vladimir V. Goncharov,et al.  Variational Inequalities and Regularity Properties of Closed Sets in Hilbert Spaces , 1999 .

[14]  G. Colombo,et al.  The Sweeping Processes without Convexity , 1999 .

[15]  M. Marques,et al.  Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction , 1993 .

[16]  Homéomorphisme entre ouverts lipschitziens , 1978 .

[17]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[18]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[19]  Hiroshi Tanaka.  Stochastic differential equations with reflecting boundary condition in convex regions , 1979 .

[20]  F. Clarke,et al.  Proximal Smoothness and the Lower{C 2 Property , 1995 .