March  2013, 18(2): 565-573. doi: 10.3934/dcdsb.2013.18.565

Equicontinuous sweeping processes

1. 

Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russian Federation

Received  October 2011 Revised  May 2012 Published  November 2012

We prove that the sweeping process on a "regular" class of convex sets is equicontinuous. Classes of polyhedral sets with a given finite set of normal vectors are regular, as well as classes of uniformly strictly convex sets. Regularity is invariant to certain operations on classes of convex sets such as intersection, finite union, arithmetic sum and affine transformation.
Citation: Alexander Vladimirov. Equicontinuous sweeping processes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 565-573. doi: 10.3934/dcdsb.2013.18.565
References:
[1]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem with applications,, Stoch. and Stoch. Rep., 35 (1991), 31.   Google Scholar

[2]

H. Frankowska, A viability approach to the Skorohod problem,, Stochastics, 14 (1985), 227.   Google Scholar

[3]

J. Kelley, "General Topology,", D. Van Nostrand Company, (1957).   Google Scholar

[4]

A. A. Vladimirov, A. F. Klepcyn, V. S. Kozyakin, M. A. Krasnosel'skii, E. A. Lifshitz and A. V. Pokrovskii, Vector hysteresis nonlinearities of von Mises-Tresca type (Russian),, Dokl. Akad. Nauk SSSR, 257 (1981), 506.   Google Scholar

[5]

M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis,", Springer-Verlag, (1988).   Google Scholar

[6]

P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakkotosho, (1996).   Google Scholar

[7]

P. Krejci and A. Vladimirov, Lipschitz continuity of polyhedral Skorokhod maps,, J. Analysis Appl., 20 (2001), 817.   Google Scholar

[8]

P. Krejci and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions,, Set-Valued Anal., 11 (2003), 91.  doi: 10.1023/A:1021980201718.  Google Scholar

[9]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process,, in, 551 (2000), 1.   Google Scholar

[10]

M. D. P. Monteiro Marques, Rafle par un convexe semi-continue inferieurement d'interieur non vide en dimension finie,, C.R.Acad.Sci, 229 (1984), 307.   Google Scholar

[11]

M. D. P. Monteiro Marques, "Differential Inclusions in Nonsmooth Mechanical Problems-- Shocks and Dry Friction,", Birkhauser, (1993).   Google Scholar

[12]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, Journ. of Dif. Eq., 20 (1997), 347.   Google Scholar

[13]

A. A. Vladimirov, Does continuity of convex-valued maps survive under intersection?,, in, (2001), 415.   Google Scholar

[14]

A. A. Vladimirov and A. F. Kleptsyn, On some hysteresis elements (Russian),, Avtomat. i Telemekh., 7 (1982), 165.   Google Scholar

show all references

References:
[1]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem with applications,, Stoch. and Stoch. Rep., 35 (1991), 31.   Google Scholar

[2]

H. Frankowska, A viability approach to the Skorohod problem,, Stochastics, 14 (1985), 227.   Google Scholar

[3]

J. Kelley, "General Topology,", D. Van Nostrand Company, (1957).   Google Scholar

[4]

A. A. Vladimirov, A. F. Klepcyn, V. S. Kozyakin, M. A. Krasnosel'skii, E. A. Lifshitz and A. V. Pokrovskii, Vector hysteresis nonlinearities of von Mises-Tresca type (Russian),, Dokl. Akad. Nauk SSSR, 257 (1981), 506.   Google Scholar

[5]

M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis,", Springer-Verlag, (1988).   Google Scholar

[6]

P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations,", Gakkotosho, (1996).   Google Scholar

[7]

P. Krejci and A. Vladimirov, Lipschitz continuity of polyhedral Skorokhod maps,, J. Analysis Appl., 20 (2001), 817.   Google Scholar

[8]

P. Krejci and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions,, Set-Valued Anal., 11 (2003), 91.  doi: 10.1023/A:1021980201718.  Google Scholar

[9]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process,, in, 551 (2000), 1.   Google Scholar

[10]

M. D. P. Monteiro Marques, Rafle par un convexe semi-continue inferieurement d'interieur non vide en dimension finie,, C.R.Acad.Sci, 229 (1984), 307.   Google Scholar

[11]

M. D. P. Monteiro Marques, "Differential Inclusions in Nonsmooth Mechanical Problems-- Shocks and Dry Friction,", Birkhauser, (1993).   Google Scholar

[12]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, Journ. of Dif. Eq., 20 (1997), 347.   Google Scholar

[13]

A. A. Vladimirov, Does continuity of convex-valued maps survive under intersection?,, in, (2001), 415.   Google Scholar

[14]

A. A. Vladimirov and A. F. Kleptsyn, On some hysteresis elements (Russian),, Avtomat. i Telemekh., 7 (1982), 165.   Google Scholar

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