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 and 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), 31-62.

[2]

H. Frankowska, A viability approach to the Skorohod problem, Stochastics, 14 (1985), 227-244.

[3]

J. Kelley, "General Topology," D. Van Nostrand Company, Inc., New York, 1957.

[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), 506-509.

[5]

M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis," Springer-Verlag, Berlin, 1988.

[6]

P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations," Gakkotosho, Tokyo, 1996.

[7]

P. Krejci and A. Vladimirov, Lipschitz continuity of polyhedral Skorokhod maps, J. Analysis Appl., 20 (2001), 817-844.

[8]

P. Krejci and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Anal., 11 (2003), 91-110. doi: 10.1023/A:1021980201718.

[9]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in "Impacts in Mechanical Systems - Analysis and Modelling," 551 of Lecture Notes in Physics, Springer, Berlin-New York, (2000), 1-60.

[10]

M. D. P. Monteiro Marques, Rafle par un convexe semi-continue inferieurement d'interieur non vide en dimension finie, C.R.Acad.Sci, Paris, Ser. I, 229 (1984), 307-310.

[11]

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

[12]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, Journ. of Dif. Eq., 20 (1997), 347-374.

[13]

A. A. Vladimirov, Does continuity of convex-valued maps survive under intersection?, in "Optimization and Related Topics," Kluwer Acad. Publ., (2001), 415-428.

[14]

A. A. Vladimirov and A. F. Kleptsyn, On some hysteresis elements (Russian), Avtomat. i Telemekh., 7 (1982), 165-169.

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), 31-62.

[2]

H. Frankowska, A viability approach to the Skorohod problem, Stochastics, 14 (1985), 227-244.

[3]

J. Kelley, "General Topology," D. Van Nostrand Company, Inc., New York, 1957.

[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), 506-509.

[5]

M. A. Krasnosel'skii and A. V. Pokrovskii, "Systems with Hysteresis," Springer-Verlag, Berlin, 1988.

[6]

P. Krejci, "Hysteresis, Convexity and Dissipation in Hyperbolic Equations," Gakkotosho, Tokyo, 1996.

[7]

P. Krejci and A. Vladimirov, Lipschitz continuity of polyhedral Skorokhod maps, J. Analysis Appl., 20 (2001), 817-844.

[8]

P. Krejci and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Anal., 11 (2003), 91-110. doi: 10.1023/A:1021980201718.

[9]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in "Impacts in Mechanical Systems - Analysis and Modelling," 551 of Lecture Notes in Physics, Springer, Berlin-New York, (2000), 1-60.

[10]

M. D. P. Monteiro Marques, Rafle par un convexe semi-continue inferieurement d'interieur non vide en dimension finie, C.R.Acad.Sci, Paris, Ser. I, 229 (1984), 307-310.

[11]

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

[12]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, Journ. of Dif. Eq., 20 (1997), 347-374.

[13]

A. A. Vladimirov, Does continuity of convex-valued maps survive under intersection?, in "Optimization and Related Topics," Kluwer Acad. Publ., (2001), 415-428.

[14]

A. A. Vladimirov and A. F. Kleptsyn, On some hysteresis elements (Russian), Avtomat. i Telemekh., 7 (1982), 165-169.

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