Differential variational inequalities. Pang, J. & Stewart, D. E. MATHEMATICAL PROGRAMMING, 113(2):345-424, SPRINGER, 233 SPRING STREET, NEW YORK, NY 10013 USA, JUN, 2008. doi abstract bibtex This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with "index″ not exceeding two and which have absolutely continuous solutions.
@article{ Pang_Stewart,
author = {Pang, Jong-Shi and Stewart, David E.},
title = {{Differential variational inequalities}},
journal = {{MATHEMATICAL PROGRAMMING}},
year = {{2008}},
volume = {{113}},
number = {{2}},
pages = {{345-424}},
month = {{JUN}},
abstract = {{This paper introduces and studies the class of differential variational
inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI
provides a powerful modeling paradigm for many applied problems in
which dynamics, inequalities, and discontinuities are present; examples
of such problems include constrained time-dependent physical systems
with unilateral constraints, differential Nash games, and hybrid
engineering systems with variable structures. The DVI unifies several
mathematical problem classes that include ordinary differential
equations (ODEs) with smooth and discontinuous right-hand sides,
differential algebraic equations (DAEs), dynamic complementarity
systems, and evolutionary variational inequalities. Conditions are
presented under which the DVI can be converted, either locally or
globally, to an equivalent ODE with a Lipschitz continuous right-hand
function. For DVIs that cannot be so converted, we consider their
numerical resolution via an Euler time-stepping procedure, which
involves the solution of a sequence of finite-dimensional variational
inequalities. Borrowing results from differential inclusions (DIs) with
upper semicontinuous, closed and convex valued multifunctions, we
establish the convergence of such a procedure for solving initial-value
DVIs. We also present a class of DVIs for which the theory of DIs is
not directly applicable, and yet similar convergence can be
established. Finally, we extend the method to a boundary-value DVI and
provide conditions for the convergence of the method. The results in
this paper pertain exclusively to systems with "index″ not
exceeding two and which have absolutely continuous solutions.}},
publisher = {{SPRINGER}},
address = {{233 SPRING STREET, NEW YORK, NY 10013 USA}},
type = {{Review}},
language = {{English}},
affiliation = {{Pang, JS (Reprint Author), Rensselaer Polytech Inst, Dept Math Sci, Troy, NY 12180 USA.
{[}Pang, Jong-Shi] Rensselaer Polytech Inst, Dept Math Sci, Troy, NY 12180 USA.
{[}Pang, Jong-Shi] Rensselaer Polytech Inst, Dept Decis Sci \& Engn Syst, Troy, NY 12180 USA.
{[}Stewart, David E.] Univ Iowa, Dept Math, Iowa City, IA 52242 USA.}},
doi = {{10.1007/s10107-006-0052-x}},
issn = {{0025-5610}},
keywords-plus = {{RIGID-BODY DYNAMICS; RIGHT-HAND SIDE; LINEAR COMPLEMENTARITY SYSTEMS;
COMPLIANT CONTACT PROBLEMS; TIME-STEPPING METHOD; GENERALIZED
EQUATIONS; SWEEPING PROCESS; MULTIBODY DYNAMICS; HILBERT-SPACES;
STABILITY}},
subject-category = {{Computer Science, Software Engineering; Operations Research \&
Management Science; Mathematics, Applied}},
author-email = {{pangj@rpi.edu
dstewart@math.uiowa.edu}},
number-of-cited-references = {{102}},
times-cited = {{20}},
journal-iso = {{Math. Program.}},
doc-delivery-number = {{256ZW}},
unique-id = {{ISI:000252769400006}}
}
Downloads: 0
{"_id":{"_str":"52744b6957266d0f1e000257"},"__v":0,"authorIDs":[],"author_short":["Pang, J.","Stewart, D.<nbsp>E."],"bibbaseid":"pang-stewart-differentialvariationalinequalities-2008","bibdata":{"html":"<div class=\"bibbase_paper\"> \n\n\n<span class=\"bibbase_paper_titleauthoryear\">\n\t<span class=\"bibbase_paper_title\"><a name=\"Pang_Stewart\"> </a>Differential variational inequalities.</span>\n\t<span class=\"bibbase_paper_author\">\nPang, J.; and Stewart, D. E.</span>\n\t<!-- <span class=\"bibbase_paper_year\">2008</span>. -->\n</span>\n\n\n\n<i>MATHEMATICAL PROGRAMMING</i>,\n\n113(2):345-424.\n\nJUN 2008.\n\n\n\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content\">\n \n \n \n <a href=\"javascript:showBib('Pang_Stewart')\">\n <img src=\"http://www.bibbase.org/img/filetypes/bib.png\" \n\t alt=\"Differential variational inequalities [bib]\" \n\t class=\"bibbase_icon\"\n\t style=\"width: 24px; height: 24px; border: 0px; vertical-align: text-top\"><span class=\"bibbase_icon_text\">Bibtex</span></a>\n \n \n\n \n \n \n \n \n\n \n <a class=\"bibbase_abstract_link\" href=\"javascript:showAbstract('Pang_Stewart')\">Abstract</a>\n \n \n</span>\n\n<!-- -->\n<!-- <div id=\"abstract_Pang_Stewart\"> -->\n<!-- This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with \"index″ not exceeding two and which have absolutely continuous solutions. -->\n<!-- </div> -->\n<!-- -->\n\n</div>\n","downloads":0,"bibbaseid":"pang-stewart-differentialvariationalinequalities-2008","role":"author","year":"2008","volume":"113","unique-id":"ISI:000252769400006","type":"Review","title":"Differential variational inequalities","times-cited":"20","subject-category":"Computer Science, Software Engineering; Operations Research \\& Management Science; Mathematics, Applied","publisher":"SPRINGER","pages":"345-424","number-of-cited-references":"102","number":"2","month":"JUN","language":"English","keywords-plus":"RIGID-BODY DYNAMICS; RIGHT-HAND SIDE; LINEAR COMPLEMENTARITY SYSTEMS; COMPLIANT CONTACT PROBLEMS; TIME-STEPPING METHOD; GENERALIZED EQUATIONS; SWEEPING PROCESS; MULTIBODY DYNAMICS; HILBERT-SPACES; STABILITY","key":"Pang_Stewart","journal-iso":"Math. Program.","journal":"MATHEMATICAL PROGRAMMING","issn":"0025-5610","id":"Pang_Stewart","doi":"10.1007/s10107-006-0052-x","doc-delivery-number":"256ZW","bibtype":"article","bibtex":"@article{ Pang_Stewart,\n author = {Pang, Jong-Shi and Stewart, David E.},\n title = {{Differential variational inequalities}},\n journal = {{MATHEMATICAL PROGRAMMING}},\n year = {{2008}},\n volume = {{113}},\n number = {{2}},\n pages = {{345-424}},\n month = {{JUN}},\n abstract = {{This paper introduces and studies the class of differential variational\n inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI\n provides a powerful modeling paradigm for many applied problems in\n which dynamics, inequalities, and discontinuities are present; examples\n of such problems include constrained time-dependent physical systems\n with unilateral constraints, differential Nash games, and hybrid\n engineering systems with variable structures. The DVI unifies several\n mathematical problem classes that include ordinary differential\n equations (ODEs) with smooth and discontinuous right-hand sides,\n differential algebraic equations (DAEs), dynamic complementarity\n systems, and evolutionary variational inequalities. Conditions are\n presented under which the DVI can be converted, either locally or\n globally, to an equivalent ODE with a Lipschitz continuous right-hand\n function. For DVIs that cannot be so converted, we consider their\n numerical resolution via an Euler time-stepping procedure, which\n involves the solution of a sequence of finite-dimensional variational\n inequalities. Borrowing results from differential inclusions (DIs) with\n upper semicontinuous, closed and convex valued multifunctions, we\n establish the convergence of such a procedure for solving initial-value\n DVIs. We also present a class of DVIs for which the theory of DIs is\n not directly applicable, and yet similar convergence can be\n established. Finally, we extend the method to a boundary-value DVI and\n provide conditions for the convergence of the method. The results in\n this paper pertain exclusively to systems with \"index″ not\n exceeding two and which have absolutely continuous solutions.}},\n publisher = {{SPRINGER}},\n address = {{233 SPRING STREET, NEW YORK, NY 10013 USA}},\n type = {{Review}},\n language = {{English}},\n affiliation = {{Pang, JS (Reprint Author), Rensselaer Polytech Inst, Dept Math Sci, Troy, NY 12180 USA.\n {[}Pang, Jong-Shi] Rensselaer Polytech Inst, Dept Math Sci, Troy, NY 12180 USA.\n {[}Pang, Jong-Shi] Rensselaer Polytech Inst, Dept Decis Sci \\& Engn Syst, Troy, NY 12180 USA.\n {[}Stewart, David E.] Univ Iowa, Dept Math, Iowa City, IA 52242 USA.}},\n doi = {{10.1007/s10107-006-0052-x}},\n issn = {{0025-5610}},\n keywords-plus = {{RIGID-BODY DYNAMICS; RIGHT-HAND SIDE; LINEAR COMPLEMENTARITY SYSTEMS;\n COMPLIANT CONTACT PROBLEMS; TIME-STEPPING METHOD; GENERALIZED\n EQUATIONS; SWEEPING PROCESS; MULTIBODY DYNAMICS; HILBERT-SPACES;\n STABILITY}},\n subject-category = {{Computer Science, Software Engineering; Operations Research \\&\n Management Science; Mathematics, Applied}},\n author-email = {{pangj@rpi.edu\n dstewart@math.uiowa.edu}},\n number-of-cited-references = {{102}},\n times-cited = {{20}},\n journal-iso = {{Math. Program.}},\n doc-delivery-number = {{256ZW}},\n unique-id = {{ISI:000252769400006}}\n}","author_short":["Pang, J.","Stewart, D.<nbsp>E."],"author-email":"pangj@rpi.edu dstewart@math.uiowa.edu","author":["Pang, Jong-Shi","Stewart, David E."],"affiliation":"Pang, JS (Reprint Author), Rensselaer Polytech Inst, Dept Math Sci, Troy, NY 12180 USA. [Pang, Jong-Shi] Rensselaer Polytech Inst, Dept Math Sci, Troy, NY 12180 USA. [Pang, Jong-Shi] Rensselaer Polytech Inst, Dept Decis Sci \\& Engn Syst, Troy, NY 12180 USA. [Stewart, David E.] Univ Iowa, Dept Math, Iowa City, IA 52242 USA.","address":"233 SPRING STREET, NEW YORK, NY 10013 USA","abstract":"This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with \"index″ not exceeding two and which have absolutely continuous solutions."},"bibtype":"article","biburl":"http://www2.imperial.ac.uk/~omakaren/literature.bib","downloads":0,"search_terms":["differential","variational","inequalities","pang","stewart"],"title":"Differential variational inequalities","year":2008,"dataSources":["Kff4nzCYaKRYGNYL7"]}