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}} }

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