Differential variational inequalities

By Jong-Shi Pang, David E. Stewart. Mathematical programming, 2008

487 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/50e2cb2ad2002f2de72b43f90eaf33b2ce512bfa

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In Computational Theory and Mathematics

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Citations in the corpus

On optimal control of a sweeping process coupled with an ordinary differential equation (2014, background): “Another view at this model is provided by [31] where, under certain condition, inclusion (1) is reformulated as a differential variational inequality, for which some theoretical results and numerous applications exist.”
Decay solutions for a class of fractional differential variational inequalities (2015, background): “However, DVIs were first systematically studied by Pang and Stewart [25].”
A class of fractional differential hemivariational inequalities with application to contact problem (2018, background): “From another point of view, they were firstly considered and systematically studied in a framework of finite-dimensional spaces by Pang and Stewart [41] in 2008.”
Dynamics and bifurcations of nonsmooth systems: A survey (2012, background): “Differential variational inequalities provide a convenient formalism for optimal control problems (see Pang-Stewart [287], Kwon-Friesz-Mookherjee-Yao-Feng [224]) and frictional con-”
Optimal control of the sweeping process over polyhedral controlled sets (2015, background): “1) can be considered as an evolution variational inequality, or a differential variational inequality in the terminology of [32].”