Differential variational inequality

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In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems.

DVIs are useful for representing models involving both dynamics and inequality constraints. Examples of such problems include, for example, mechanical impact problems, electrical circuits with ideal diodes, Coulomb friction problems for contacting bodies, and dynamic economic and related problems such as dynamic traffic networks and networks of queues (where the constraints can either be upper limits on queue length or that the queue length cannot become negative). DVIs are related to a number of other concepts including differential inclusions, projected dynamical systems, evolutionary inequalities, and parabolic variational inequalities.

Differential variational inequalities were first formally introduced by Pang and Stewart, whose definition should not be confused with the differential variational inequality used in Aubin and Cellina (1984).

Differential variational inequalities have the form to find ${\displaystyle u(t)\in K}$ such that

${\displaystyle \langle v-u(t),F(t,x(t),u(t))\rangle \geq 0}$

for every ${\displaystyle v\in K}$ and almost all t; K a closed convex set, where

${\displaystyle {\frac {dx}{dt}}=f(t,x(t),u(t)),\quad x(t_{0})=x_{0}.}$

Closely associated with DVIs are dynamic/differential complementarity problems: if K is a closed convex cone, then the variational inequality is equivalent to the complementarity problem:

${\displaystyle K\ni u(t)\quad \perp \quad F(t,x(t),u(t))\in K^{*}.}$