Differential variational inequality

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Differential variational inequalities (DVI's) are dynamical systems that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVI's 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). DVI's are related to a number of other concepts including differential inclusions, projected dynamical systems, evolutionary inequalities, and parabolic variational inequalities.

Differential variational inequalities (DVI's)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 u(t)\in K such that

\langle v-u(t), F(t,x(t),u(t))\rangle\geq 0

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

\frac{dx}{dt}=f(t,x(t),u(t)), x(t0)=x0

Closely associated with DVI's are Dynamic/Differential Complementarity Problems: if K is a closed convex cone, then the variational inequality is equivalent to the complementarity problem:

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

[edit] Index

The concept of the index of a DVI is important and determines many questions of existence and uniqueness of solutions to a DVI. This concept is closely related to the concept of index for differential algebraic equations (DAE's), which is the number of times the algebraic equations of a DAE must be differentiated in order to obtain a complete system of differential equations for all variables. For a DVI, the index is the number of differentiations of F(t,x,u) = 0 needed in order to locally uniquely identify u as a function of t and x.

[edit] See also

[edit] References

  • Pang and Stewart (2006) "Differential Variational Inequalities" to appear.
  • Aubin and Cellina (1984) Differential Inclusions Springer-Verlag.
  • Acary and Brogliato and Goeleven (2006) "Higher order Moreau's sweeping process. Mathematical formulation and numerical formulation", Mathematical Programming A.
  • Avi Mandelbaum (1989) "Dynamic Complementarity Problems", unpublished manuscript.
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