Differential variational inequality

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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 $\frac{dx}{dt}=f(t,x(t),u(t))$ , x(t₀)=x₀, where u(t) solves the variational inequality: 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.

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

Variational inequality

[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Avi Mandelbaum (1989) "Dynamic Complementarity Problems", unpublished manuscript.