Differential Variational Inequality - 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.

This index can be computed for the above examples. For the mechanical impact example, if we differentiate once we have, which does not yet explicitly involve . However, if we differentiate once more, we can use the differential equation to give, which does explicitly involve . Furthermore, if, we can explicitly determine in terms of .

For the ideal diode systems, the computations are considerably more difficult, but provided some generally valid conditions hold, the differential variational inequality can be shown to have index one.

Differential variational inequalities with index greater than two are generally not meaningful, but certain conditions and interpretations can make them meaningful (see the references Acary, Brogliato and Goeleven, and Heemels, Schumacher, and Weiland below).