Uniqueness for solutions of differential complementarity problems

Abstract

In this paper we consider the question of which matrices M give unique solutions for Differential Complementarity Problems (Mandelbaum 1989, unpublished manuscript) of the form

$$\begin{array}{ll}&\frac{dw}{dt} = M\, z + q(t),\quad w(0) = w_{0},\\ K \ni&z(t) \perp w(t) \in K^{*} \quad {\rm for\,all}\,t,\end{array}$$

for all q and w ₀ ∈ K ^* where K is a closed convex cone. Explicit descriptions of the set of such matrices are given for the 2 × 2 case; the set of such M’s independent of K is a strict subset of the set of positive definite matrices (v ^T Mv >  0 for all v ≠  0) but strictly contains the set of symmetric positive definite matrices. These results have implications for a range of different formulations of dynamic systems with complementarity constraints.