Convolution complementarity problems with application to impact problems

^** Email: dstewart{at}math.uiowa.edu. Part of this work was carried out while visiting CMAF at the University of Lisbon and while visiting the University of Lyons 1. Convolution complementarity problems (CCPs) have the followingform: given a matrix-valued function k and a vector-valued functionq, find a vector-valued function u satisfying 0 u(t) (k*u)(t)+ q(t) 0 for all t. In this paper CCPs are applied to a mechanicalimpact problem, but they can also be applied to other dynamicproblems with hard constraints. CCPs are shown to have solutionsprovided q(0) 0 and q is sufficiently regular, k has locallybounded variation and k(0⁺) is a P-matrix. Uniqueness also holdsprovided, in addition, k(0⁺) is symmetric positive definite.This theory shows that the impact problem studied here has aunique solution, and that energy is conserved. Numerical methodshave been devised and implemented for the impact problem, andthe results are presented.