Convolution complementarity problems with application to impact problems

Convolution complementarity problems (CCPs) have the following form: given a matrix-valued function k and a vector-valued function q, 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 mechanical impact problem, but they can also be applied to other dynamic problems with hard constraints. CCPs are shown to have solutions provided q(0) ≥ 0 and q is sufficiently regular, k has locally bounded variation and k(0⁺) is a P-matrix. Uniqueness also holds provided, in addition, k(0⁺) is symmetric positive definite. This theory shows that the impact problem studied here has a unique solution, and that energy is conserved. Numerical methods have been devised and implemented for the impact problem, and the results are presented.