Equivalence of Linear Complementarity Problems: Theory and Application to Nonsmooth Bifurcations

Linear complementarity problems provide a powerful framework to model nonsmooth phenomena in dynamical control systems. Mimicking the general strategy that led to the foundation of bifurcation theory in smooth maps, we introduce a novel notion of equivalence between linear complementarity problems that sets the basis for a theory of bifurcations in a large class of nonsmooth maps, including steady-state bifurcations in linear complementarity systems. Our definition leads to constructive algebraic conditions for identifying and classifying the nonsmooth singularities associated with nonsmooth bifurcations. We thoroughly illustrate our theory on an extended applied example and on the identification and classification of all possible equivalence classes in two-dimensional linear complementarity problems.