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

TL;DR

This paper introduces a new notion of equivalence for linear complementarity problems, enabling a bifurcation theory for nonsmooth maps and providing algebraic tools to classify nonsmooth singularities in applications like electrical networks.

Contribution

It develops a novel equivalence concept for linear complementarity problems, laying the foundation for bifurcation analysis in nonsmooth systems and offering algebraic criteria for classifying singularities.

Findings

Established a new equivalence framework for LCPs

Derived algebraic conditions for nonsmooth bifurcations

Classified all equivalence classes in 2D LCPs

Abstract

Linear complementarity problems provide a powerful framework to model nonsmooth phenomena in a variety of real-world applications. In dynamical control systems, they appear coupled to a linear input-output system in the form of linear complementarity systems. Mimicking the program 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, but not restricted to, steadystate bifurcations in linear complementarity systems. Our definition exploits the rich geometry of linear complementarity problems and 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, the design of bistability in an electrical network, and a more theoretical one, the identification and classification of all possible equivalence classes in two-dimensional linear complementarity problems.