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

• 成果类型: Article
• 署名作者: Miranda-Villatoro, Felix A.; Castanos, Fernando; Franci, Alessio
• 署名单位: Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); CINVESTAV - Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional; University of Liege; Universidad Nacional Autonoma de Mexico
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3338981
• 发表日期: 2024
• 页码: 4570-4582
• 关键词: bifurcation; transistors; control theory; control systems; Behavioral sciences; steady-state; Limit-cycles; bifurcations; linear complementarity problems (LCPs); nonsmooth dynamics; piecewise linear equations

摘要

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 2-D linear complementarity problems.