Convergence of Time-Stepping Schemes for Passive and Extended Linear Complementarity Systems

Lanshan Han*, Alok Tiwari, M. Kanat Camlibel, Jong-Shi Pang

*Corresponding author for this work

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Abstract

Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established.

Original languageEnglish
Pages (from-to)3768-3796
Number of pages29
JournalSiam journal on numerical analysis
Volume47
Issue number5
DOIs
Publication statusPublished - 2009

Keywords

  • linear complementarity systems
  • time stepping
  • convergence analysis
  • passivity
  • CONTROLLABILITY
  • NETWORKS

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