Stability analysis of switched systems using variational principles: An introduction

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Many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode. Switched systems provide a suitable mathematical model for such processes, and their stability analysis is important for both theoretical and practical reasons. We review a specific approach for stability analysis based on using variational principles to characterize the "most unstable" solution of the switched system. We also discuss a link between the variational approach and the stability analysis of switched systems using Lie-algebraic considerations. Both approaches require the use of sophisticated tools from many different fields of applied mathematics. The purpose of this paper is to provide an accessible and self-contained review of these topics, emphasizing the intuitive and geometric underlying ideas.

Original languageEnglish
Pages (from-to)2059-2077
Number of pages19
Issue number12
StatePublished - Dec 2006


  • Absolute stability
  • Bang-bang control
  • Bilinear systems
  • Differential inclusions
  • Dynamic programming
  • Geometric control theory
  • Global asymptotic stability
  • Hamilton-Jacobi-Bellman equation
  • Hybrid systems
  • Lie algebra
  • Lie bracket
  • Maximum principle
  • Nilpotent control systems
  • Reachability with nice controls
  • Stability under arbitrary switching
  • Switched controllers


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