From Partial and Horizontal Contraction to <italic>K</italic>-Contraction

Chengshuai Wu, Dimos V. Dimarogonas

Research output: Contribution to journalArticlepeer-review


A geometric generalization of contraction theory called&#x00A0;<inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-contraction was recently developed using <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-compound matrices. In this note, we focus on the relations between <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-contraction and two other generalized contraction frameworks: partial contraction (also known as virtual contraction) and horizontal contraction. We show that in general these three notions of contraction are different. We here provide new sufficient conditions guaranteeing that partial contraction implies horizontal contraction, and that horizontal contraction implies <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-contraction. We use the Andronov-Hopf oscillator to demonstrate some of the theoretical results.

Original languageEnglish
Pages (from-to)1-8
Number of pages8
JournalIEEE Transactions on Automatic Control
StateAccepted/In press - 2023
Externally publishedYes


  • <inline-formula xmlns:ali="" xmlns:mml="" xmlns:xlink="" xmlns:xsi=""> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-contraction
  • Andronov-Hopf oscillator
  • Behavioral sciences
  • compound matrix
  • Compounds
  • horizontal contraction
  • Manifolds
  • Oscillators
  • partial contraction
  • Sufficient conditions
  • Synchronization
  • Trajectory
  • virtual contraction

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