Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

Peter Giesl*, Sigurdur Hafstein

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem. In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices. The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.

Original languageEnglish
Pages (from-to)114-134
Number of pages21
JournalNonlinear Analysis, Theory, Methods and Applications
Publication statusPublished - 2013

Bibliographical note

Funding Information:
This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/J014532/1 ].

Other keywords

  • Basin of attraction
  • Borg's criterium
  • Contraction metric
  • CPA function
  • Periodic differential equation
  • Periodic orbit
  • Semidefinite optimization


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