Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 43, Issue 3, Pages 1122-1144Publisher
SIAM PUBLICATIONS
DOI: 10.1137/090779450
Keywords
delayed feedback control; stability; delay differential equations; Floquet theory
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Funding
- Japan Society for the Promotion of Science [16540187, 22540223]
- Grants-in-Aid for Scientific Research [22540223] Funding Source: KAKEN
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The delayed feedback control (DFC) is a control method for stabilizing unstable periodic orbits in nonlinear autonomous differential equations. We give an important relationship between the characteristic multipliers of the linear variational equation around an unstable periodic solution of the equation and those of its delayed feedback equation. The key of our proof is a result about the spectrum of a matrix which is a difference of commutative matrices. The relationship, moreover, allows us to design control gains of the DFC such that the unstable periodic solution is stabilized. In other words, the validity of the DFC is proved mathematically. As an application for the Rossler equation, we determine the best range of k such that the unstable periodic orbit is stabilized by taking a feedback gain K = kE.
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