Linear and Multilinear Algebra
We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizingT-cycles of a differentiable functionf:R→Rof the form x(k+1)=f(x(k))+u(k)
with a1+...+aN=1. Following an approach of Morgül, we construct a map F:RT+1→RT+1 whose fixed points correspond to T-cycles of f. We then analyze the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrum points of F. We associate to each periodic orbit of f an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. An example indicating the efficacy of this method is provided.
Dmitrishin, Dmitriy, Paul Hagelstein, Anna Khamitova, Alexander M. Stokolos.
"On the Stability of Cycles by Delayed Feedback Control."
Linear and Multilinear Algebra, 64 (8): 1538-1549.