We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function f : R → R of the form x(k + 1) = f(x(k)) + u(k) where u(k) = (a1−1)f(x(k))+a2f(x(k−T))+· · ·+aN f(x(k−(N −1)T)) , with a1 + · · · + aN = 1. Following an approach of Morgul, we associate to each periodic orbit of f, N ∈ N, and a1, . . . , aN an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of f, there exist N and a1, . . ., aN whose associated polynomial is Schur stable, and we find the minimal N that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fejer kernels found in classical harmonic analysis.
Dmitrishin, Dmitriy, Paul Hagelstein, Anna Khamitova, Anatolii Korenovskyi, Alexander M. Stokolos.
"Fejér Polynomials and Control of Nonlinear Discrete Systems."