Fejér Polynomials and Control of Nonlinear Discrete Systems

We consider optimization problems associated with 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))+⋯+aNf(x(k-(N-1)T)),with a₁+ ⋯ + a_N= 1. Following an approach of Morgül, we associate with each periodic orbit of f, N∈ N, and a₁,.., a_N 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, … , a_N 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 Fejér kernels found in classical harmonic analysis.