Fejer Polynomials and Control of Nonlinear Discrete Systems

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 $f:\mathbb{R}\to \mathbb{R}$ of the form $$x(k+1)=f(x(k))+u(k)$$ where $$u(k)=(a_1-1)f(x(k))+a_{2}f(x(k-T))+\cdots +a_{N}f(x(k-(N-1)T)),$$ with $a_1+\cdots +a_N=1$. Following an approach of Morg"ul, we associate to each periodic orbit of $f$, $N\in \mathbb{N}$, and $a_1,\dots ,a_N$ 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 $a_1,\dots ,a_N$ 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 Fej'er kernels found in classical harmonic analysis.