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 $$f: \mathbb {R}\rightarrow \mathbb {R}$$ f : R → R of the form $$\begin{aligned} x(k+1) = f(x(k)) + u(k), \end{aligned}$$ x ( k + 1 ) = f ( x ( k ) ) + u ( k ) , where $$\begin{aligned} u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + \cdots + a_N f(x(k-(N-1)T)), \end{aligned}$$ u ( k ) = ( a 1 - 1 ) f ( x ( k ) ) + a 2 f ( x ( k - T ) ) + ⋯ + a N f ( x ( k - ( N - 1 ) T ) ) , with $$a_1 + \cdots + a_N = 1$$ a 1 + ⋯ + a N = 1 . Following an approach of Morgül, we associate with each periodic orbit of f , $$N \in \mathbb {N}$$ N ∈ N , and $$a_1$$ a 1 , ..., $$a_N$$ 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 $$a_1$$ a 1 , $$\ldots $$ … , $$a_N$$ 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.

中文翻译：

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Fejér 多项式和非线性离散系统的控制

我们考虑与用于稳定一维离散时间系统循环的延迟反馈控制 (DFC) 机制相关的优化问题。特别是，我们考虑了一种延迟反馈控制，用于稳定可微函数 $$f 的 T 循环： \mathbb {R}\rightarrow \mathbb {R}$$ f : R → R 形式 $$\begin{aligned } x(k+1) = f(x(k)) + u(k), \end{aligned}$$ x ( k + 1 ) = f ( x ( k ) ) + u ( k ) ，其中$ $\begin{aligned} u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(kT)) + \cdots + a_N f(x(k-(N-1)T) ), \end{aligned}$$ u ( k ) = ( a 1 - 1 ) f ( x ( k ) ) + a 2 f ( x ( k - T ) ) + ⋯ + a N f ( x ( k - ( N - 1 ) T ) ) ，其中 $$a_1 + \cdots + a_N = 1$$ a 1 + ⋯ + a N = 1 。遵循 Morgül 的方法，我们与 f 的每个周期轨道相关联， $$N \in \mathbb {N}$$ N ∈ N ，和 $$a_1$$ a 1 ，...，$$a_N$$ a N 一个显式多项式，其舒尔稳定性对应于该轨道上 DFC 的稳定性。我们证明，给定 f 的任何 1-或 2-循环，存在 N 和 $$a_1$$ a 1 ， $$\ldots $$ ... ， $$a_N$$ a N，其关联多项式是 Schur 稳定的，并且我们找到了保证这种稳定性的最小 N。证明技术将利用经典谐波分析中发现的 Fejér 核的极值特性。