In this work, we show that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportional-derivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the system’s solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator. It is worth mentioning that this work represents a natural continuation of Amrane et al. (2018) and Bedouhene et al. (2020), addressing the problem of coexisting real spectral values for linear dynamical systems including delays in their models.

中文翻译：

---

对二阶延迟系统中均匀分布实谱值的见解：部分极点配置的视角

在这项工作中，我们证明了通用单延迟延迟二阶微分方程的最大实谱值数量的共存保证了最右边谱值的真实性。从控制理论的角度来看，这意味着延迟比例微分 (PD) 控制器可以稳定延迟二阶微分方程。通过将最大负根数分配给相应的特征函数（拟多项式），我们建立了渐近稳定性的条件。如果指定的真实光谱值是均匀分布的，我们指定最右根的 a 为负，从而保证系统解的指数衰减率。我们阐述了阻尼谐振子延迟 PD 控制中提出的设计方法。值得一提的是，这项工作代表了 Amrane 等人的自然延续。(2018) 和 Bedouhene 等人。（2020），解决线性动力系统共存实谱值的问题，包括模型中的延迟。