Stabilization of unstable stationary points in equations with delayed argument (Q1578494)

Consider the differential-delay system \[ dx(t)/dt =F(x(t),x(t-\tau_1), \dots , x(t-\tau _m), \mu) \tag \(*\) \] for \(x \in \mathbb{R}^n\), \(\mu \in \mathbb{R}\), \(0 \leq \tau _1 \leq \dots \leq \tau_m\). Let \(x^* (\mu)\) be a stationary solution to \((*)\) which is stable for \(\mu \leq \mu^*\), and unstable for \(\mu > \mu^*\). The authors describe a method to stabilize \(x^* (\mu)\) for \(\mu > \mu^*\). The main idea is to replace \((*)\) by the system \[ \begin{aligned} dx/dt & = F(x(t), x(t-\tau_1), \dots, x(t-\tau_m),\mu)+ \varepsilon (q(t)-\mu), \\ dq/dt & =G(x(t), x(t-\tau_1), \dots, x(t-\tau_m),\mu)+ \beta (q(t)-\mu),\end{aligned} \] where \(G\) satisfies \(G(x^* (\mu), x^* (\mu),\dots ,x^* (\mu),\mu)\equiv 0,\) and to choose \(\varepsilon \in \mathbb{R}^n\) and \(\beta \in \mathbb{R}\) in such a way as to guarantee the desired stability condition. The paper extends an idea of the second author [Dokl. Math. 55, No. 1, 160--162 (1997); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 352, No. 5, 610--612 (1997; Zbl 0967.34040)].