The methods of testing the robust \(D\)-stability of polynomials in a particular case (Q1422763)

Let \(D\subset\mathbb{C}\) and \(p(s)= a_0s^n+\cdots+ a_n\) be a real polynomial with roots in \(D\) (i.e., \(p\) is \(D\)-stable). Consider the \(n\)-dimensional family \(P\) of polynomials \(p()+ \delta_1 s^{n-1}+\cdots+ \delta_n\), where \(|\delta_1|+\cdots+ |\delta_n|\leq \gamma\) and \(\gamma> 0\) is a given constant. The set \(T\subset P\) is said to be a testing subfamily if the \(D\)-stability of \(T\) implies \(D\)-stability of \(P\). The author analyzes 5 typical regions \(D\) (including open half-planes, discs and sectors) and the possibility to minimize the corresponding testing subfamilies.