Stability of imbedded families of polynomials
The article is dedicated to the problem of robust stability, i.e., defining conditions such that the roots of the polynomial family \[ p(s)= a_0s^n+ a_1s^{n- 1}+\cdots+ a_n,\quad a_0\neq 0\quad (a_0,a_1,\dots, a_n)\in Q \] are situated left from the imaginary axis. It is assumed that \(Q\) is a convex polytope or a ball. Theorem (edge theorem). Let \(Q\) be a convex polytope, and all polynomials of the family have the degree \(n\). The polynomials have roots with \(\text{Re }\lambda_i<0\), \(i= 1,\dots,n\), if and only if this property holds for polynomials of the set \(E_Q\) (points of the edges of the polytope \(Q\)). The special case of the interval polynomial, i.e. \[ \underline a_j\leq a_j\leq\overline a_j,\quad j=0,\dots, n, \] is considered. Four polytopes of a special type are constructed. \[ \begin{aligned} p_1(s) & = \underline a_n+\underline a_{n- 1}s+\overline a_{n- 2}s^2+\overline a_{n- 3}s^3+\underline a_{n- 4}s^4,\\ p_2(s) & = \overline a_n+\underline a_{n- 1}s+\underline a_{n- 2}s^2+\overline a_{n-3}s^3+\overline a_{n- 4}s^4,\\ p_3(s) & = \overline a_n+\overline a_{n- 1}s+\underline a_{n- 2}s^2+\underline a_{n- 3}s^3+\overline a_{n- 4}s^4,\\ p_4(s) & = \underline a_n+\overline a_{n- 1}s+\overline a_{n- 2}s^2+\underline a_{n- 3}s^3+\underline a_{n- 4}s^4.\end{aligned} \] Theorem. The family of polytopes is interval stable if and only if the four polytopes are stable.
scientific article; zbMATH DE number 1290394