D-稳定性区域的单连通性与多项式族的D-稳定性(英文)

Simple connectivity of the D-stability Domain and D-stability of polynomial families

证明了其根位于复平面中某开域的所有ｎ阶多项式的全体构成多项式参数空间中的一个单连通集，并基于这一结果进一步提出了关于一般多面体多项式族Ｄ－稳定性的棱边定理和关于一般多项式紧集Ｄ－稳定性的边界定理．所得结果包含了多项式参数无关和相关两种情形．不同于已有结果，作者提出的棱边定理和边界定理不要求多项式集的凸性与单连通性．另外，关于相关参数的多项式集的边界定理还突破了现有结果中关于参数相关关系为仿射映射的限制．

It is proved that all monic polynomials of order n with roots lying in some open region on the complex plane forms a simply connected set in the polynomial parameter space. Based on this re- sult, Edge Theorems for D-stability of general polyhedrons of polynomials and Boundary Theorems for D-stability of compact sets of polynomials are obtained. Both cases of families of polynomials with dependent and independent coefficients are considered. Different from the previous ones, our Edge Theorems and Boundary Theorems do not require the convexity orthe connectivity of the set of polynomials. Moreover, our Boundary Theorem for families of polynodrials with dependent coefficients does not require the coefficient dependency relation to be affine.