Abstract :
We consider a parametric family F ≔ {ƒp(z) : p ∈ P} of functions ƒp(z) holomorphic in a closed region D ⊂ C depending continuously on a finite dimensional real or complex parameter vector p ≔ (p1, p2,..., pn) ranging over a closed region P in parameter space. The problem is to establish criteria which guarantee that ƒp(z) ≠ 0 for z ∈ D and p ∈ p. (1) Such criteria are needed in order to reduce the amount of checks to be performed when verifying classwise stability. In the case of linear parameter dependence, ƒp(z) ≔ ƒ0(z) + ƒ1(z)p1 + ƒ2(z)p2 +•••+ ƒn(z)pn, and an interval P ≔ P1 × P2 ו••× Pn where pk ∈ Pk ≔ [pk, pk] ⊂ of R, we construct a real test function T(t) of a real variable t ∈ [a, b] such that (1) holds if (a) there is an ƒp ∈ F such that (1) is true and (b) T(t) > 0 for t ∈ [a, b]. As an application, Kharitonov′s theorem [Differentsial′nye Uravneniya14 No. 11 (1978) 2086-2088] is generalized to real exponential polynomes with some coefficients interval-valued. The problem is treated as one belonging to the theory of functions of one complex variable.
