Solvability of a transcendental problem in system theory

The author gives a complete solution for a special transcendental problem in system theory: given three polynomials F₁(s), F₂(s), and F₃(s), when do there exist two Hurwitz polynomials H₁(s), H₂( s) and one polynomial H´₃(s) having no real roots in the closed right half of the complex plane such that F₁(s) H₁(s)+F₂(s) H₂(s)+F₃(s) H´₃(s)=0? It is found that the solvability of the transcendental problem depends solely on the real unstable roots of the three given polynomials. More precisely, those roots must have a special interlacing property called the strong interlacing property, which is introduced. For a triplet of polynomials having the strong interlacing property, a computation procedure for solving the problem is provided and illustrated via an example