On zeroes of some transcendental equations.

*(English)*Zbl 0603.34069The equations treated in this paper are of the form (*) \(P(z)+Q(z)e^{- Tz}=0\) and arise from the consideration of delay-differential equations with a single delay T, \(T\geq 0\). The equation is called stable if all zeroes lie in \(Re(z)<0\) and unstable if at least one zero lies in \(Re(z)>0\). A stability switch is said to occur if the equation changes from stable to unstable, or vice-versa, as T varies through particular values. The main result is the following theorem.

Assume that: (i) P(z), Q(z) are analytic functions in \(Re(z)>\delta\) \((\delta >0)\) which have no common imaginary zero; (ii) the conjugates of P(-iy) and Q(-iy) are P(iy) and Q(iy) for real y; (iii) \(P(0)+Q(0)\neq 0;\) (iv) there are at most a finite number of zeroes of P(z) \(+Q(z)\) in the right half-plane; (v) \(F(y)\equiv | P(iy)|^ 2-| Q(iy)|^ 2\) for real y, has at most a finite number of real zeroes.

Under these conditions, the following statements are true: (a) If the equation \(F(y)=0\) has no positive roots, then if (*) is stable [unstable] at \(T=0\) it remains stable [unstable] for all \(T\geq 0.\)

(b) Suppose that \(F(y)=0\) has at least one positive root and each positive root is simple. As T increases, stability switches may occur. There exists a positive \(T^*\) such that (*) is unstable for all \(T>T^*\). As T varies from 0 to \(T^*\), at most a finite number of stability switches may occur.

Assume that: (i) P(z), Q(z) are analytic functions in \(Re(z)>\delta\) \((\delta >0)\) which have no common imaginary zero; (ii) the conjugates of P(-iy) and Q(-iy) are P(iy) and Q(iy) for real y; (iii) \(P(0)+Q(0)\neq 0;\) (iv) there are at most a finite number of zeroes of P(z) \(+Q(z)\) in the right half-plane; (v) \(F(y)\equiv | P(iy)|^ 2-| Q(iy)|^ 2\) for real y, has at most a finite number of real zeroes.

Under these conditions, the following statements are true: (a) If the equation \(F(y)=0\) has no positive roots, then if (*) is stable [unstable] at \(T=0\) it remains stable [unstable] for all \(T\geq 0.\)

(b) Suppose that \(F(y)=0\) has at least one positive root and each positive root is simple. As T increases, stability switches may occur. There exists a positive \(T^*\) such that (*) is unstable for all \(T>T^*\). As T varies from 0 to \(T^*\), at most a finite number of stability switches may occur.

##### MSC:

34K20 | Stability theory of functional-differential equations |

30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |