DELAY-INDEPENDENT STABILITY OF A SPECIAL SEQUENCE OF NEUTRAL DIFFERENCE DIFFERENTIAL-EQUATIONS WITH ONE DELAY

The equations of the title for the state X: [-T, ∞] → RL[X]: = [(D - a)(D - a)]n X(t) + B·[(D - b)(D - b)]n X(t - T) = 0, D: = d dt, (1) are considered in the parameter space (a, b, B, n) ε{lunate} P where P is defined by Re(a) < 0, Im(a) ≥ 0, Im(b) ≥ 0, Bε{lunate}R, nε{lunate}N. If a ε{lunate} R, then a is a second real point a ≠ a. For a ε{lunate} R, a is the complex conjugate to a ε{lunate} C. The same notation applies for b. By a tedious but elementary analysis, the set S0 ⊂ P for which (1) has for all T ≥ 0 only asymptotically stable solutions in the sense of Liapunov is explicitly determined in the form |B| < Mn(a, b) ≤ 1. (2) The stability boundary decays exponentially with respect to the multiplicity, n, of the zeros of the coefficient polynomials of (1). We generalize a theorem of F. Brauer [J. Differential Equations 69 (1987), 185-191] dealing with delay-independent stability for characteristic equations of the form H(z): = A(z) + B(z) exp(-zT), T ≥ 0, (3) where A(z), B(z) are holomorphic in Re(z) ≥ 0 in such a way that the neutral case can be dealt with. © 1991.