A stability result for a scalar neutral equation with multiple delays

For linear delay-differential equations a question of ongoing interest is to determine delay-independent stability conditions, which are conditions on the equation parameters that guarantee exponential stability of solutions for all delays. Most of the delay-independent stability conditions found in the literature are conservative in the sense that they are not sharp even when applied to the scalar case. We derive a sharp delayindependent stability condition for the case of a scalar, multiple-delay neutral equation with delays only in the derivative. Furthermore, our method involves construction of an appropriate equivalent norm, and we show how to use it to modify a recently developed semidiscrete approximation scheme for delay equations with multiple delays. This yields a convergent approximation scheme which also preserves exponential stability uniformly in the discretization parameter. An illustrative numerical example is given.