Geometric and probabilistic stability criteria for delay systems

A new approach to the study of the stability of delay systems is developed. The method is applicable to biological control systems and other systems where little information about time delays is available. The view proposed is that stability information can be deduced from the statistical properties of the probability distribution that encodes the structure of the time delay. The main statistical variables used are the usual expectation parameter E and a modified variance, called relative variance and denoted R, that is invariant under time-scale changes. In many cases, the stability of a model improves as R increases while E remains fixed. The statistical approach is shown to be closely related to a geometric method of Walther and Cushing that establishes stability in the case of a convex delay distribution function. In fact, it is shown that convex and concave distributions have R values respectively greater than and less than 1/2. A generalized version of the geometric theory is presented that relaxes the smoothness hypothesis on the density function; this brings it more into correspondence with statistical theory, which applies to general distributions irrespective of their smoothness.