Stability Crossing Curves of Shifted Gamma-Distributed Delay Systems

This paper characterizes the stability crossing curves of a class of linear systems with gamma-distributed delay with a gap. First, we describe the crossing set, i.e., the set of frequencies where the characteristic roots may cross the imaginary axis as the parameters change. Then, we describe the corresponding stability crossing curves, i.e., the set of parameters such that there is at least one pair of characteristic roots on the imaginary axis. Such stability crossing curves divide the parameter space $\mathbb{R}_{+}^{2}$ defined by the mean delay and the gap into different regions. Within each such region, the number of characteristic roots on the right half complex plane is fixed. This naturally describes the regions of parameters where the system is stable. The classification of the stability crossing curves is also discussed. Some illustrative examples (Cushing equation in biology, traffic flow models in transportation systems, and control over networks of a simplified helicopter model) are also pr...