Direct stability boundary method for distributed systems with discrete delay

The distributed systems considered are those with a characteristic equation expressed in terms of u = √s, where s is the usual Laplace transform variable. The discrete delay h enters the characteristic equation as a term or terms with a factor exp(−sh). The stability boundary method requires four steps: (a) the stability or instability at h = 0 is determined; (b) values of u taking the form u = exp (iπ/4)v, v real, at which a change of stability may take place, are determined by solving a polynomial equation in v; (c) the sign of the gradient d Res/dh is deduced simply at each of these values; (d) finally, the h values corresponding to the critical values of v are found (or possibly just arranged in order, if qualitative results suffice). Stability changes require a consistent sequence of the signs of d Res/dh as h increases.