Exact stability test of neutral delay differential equations via a rough estimation of the testing integral

Abstract

In this paper, we study the problem of stability test of neutral delay differential equations. Firstly we prove that the Mikhailov stability criterion and its equivalent integral form for ordinary differential equations hold for neutral delay differential equations in general form. The criteria are simple in form that is easy for numerical implementation. However, the criteria are characterized by an auxiliary function associated with the characteristic function, not by the characteristic function itself. To reduce the computational complexity, we further prove that a Mikhailov-type criterion in terms of the characteristic function holds. With this new criterion, the stability of a given neutral delay differential equation can be tested with a rough estimation of the testing integral. Thus, the computational complexity and computational cost can be greatly reduced. As two applications of these criteria, we firstly propose a numerical scheme for calculating the rightmost characteristic root(s) as well as the characteristic roots other than the rightmost roots of a given neutral delay differential equation, demonstrated with two examples. Then we derive a graphical stability criterion. With this graphical stability criterion, it is not required to know the exact curve of the Nyquist plot, but just to know whether the Nyquist plot encircles the origin of the complex plane or not.