Unconditional stability in kinds of 3rd-order delay difference equations (Q2748252)

A necessary and sufficient condition is given for the differential equation NEWLINE\[NEWLINEx'''(t)+ax''(t)+ bx'(t)+cx(t)+dx(t-\tau)=0NEWLINE\]NEWLINE to be asymptotically stable. The condition is cumbersome to write down, but it is a set of algebraic conditions involving the coefficients \(a,b,c,d\) and the delay \(\tau\). The technique for obtaining this set of conditions is by investigating the properties of the roots of the characteristic equation NEWLINE\[NEWLINE\lambda^3+ a\lambda^2 +b\lambda+c+ de^{-\lambda\tau}=0.NEWLINE\]