Sufficient conditions for α-stability of linear time-delay systems

The asymptotic stability of time-delayed systems subject to multiple bounded point delays has received important attention in past years (see, for instance, Bourles, H. 1994. International Journal of Control, 59(2): 529-541; De la Sen, M. 2000. Electronics Letters, 36(4): 373-374; Xu, B. 2000. Journal of Mathematical Analysis and Applications, 282: 484-494). It is basically proved that the local a-stability in the delays (i.e., all the, eigenvalues have prefixed strictly negative real parts located in Res less than or equal to -alpha < 0) may be tested for a set of admissible delays including possible zero delays either through a set of Lyapunov matrix inequalities or, equivalently, by checking that an identical number of matrices related to the delayed dynamics are all stability matrices. The result may be easily extended to check the epsilon-asymptotic stability, independent of the delays; that is, for all the delays having any values, the eigenvalues are stable and located in Res <= epsilon --> 0(-) (De la. Sen, 2000; Xu, 2000). This number is 2(r) for a set of r distinct point delays and includes all possible cases of alternate signs for summations for all the matrices of delayed dynamics (Xu, 2000).