Robustness of nonlinear delay equations with respect to input perturbations: a trajectory-based approach.
The authors study the effects of input perturbations \(\eta\) on the stability properties of the delay differential equation (DDE) \[ \dot x(t)=f(t,x(t),x(t-\tau))+\psi(t,x(t),x(t-\tau))\eta(t), \] under the assumptions that the unperturbed DDE \(\dot x(t)=f(t,x(t),x(t-\tau))\) has a globally uniformly asymptotically stable null-solution. Although stability results for delay differential equations are usually obtained within a Lyapunov framework, they study stability and boundedness properties by means of an alternative approach, which is based on an analysis of trajectories by means of the Gronwall Lemma. This alternative approach may be seen as an extension of the trajectory-based approach developed e.g. in [\textit{L. Moreau} and \textit{D. Aeyels}, SIAM J. Control Optimization 41, No. 6, 1922--1945 (2003; Zbl 1054.34084)] for ordinary differential equations to delay differential equations. Within this framework they have generalized some stability and boundedness results from [\textit{W. Michiels, R. Sepulchre} and \textit{D. Roose}, SIAM J. Control Optimization 40, No. 3, 661--680 (2001; Zbl 1002.34069)] to a much broader class of delay differential equations.
scientific article; zbMATH DE number 1890503