Robust stability for a class of systems including nonlinear delayed perturbation (Q2707026)

The authors consider the system NEWLINE\[NEWLINE\dot x(t)=Ax(t)+ f\bigl(x(t-h(t)),t\bigr),\;t\in[t_0,\infty),NEWLINE\]NEWLINE where \(A\in\mathbb{R}^{n\times n}\), \(x\in\mathbb{R}^n\), \(h(t)\) and \(f\) are continuous functions with NEWLINE\[NEWLINE0\leq h(t)\leq\overline h\quad\text{and}\quad \bigl\|f(x,t)\bigr\|\leq\beta\|x\|.NEWLINE\]NEWLINE The main result is that the above system is asymptotically stable if NEWLINE\[NEWLINE\beta< \delta_{\min} (Q^{ 1\over 2})/\delta_{\max}(Q^{-{1\over 2}}P),NEWLINE\]NEWLINE where \(\delta_{\min} (\cdot)\) and \(\delta_{\max}(\cdot)\) denote the minimum and the maximum eigenvalue of the matrix \((\cdot)\), respectively, \(Q\) and \(P\) are positive definite matrices that satisfy NEWLINE\[NEWLINEA^TP+PA= -2Q.NEWLINE\]NEWLINE The condition should be slightly revised since there are some gaps in the original result in [\textit{T.-J. Su} and \textit{C.-G. Huang}, IEEE Trans. Autom. Control 37, 1656-1659 (1992; Zbl 0770.93077)] used by the authors [see \textit{B. Xu}, IEEE Trans. Autom. Control 39, 2365 (1994; Zbl 0825.93605) and ibid. 42, 430 (1997; Zbl 0875.93477)].