Robust stability for a class of linear systems with time-varying delay and nonlinear perturbations
The author investigates the robust stability of uncertain systems with a single time-varying discrete delay \[ x'(t)=Ax(t)+Bx(t-h(t))+f(x(t),t)+g(x(t-h(t)),t),\tag{1} \] where \(f\) and \(g\) represent unknown perturbations and obtains a stability criteria. Numerical examples show significant improvements over some existing results. Reviewer's remarks: 1) Describing the system (1) the author should indicate the interval where the system is defined, i.e., on the semi-axis \([0,\infty)\). 2) The second line after formula (1) is formally correct, but usually it is written \(f(u,t)\), where \(u\in{\mathbb R}^n\), \(t\geq 0\), instead of \(f(x(t),t)\) that is superposition. 3) In the ``Definition 1'' it is not clear what is meant by \({\mathbb Z}=\{z\}\). 4) The conditions (6a), (6b) are not written in terms of solutions, but in terms of functions \(f\), \(g\), for instance in the form: \[ \|f(u,t)\|\leq\alpha\|u\|, \qquad \|g(u,t)\|\leq\beta\|u\|. \tag{(1)} \] 5) The same remark for inequalities (7a), (7b), in the line before the ``Corollary 1'', after formula (16), and also in the ``Example 1''. 6) Inequality (8): Maybe the author means a negative defined matrix? 7) I suggest to the author to indicate the dimensions of the matrices introduced in ``Theorem 1''. 8) The formulation of ``Lemma 1'' is not concise because there is not the phrase ``there exists'' after \(\gamma\) or after \(\omega\). In (9) instead matrix \(M\) is written the matrix \(Q\) that is not defined above, also \(r_M\), and \(r(t)\) were not defined before. 9) In page 1204, line 7 from bottom, instead ``are the solution of (8)'' I think is better to write ``satisfy (8)''. 10) I did not understand why in the ``Remark 3'' the constant \(h_M\) is linear?. 11) The second phrase from the beginning of Section 4. I did not understand why from bounded by norm \(f\), \(g\) follows the formula (17) (nonlinear equation became linear). 12) Next, in (18) is not explained what means the symbol \(\sigma_{\max}\), perhaps it is the maximal eigenvalue. In the next paragraph ``System (15)'' the citation number is wrong. 13) The phrase after (20) is not adequate: ``rewrite (17), (18) as (21a), (21b)''. Indeed, (18) as I understood, means that the spectral radius \(F(t)\) not more than 1 and (21b) means that \(\|F(t)y\|\leq \|y\|\), i.e., \(\|F(t)\|\leq 1\). But this condition is more strong than (18), i.e., (18)\(\Leftarrow\)(21b), however (18)\(\not\Rightarrow\)(21b). Thus, the word ``rewrite'' is not exact.
scientific article; zbMATH DE number 2142379