Robust stability for a class of linear systems with time-varying delay and nonlinear perturbations.

*(English)*Zbl 1154.93408The 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.

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.

Reviewer: M. J. Alves (Maputo)

##### MSC:

93D09 | Robust stability |

93C23 | Control/observation systems governed by functional-differential equations |

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\textit{Q.-L. Han}, Comput. Math. Appl. 47, No. 8--9, 1201--1209 (2004; Zbl 1154.93408)

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