Fundamental properties of linear control systems with after-effect. I: The continuous case.

*(English)*Zbl 0671.93004The paper contains considerations of some special questions of qualitative control theory for after-effect systems. The author starts with the problems of controllability which can be interpreted as modifications of approximate controllability. These controllability properties are investigated by using standard methods. The results obtained are applied to the local (“sensitivity”) parametrical identifiability.

Reviewer’s remarks:

(1) The concept of state used by the author is not minimal. [For the minimal state see, e.g., the reviewer’s paper, Dokl. Akad. Nauk BSSR 28, 297-300 (1984; Zbl 0568.93038)].

(2) The author states that the system \(\dot x(t)=A(p)x(t)+Bu(t)\), where \(A(p)=\sum^{\ell}_{j=0}A_ je^{-pjh}\), \(p=d/dt\), is null- completely controllable iff a) rank\((B,A(\lambda)B,...,(A(\lambda))^{n- 1}B)=n\) for some \(\lambda\in {\mathbb{C}}\); b) rank\((I_ n-A(\lambda),B)=n\) for all \(\lambda\in {\mathbb{C}}\) for which the condition a) is not valid. It is known [see the reviewer’s paper, Dokl. Akad. Nauk SSSR 236, 1083-1086 (1977; Zbl 0437.93004)] that the condition a) follows from the condition b). One can find more exact results there [see also the paper of A. Manitius, SIAM J. Control Optimization 19, 516-532 (1981; Zbl 0477.93011)].

(3) The Proposition 2.4: rank\((b,\bar A(\lambda)b,...,(\bar A(\lambda))^{n-1}b)<n\) for some \(\lambda\in {\mathbb{C}}\) \(\Rightarrow\) rank\((I_ n-\bar A(\lambda),b)\leq n\) for \(\lambda\in {\mathbb{C}}\) is not very interesting because the second condition is always valid.

(4) Some results formulated in the paper have already been published.

Reviewer’s remarks:

(1) The concept of state used by the author is not minimal. [For the minimal state see, e.g., the reviewer’s paper, Dokl. Akad. Nauk BSSR 28, 297-300 (1984; Zbl 0568.93038)].

(2) The author states that the system \(\dot x(t)=A(p)x(t)+Bu(t)\), where \(A(p)=\sum^{\ell}_{j=0}A_ je^{-pjh}\), \(p=d/dt\), is null- completely controllable iff a) rank\((B,A(\lambda)B,...,(A(\lambda))^{n- 1}B)=n\) for some \(\lambda\in {\mathbb{C}}\); b) rank\((I_ n-A(\lambda),B)=n\) for all \(\lambda\in {\mathbb{C}}\) for which the condition a) is not valid. It is known [see the reviewer’s paper, Dokl. Akad. Nauk SSSR 236, 1083-1086 (1977; Zbl 0437.93004)] that the condition a) follows from the condition b). One can find more exact results there [see also the paper of A. Manitius, SIAM J. Control Optimization 19, 516-532 (1981; Zbl 0477.93011)].

(3) The Proposition 2.4: rank\((b,\bar A(\lambda)b,...,(\bar A(\lambda))^{n-1}b)<n\) for some \(\lambda\in {\mathbb{C}}\) \(\Rightarrow\) rank\((I_ n-\bar A(\lambda),b)\leq n\) for \(\lambda\in {\mathbb{C}}\) is not very interesting because the second condition is always valid.

(4) Some results formulated in the paper have already been published.

Reviewer: V.Marcenko

##### MSC:

93B05 | Controllability |

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

##### Keywords:

time-delay; after-effect systems; approximate controllability; parametrical identifiability
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\textit{M. De La Sen}, Math. Comput. Modelling 10, No. 7, 473--489 (1988; Zbl 0671.93004)

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