| Title:
|
Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems (English) |
| Author:
|
Reitmann, Volker |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
136 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
185-194 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the stability and instability investigation of a class of nonlinear Volterra integral equations in a Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in a weighted Sobolev space, which has the same stability properties as the Volterra equation. (English) |
| Keyword:
|
infinite dimensional Volterra integral equation |
| Keyword:
|
realization theory |
| Keyword:
|
absolute instability |
| Keyword:
|
frequency-domain method |
| MSC:
|
34B16 |
| MSC:
|
34C25 |
| MSC:
|
45D05 |
| MSC:
|
47G10 |
| MSC:
|
47H14 |
| MSC:
|
93B15 |
| MSC:
|
93D10 |
| idZBL:
|
Zbl 1224.93025 |
| idMR:
|
MR2856135 |
| DOI:
|
10.21136/MB.2011.141581 |
| . |
| Date available:
|
2011-06-07T11:30:33Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141581 |
| . |
| Reference:
|
[1] Berezanskii, Yu. M.: Eigenfunction Expansion of Self-Adjoint Operators.Naukova Dumka, Kiev (1965), Russian. |
| Reference:
|
[2] Boichenko, V. A., Leonov, G. A., Reitmann, V.: Dimension Theory for Ordinary Differential Equations.Teubner-Texte zur Mathematik 141, Stuttgart (2005). Zbl 1094.34002, MR 2381409 |
| Reference:
|
[3] Brusin, V. A.: Apparatus of abstract differential equations in the investigation of integral equations of Volterra type.Sibirskii Mat. Zhurnal 18 (1977), 1246-1258 Russian. MR 0477622 |
| Reference:
|
[4] Gripenberg, G., Londen, S.-O., Staffans, O. J.: Volterra Integral and Functional Equations.Cambridge University Press, Cambridge (1990). Zbl 0695.45002, MR 1050319 |
| Reference:
|
[5] Lions, J. L.: Optimal Control of Systems Governed by Partial Differential Equations.Springer, Berlin (1971). Zbl 0203.09001, MR 0271512 |
| Reference:
|
[6] Reitmann, V., Kantz, H.: Stability investigation of Volterra integral equations by realization theory and frequency-domain methods.Preprint 61' (2004), Preprint series of the DFG priority program 1114 “Mathematical methods for time series analysis and digital image processing”. Available electronically via http://www.math.uni-bremen.de/zetem/DFG-Schwerpunkt/. MR 2086940 |
| Reference:
|
[7] Salamon, D.: Realization theory in Hilbert space.Math. Systems Theory 21 (1989), 147-164. Zbl 0668.93018, MR 0977021, 10.1007/BF02088011 |
| Reference:
|
[8] Wloka, J.: Partial Differential Equations.Cambridge University Press, Cambridge (1987). Zbl 0623.35006, MR 0895589 |
| Reference:
|
[9] Yakubovich, V. A.: Frequency-domain conditions for stability of nonlinear integral equations of control theory.Vestn. Leningr. Univ. 7 (1967), 109-125 Russian. |
| . |
