Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems

Volker Reitmann

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 2, page 185-194
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Reitmann, Volker. "Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems." Mathematica Bohemica 136.2 (2011): 185-194. <http://eudml.org/doc/197211>.

@article{Reitmann2011,
abstract = {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.},
author = {Reitmann, Volker},
journal = {Mathematica Bohemica},
keywords = {infinite dimensional Volterra integral equation; realization theory; absolute instability; frequency-domain method; infinite dimensional Volterra integral equation; realization theory; absolute instability; frequency-domain method},
language = {eng},
number = {2},
pages = {185-194},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems},
url = {http://eudml.org/doc/197211},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Reitmann, Volker
TI - Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 2
SP - 185
EP - 194
AB - 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.
LA - eng
KW - infinite dimensional Volterra integral equation; realization theory; absolute instability; frequency-domain method; infinite dimensional Volterra integral equation; realization theory; absolute instability; frequency-domain method
UR - http://eudml.org/doc/197211
ER -

References

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  1. Berezanskii, Yu. M., Eigenfunction Expansion of Self-Adjoint Operators, Naukova Dumka, Kiev (1965), Russian. (1965) 
  2. Boichenko, V. A., Leonov, G. A., Reitmann, V., Dimension Theory for Ordinary Differential Equations, Teubner-Texte zur Mathematik 141, Stuttgart (2005). (2005) Zbl1094.34002MR2381409
  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. (1977) MR0477622
  4. Gripenberg, G., Londen, S.-O., Staffans, O. J., Volterra Integral and Functional Equations, Cambridge University Press, Cambridge (1990). (1990) Zbl0695.45002MR1050319
  5. Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin (1971). (1971) Zbl0203.09001MR0271512
  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/. (2004) MR2086940
  7. Salamon, D., 10.1007/BF02088011, Math. Systems Theory 21 (1989), 147-164. (1989) Zbl0668.93018MR0977021DOI10.1007/BF02088011
  8. Wloka, J., Partial Differential Equations, Cambridge University Press, Cambridge (1987). (1987) Zbl0623.35006MR0895589
  9. Yakubovich, V. A., Frequency-domain conditions for stability of nonlinear integral equations of control theory, Vestn. Leningr. Univ. 7 (1967), 109-125 Russian. (1967) 

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