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infinite dimensional Volterra integral equation; realization theory; absolute instability; frequency-domain method
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.
[1] Berezanskii, Yu. M.: Eigenfunction Expansion of Self-Adjoint Operators. Naukova Dumka, Kiev (1965), Russian.
[2] Boichenko, V. A., Leonov, G. A., Reitmann, V.: Dimension Theory for Ordinary Differential Equations. Teubner-Texte zur Mathematik 141, Stuttgart (2005). MR 2381409 | Zbl 1094.34002
[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
[4] Gripenberg, G., Londen, S.-O., Staffans, O. J.: Volterra Integral and Functional Equations. Cambridge University Press, Cambridge (1990). MR 1050319 | Zbl 0695.45002
[5] Lions, J. L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971). MR 0271512 | Zbl 0203.09001
[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 MR 2086940
[7] Salamon, D.: Realization theory in Hilbert space. Math. Systems Theory 21 (1989), 147-164. DOI 10.1007/BF02088011 | MR 0977021 | Zbl 0668.93018
[8] Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987). MR 0895589 | Zbl 0623.35006
[9] Yakubovich, V. A.: Frequency-domain conditions for stability of nonlinear integral equations of control theory. Vestn. Leningr. Univ. 7 (1967), 109-125 Russian.
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