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

Volker Reitmann (2011)

Mathematica Bohemica

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.

Resolvents, integral equations, limit sets

Theodore Allen Burton, D. P. Dwiggins (2010)

Mathematica Bohemica

In this paper we study a linear integral equation x ( t ) = a ( t ) - 0 t C ( t , s ) x ( s ) d s , its resolvent equation R ( t , s ) = C ( t , s ) - s t C ( t , u ) R ( u , s ) d u , the variation of parameters formula x ( t ) = a ( t ) - 0 t R ( t , s ) a ( s ) d s , and a perturbed equation. The kernel, C ( t , s ) , satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of C and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations.

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