In this paper the class of distributed bilinear systems in a nonlinear feedback loop is considered. First of all the existence and uniqueness of solution is stated when input and state belong to L2 spaces. Then the existence of the Volterra series expansion for the input-output map is proved for inputs in any sphere around the origin; the proof is based on a new version of the Weierstrass approximation theorem.
Volterra Series Expansion for Bilinear Systems in a Nonlinear Feedback Loop
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Bertuzzi, A., Bruni, C., Gandolfi, A., and Germani, A. (June 1, 1981). "Volterra Series Expansion for Bilinear Systems in a Nonlinear Feedback Loop." ASME. J. Dyn. Sys., Meas., Control. June 1981; 103(2): 84–88. https://doi.org/10.1115/1.3139659
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