Abstract
Recently, two of the authors have introduced a combinatorial resolution of systems of forced differential equations [8,9,10]. In particular this approach allows to obtain a formal expansion for these equations with a richer structure than the Fliess generating power series [4]. In this paper we obtain approximations by imposing a bound p on the possible widths of trees and hedges. We show that these approximants satisfy a bilinear system and possess the same Volterra kernels as the exact solution, up to order p. They are in fact equivalent to approximants introduced by Brockett [1] using a truncated Carleman linearization [2]. Recall that using these approximations, Krener has shown then that a nonlinear system with control entering linearly is locally almost bilinear [7]. Besides the mathematical interest, the relevance of this combinatorial approach is that it provides a clear iterative scheme in order to find the functional expansion of the solution and that it should lead to efficient computer tools for analyzing the behavior of the solution around equilibrium points.
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partially supported by “Action Incitative” CNRS/NSF no 0693
with the financial support of FCAR (Quebec) and NSERC (Canada)
partially supported by PRC “Mathématiques et Informatique”.
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© 1991 Birkhäuser Boston
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Lamnabhi-Lagarrigue, F., Leroux, P., Viennot, X.G. (1991). Combinatorial Approximations of Volterra Series by Bilinear Systems. In: Bonnard, B., Bride, B., Gauthier, JP., Kupka, I. (eds) Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory, vol 8. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3214-8_27
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DOI: https://doi.org/10.1007/978-1-4612-3214-8_27
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