Summary
Approximations for the function ϕ implicitly defined by ϕ(u)=Φ(u, ϕ(u)) are obtained via the iterative scheme ϕn(u)=Φ(u, ϕn−1(u)). In this paper the uniform convergence of high order derivatives of ϕn to the corresponding derivatives of ϕ is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.
Article PDF
Similar content being viewed by others
References
J. Dieudonné,Foundations of Modern Analysis, Academic Press, New York, 1960.
M. Fliess,Fonctionnelles causales non linéaires et indéterminées non commutatives, Bull. Soc. Math. France,109 (1981), pp. 3–40.
P. J.Olver,Applications of Lie groups to differential equations, Mathematical Institute, Un. of Oxford, Lecture Notes.
L. V. Ovsiannikov,Group analysis of differential equations, Academic Press, New York, 1982.
F. Treves,Topological vector spaces, distributions and kernels, Academic Press, New York, 1967.
Author information
Authors and Affiliations
Additional information
Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.
Rights and permissions
About this article
Cite this article
Bressan, A. High order approximation of implicitly defined maps. Annali di Matematica pura ed applicata 137, 163–173 (1984). https://doi.org/10.1007/BF01789393
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01789393