# High order approximation of implicitly defined maps

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

Approximations for the function φ{symbol} implicitly defined by φ{symbol}(u)=Φ(u, φ{symbol}(u)) are obtained via the iterative scheme φ{symbol}n(u)=Φ(u, φ{symbol}n-1(u)). In this paper the uniform convergence of high order derivatives of φ{symbol}n to the corresponding derivatives of φ{symbol} 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.

Original language English (US) 163-173 11 Annali di Matematica Pura ed Applicata 137 1 https://doi.org/10.1007/BF01789393 Published - Dec 1 1984

### Fingerprint

Higher Order Approximation
Derivatives
Nonlinear control systems
Iterated integral
Nonlinear Control Systems
Higher order derivative
Approximation Theorem
Uniform convergence
Iterative Scheme
Linear Combination
Derivative
Output
Approximation

### All Science Journal Classification (ASJC) codes

• Applied Mathematics

### Cite this

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title = "High order approximation of implicitly defined maps",
abstract = "Approximations for the function φ{symbol} implicitly defined by φ{symbol}(u)=Φ(u, φ{symbol}(u)) are obtained via the iterative scheme φ{symbol}n(u)=Φ(u, φ{symbol}n-1(u)). In this paper the uniform convergence of high order derivatives of φ{symbol}n to the corresponding derivatives of φ{symbol} 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.",
author = "Alberto Bressan",
year = "1984",
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doi = "10.1007/BF01789393",
language = "English (US)",
volume = "137",
pages = "163--173",
journal = "Annali di Matematica Pura ed Applicata",
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publisher = "Springer Verlag",
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}

In: Annali di Matematica Pura ed Applicata, Vol. 137, No. 1, 01.12.1984, p. 163-173.

Research output: Contribution to journalArticle

TY - JOUR

T1 - High order approximation of implicitly defined maps

AU - Bressan, Alberto

PY - 1984/12/1

Y1 - 1984/12/1

N2 - Approximations for the function φ{symbol} implicitly defined by φ{symbol}(u)=Φ(u, φ{symbol}(u)) are obtained via the iterative scheme φ{symbol}n(u)=Φ(u, φ{symbol}n-1(u)). In this paper the uniform convergence of high order derivatives of φ{symbol}n to the corresponding derivatives of φ{symbol} 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.

AB - Approximations for the function φ{symbol} implicitly defined by φ{symbol}(u)=Φ(u, φ{symbol}(u)) are obtained via the iterative scheme φ{symbol}n(u)=Φ(u, φ{symbol}n-1(u)). In this paper the uniform convergence of high order derivatives of φ{symbol}n to the corresponding derivatives of φ{symbol} 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.

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JF - Annali di Matematica Pura ed Applicata

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