A philosophy for the modelling of realistic nonlinear systems

Authors:
Phil Howlett, Anatoli Torokhti and Charles Pearce

Journal:
Proc. Amer. Math. Soc. **132** (2004), 353-363

MSC (2000):
Primary 47H99, 47A58; Secondary 37M05

Published electronically:
August 28, 2003

MathSciNet review:
2022356

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Abstract | References | Similar Articles | Additional Information

Abstract: A nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of *realistic* dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.

**1.**B. Russell,*On the notion of cause*, Proc. Aristotelian Soc.**13**(1913), 1-25.**2.**Raymond E. A. C. Paley and Norbert Wiener,*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142****3.**Y. Fourès and I. E. Segal,*Causality and analyticity*, Trans. Amer. Math. Soc.**78**(1955), 385–405. MR**0069401**, 10.1090/S0002-9947-1955-0069401-5**4.**P. L. Falb and M. I. Freedman,*A generalized transform theory for causal operators*, SIAM J. Control**7**(1969), 452–471. MR**0682782****5.**Jan C. Willems,*Stability, instability, invertibility and causality*, SIAM J. Control**7**(1969), 645–671. MR**0275936****6.**I. C. Gohberg and M. G. Kreĭn,*Theory and applications of Volterra operators in Hilbert space*, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. MR**0264447****7.**Irwin W. Sandberg and Lilian Xu,*Uniform approximation of multidimensional myopic maps*, IEEE Trans. Circuits Systems I Fund. Theory Appl.**44**(1997), no. 6, 477–485. MR**1452221**, 10.1109/81.585959**8.**A. P. Torokhti and P. G. Howlett,*On the constructive approximation of non-linear operators in the modelling of dynamical systems*, J. Austral. Math. Soc. Ser. B**39**(1997), no. 1, 1–27. MR**1462806**, 10.1017/S0334270000009188**9.**P. G. Howlett and A. P. Torokhti,*A methodology for the constructive approximation of nonlinear operators defined on noncompact sets*, Numer. Funct. Anal. Optim.**18**(1997), no. 3-4, 343–365. MR**1448895**, 10.1080/01630569708816764**10.**P. G. Howlett and A. P. Torokhti,*Weak interpolation and approximation of non-linear operators on the space 𝒞([0,1])*, Numer. Funct. Anal. Optim.**19**(1998), no. 9-10, 1025–1043. MR**1656408**, 10.1080/01630569808816872**11.**P. M. Prenter,*A Weierstrass theorem for real, separable Hilbert spaces*, J. Approximation Theory**3**(1970), 341–351. MR**0433214****12.**Vincent J. Bruno,*A Weierstrass approximation theorem for topological vector spaces*, J. Approx. Theory**42**(1984), no. 1, 1–3. MR**757098**, 10.1016/0021-9045(84)90049-2**13.**G. Cybenko,*Approximation by superpositions of a sigmoidal function*, Math. Control Signals Systems**2**(1989), no. 4, 303–314. MR**1015670**, 10.1007/BF02551274**14.**I. K. Daugavet,*On operator approximation by causal operators and their generalizations. II: Nonlinear case (Russian)*, Methods of Optimiz. and their Applic., Irkutsk. Sib. Energ. Institut (1988), 166-178.**15.**Anatoli P. Torokhti and Phil G. Howlett,*On the best quadratic approximation of nonlinear systems*, IEEE Trans. Circuits Systems I Fund. Theory Appl.**48**(2001), no. 5, 595–602. MR**1854954**, 10.1109/81.922461**16.**A. Torokhti and P. Howlett,*Optimal Fixed Rank Transform of the Second Degree*, IEEE Trans. on Circuits and Systems. Part II, Analog and Digital Signal Processing,**48**, No. 3 (2001), 309-315.

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Additional Information

**Phil Howlett**

Affiliation:
Centre for Industrial and Applied Mathematics, University of South Australia, Mawson Lakes, SA 5095, Australia

Email:
p.howlett@unisa.edu.au

**Anatoli Torokhti**

Affiliation:
Centre for Industrial and Applied Mathematics, University of South Australia, Mawson Lakes, SA 5095, Australia.

Email:
a.torokhti@unisa.edu.au

**Charles Pearce**

Affiliation:
Department of Applied Mathematics, University of Adelaide, Adelaide, SA 5005, Australia

Email:
cpearce@maths.adelaide.edu.au

DOI:
http://dx.doi.org/10.1090/S0002-9939-03-07164-8

Keywords:
Operator approximation,
realistic nonlinear systems

Received by editor(s):
September 8, 2000

Published electronically:
August 28, 2003

Additional Notes:
This research was supported by Australian Research Council Grant #A49943121

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2003
American Mathematical Society