Journal of Dynamical and Control Systems

, Volume 23, Issue 2, pp 317–335 | Cite as

Dynamics Identification in Evolution Models Using Radial Basis Functions

  • Juri MergerEmail author
  • Alfio Borzì


The problem of identifying an unknown function of the state of an evolution model with differential equations is considered in the framework of a minimization problem. The well-posedness of this minimization problem as well as unique solvability is proven. The analysis of the dependence of the identified function on the data is presented by means of the derivative of the “data–to–function” mapping. Moreover, the infinite dimensional function space, where the unknown function is sought, is discretized by suitable radial basis functions that are chosen such that optimal approximation results are obtained. The numerical treatment of a representative evolution model and the application to a bio-chemical model illustrate the proposed approach.


Function identification Infinite dimensional optimization Radial basis functions 



We cordially thank Dr. Christian von Wallbrunn and his team at the Departments of Microbiology & Biochemistry of the Geisenheim University for providing us the experimental data of the wine fermentation.


  1. 1.
    Bard Y. Nonlinear parameter estimation. New York-London: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers]; 1974.zbMATHGoogle Scholar
  2. 2.
    Bock HG. 1987. Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen. Bonner Mathematische Schriften [Bonn Mathematical Publications], 183: Universität Bonn, Mathematisches Institut, Bonn. Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1985.Google Scholar
  3. 3.
    Biegler L, Damiano J, Blau G. AIChE J 1986;32(1):29.CrossRefGoogle Scholar
  4. 4.
    Li Z, Osborne MR, Prvan T. IMA J Numer Anal 2005;25(2):264. doi: Scholar
  5. 5.
    Lorenzi A. Annali di Matematica Pura ed Applicata. Serie Quarta 1982;131:145. doi: 10.1007/BF01765150.Google Scholar
  6. 6.
    Egger H, Pietschmann JF, Schlottbom M. Inverse problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data 2014;30(2):025004, 14. doi: 10.1088/0266-5611/30/2/025004.Google Scholar
  7. 7.
    Egger H, Pietschmann JF, Schlottbom M. Inverse problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data 2014;30(3):035009, 8. doi: 10.1088/0266-5611/30/3/035009.Google Scholar
  8. 8.
    Egger H, Pietschmann JF, Schlottbom M 2014.Google Scholar
  9. 9.
    Egger H, Pietschmann JF, Schlottbom M. SIAM J Appl Math 2015;75(2): 275. doi: 10.1137/140967222.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Burger M, Pietschmann JF, Wolfram MT. Inverse Problems and Imaging 2013;7 (4):1157. doi: 10.3934/ipi.2013.7.1157.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Onyango TTM, Ingham DB, Lesnic D. Comput Math Appl Int J 2008;56 (1):114. doi: 10.1016/j.camwa.2007.11.038.CrossRefGoogle Scholar
  12. 12.
    Engl HW, Fusek P, Pereverzev SV. Journal of Inverse and Ill-Posed Problems 2005;13(3–6):567. doi: 10.1163/156939405775199497. Inverse problems: modeling and simulation.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rösch A, Tröltzsch F. Polish academy of sciences. Committee of automatic control and robotics. Archives of Control Sciences 1992;1(3–4):183.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Rösch A. Numer Funct Anal Optim Int J 1994;15(3-4):417. doi: 10.1080/01630569408816573.CrossRefGoogle Scholar
  15. 15.
    Rösch A. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and its Applications 1996;15(3):603. doi: 10.4171/ZAA/718.MathSciNetGoogle Scholar
  16. 16.
    Rösch A. Inverse problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data 1996;12(5):743. doi: 10.1088/0266-5611/12/5/015.MathSciNetGoogle Scholar
  17. 17.
    Rösch A. Control and estimation of distributed parameter systems (vorau, 1996), internat. Ser. Numer. Math, vol. 126, Birkhäuser, Basel; 1998. p. 237–246.Google Scholar
  18. 18.
    Rösch A. Optimal control of complex structures (Oberwolfach, 2000), Internat. Ser. Numer. Math. Birkhäuser: Basel; 2002. p. 217–230.Google Scholar
  19. 19.
    Hào DN, Huong BV, Thanh PX, Lesnic D. Applicable analysis. an International Journal 2015;94(9):1784. doi: 10.1080/00036811.2014.948425.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kügler P, Engl HW. Journal of inverse and ill-posed problems 2002;10(1):67. doi: 10.1515/jiip.2002.10.1.67.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kügler P. SIAM J Numer Anal 2003;41(4):1543. doi: 10.1137/S0036142902415900.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hömberg D, Liu J, Togobytska N. Math Methods Appl Sci 2012;35(5):497. doi: 10.1002/mma.1585.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hömberg D, Lu S, Sakamoto K, Yamamoto M. Inverse problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data 2014;30(3):035003, 24. doi: 10.1088/0266-5611/30/3/035003.Google Scholar
  24. 24.
    Buhmann MD, Vol. 12. Radial basis functions: theory and implementations, Cambridge Monographs on Applied and Computational Mathematics. Cambridge: Cambridge University Press; 2003.Google Scholar
  25. 25.
    Wendland H. Adv Comput Math 1995;4(4):389. doi: 10.1007/BF02123482.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Adams RA, Fournier JJF. Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol 140, 2nd ed. Amsterdam: Elsevier/Academic Press; 2003.Google Scholar
  27. 27.
    Evans LC. Partial differential equations, Graduate Studies in Mathematics, vol 19, 2nd ed. Providence, RI: American Mathematical Society; 2010.Google Scholar
  28. 28.
    Tröltzsch F. Optimal control of partial differential equations, Graduate Studies in Mathematics, vol 112. Providence: American Mathematical Society; 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.Google Scholar
  29. 29.
    Kosmol P. Optimierung und Approximation, expanded edn. Berlin: Walter de Gruyter & Co.; 2010.CrossRefGoogle Scholar
  30. 30.
    Ciarlet PG. Linear and nonlinear functional analysis with applications. Philadelphia: Society for Industrial and Applied Mathematics; 2013.zbMATHGoogle Scholar
  31. 31.
    Wendland H. Journal of Approximation Theory 1998; 93 (2): 258. doi: 10.1006/jath.1997.3137.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wu ZM, Schaback R. IMA J Numer Anal 1993; 13 (1): 13. doi: 10.1093/imanum/13.1.13.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Murray JD. Mathematical biology, Biomathematics, vol 19, 2nd ed. Berlin: Springer; 1993.Google Scholar
  34. 34.
    Nocedal J, Wright SJ. Numerical optimization, 2nd ed. Springer Series in Operations Research and Financial Engineering. New York: Springer; 2006.Google Scholar
  35. 35.
    Borzì A, Merger J, Müller J, Rosch A, Schenk C, Schmidt D, Schmidt S, Schulz V, Velten K, von Wallbrunn C, et al. arXiv:1412.6068 2014.
  36. 36.
    Merger J, Borzì A, Herzog R. Optim Control Appl Meth 2016. doi: 10.1002/oca.2246.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.University of WürzburgWürzburgGermany

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