Abstract
To predict the future state of a physical system, we must know the differential equations \(\dot x=f(x)\) that describe how this state changes with time. In many practical situations, we can observe individual trajectories x(t). By differentiating these trajectories with respect to time, we can determine the values of f(x) for different states x; if we observe many such trajectories, we can reconstruct the function f(x). However, in many other cases, we do not observe individual systems, we observe a set X of such systems. We can observe how this set X changes, but not how individual states change. In such situations, we need to reconstruct the function f(x) based on the observations of such “set trajectories” X(t). In this paper, we show how to extend the standard differentiation techniques of reconstructing f(x) from vector-valued trajectories x(t) to general set-valued trajectories X(t).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Assev, S.M.: Quasilinear operators and their application in the theory of multivalued mappings. Proceedings of the Steklov Institute of Mathematics 2, 23–52 (1986)
Aubin, J.P., Franskowska, H.: Set-Valued Analysis. Birkhäuser, Basel (1990)
Aubin, J.P., Franskowska, H.: Introduction: Set-valued analysis in control theory. Set-Valued Analysis 8, 1–9 (2000)
Banks, H.T., Jacobs, M.Q.: A differential calculus for multifunctions. Journal of Mathematical Analysis and Applications 29, 246–272 (1970)
Bede, B.: Note on Numerical solutions of fuzzy differential equations by predictor-corrector method. Information Sciences 178, 1917–1922 (2008)
Bede, B., Gal, S.G.: Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equation. Fuzzy Sets and Systems 151, 581–599 (2005)
Bede, B., Rudas, I.J., Bencsik, A.L.: First order linear fuzzy differential equations under generalized differentiability. Information Sciences 177, 1648–1662 (2007)
Chalco-Cano, Y., Román-Flores, H.: On the new solution of fuzzy differential equations. Chaos, Solitons & Fractals 38, 112–119 (2008)
Chalco-Cano, Y., Román-Flores, H.: Comparation between some approaches to solve fuzzy differential equations. Fuzzy Sets and Systems 160, 1517–1527 (2008)
De Blasi, F.S.: On the differentiability of multifunctions. Pacific Journal of Mathematics 66, 67–81 (1976)
Hukuhara, M.: Integration des applications mesurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj 10, 205–223 (1967)
Ibrahim, A.-G.M.: On the differentiability of set-valued functions defined on a Banach space and mean value theorem. Applied Mathematics and Computers 74, 76–94 (1996)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall, Upper Saddle River (1995)
Li, D.G., Chen, X.E.: Improvement of definitions of one-element rough function and binary rough function and investigation of their mathematical analysis properties. Journal of Shanxi University 23(4), 318–321 (2000)
Li, D.G., Hu, G.R.: Definitions of n-element rough function and investigation of its mathematical analysis properties. Journal of Shanxi University 24(4), 299–302 (2001)
Nguyen, H.T., Kreinovich, V.: How to divide a territory? a new simple differential formalism for optimization of set functions. International Journal of Intelligent Systems 14(3), 223–251 (1999)
Nguyen, H.T., Walker, E.A.: A First Course in Fuzzy Logic. Chapman & Hall/CRC, Boca Raton (2006)
Pawlak, Z.: Rough functions. Bulletin of the Polish Academy of Sciences, Technical Series PAS Tech. Ser. 355(5-6), 249–251 (1997)
Pawlak, Z.: Rough sets, rough function and rough calculus. In: Pal, S.K., Skowron, A. (eds.) Rough Fuzzy Hybridization, pp. 99–109. Springer, Berlin (1999)
Pawlak, Z., Pal, S.K., Skowron, A.: Rough-Fuzzy Hybridization: A New Trend in Decision-Making. Springer, New York (1999)
Stefanini, L., Bede, B.: Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Analysis 71, 1311–1328 (2009)
Wang, Y., Guan, Y.Y., Wang, H.: Rough derivatives in rough function model and their application. In: Proc. of the 4th International Conference on Fuzzy Systems and Knowledge Discovery, vol. 3, pp. 193–197 (2007)
Wang, Y., Wang, J.M., Guan, Y.Y.: The theory and application of rough integration in rough function model. In: Proc. of the 4th International Conference on Fuzzy Systems and Knowledge Discovery, vol. 3, pp. 224–228 (2007)
Wang, Y., Xu, X., Yu, Z.: Notes for rough derivatives and rough continuity in rough function model. In: Proceedings of the 7th International Conference on Fuzzy Systems and Knowledge Discovery FSKD 2010, Yantai, China, August 10-12, pp. 245–247. IEEE Press, Los Alamitos (2010)
Zhuo, Z.Q.: Improvement of definition of rough function and proof of rough derivatives properties in rough set theory. Journal of Huaibei Coal Normal University 23(3), 10–15 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Villaverde, K., Kosheleva, O. (2011). How to Reconstruct the System’s Dynamics by Differentiating Interval-Valued and Set-Valued Functions. In: Kuznetsov, S.O., Ślęzak, D., Hepting, D.H., Mirkin, B.G. (eds) Rough Sets, Fuzzy Sets, Data Mining and Granular Computing. RSFDGrC 2011. Lecture Notes in Computer Science(), vol 6743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21881-1_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-21881-1_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21880-4
Online ISBN: 978-3-642-21881-1
eBook Packages: Computer ScienceComputer Science (R0)