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Continuous Approximations of a Class of Piecewise Continuous Systems

Marius-F. Danca

Department of Mathematics and Computer Science, Emanuel University of Oradea, Str. Nufarului nr. 87, 410597 Oradea, Romania

Romanian Institute of Science and Technology, Str. Ciresilor nr. 29, 400487 Cluj-Napoca, Romania

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https://doi.org/10.1142/S0218127415501461Cited by:9 (Source: Crossref)

Abstract

In this paper, we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piecewise continuous functions. By using techniques from the theory of differential inclusions, the underlying piecewise functions can be locally or globally approximated. The approximation results can be used to model piecewise continuous-time dynamical systems of integer or fractional-order. In this way, by overcoming the lack of numerical methods for differential equations of fractional-order with discontinuous right-hand side, unattainable procedures for systems modeled by this kind of equations, such as chaos control, synchronization, anticontrol and many others, can be easily implemented. Several examples are presented and three comparative applications are studied.

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References

V. Acary and B. Brogliato , Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics ( Springer-Verlag , NY , 2008 ) . Crossref, Google Scholar
J.-P. Aubin and A. Cellina , Differential Inclusions: Set-Valued Maps and Viability Theory ( Springer , Berlin , 1984 ) . Crossref, Google Scholar
J.-P. Aubin and H. Frankowska , Set-Valued Analysis ( Birkhäuser , Boston , 1990 ) . Google Scholar
M. A. Aziz-Alaoui and G. Chen , Int. J. Bifurcation and Chaos 12 , 147 ( 2002 ) . Link, Web of Science, Google Scholar
M. Bernardo et al. , Piecewise-Smooth Dynamical Systems ( Springer , NY , 2008 ) . Google Scholar
R. Brawn , IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 40 , 878 ( 1993 ) . Crossref, Web of Science, Google Scholar
R. Caponetto et al. , Fractional Order Systems: Modeling and Control Applications , World Scientific Series on Nonlinear Science, Series A 72 ( World Scientific , Singapore , 2010 ) . Link, Google Scholar
Caputo, M. [1967] "Linear models of dissipation whose Q is almost frequency independent-II," Geophys. J. R. Astron. Soc. 13, 529–539. Reprinted [2007] Fract. Calc. Appl. Anal. 10, 309–324 . Google Scholar
Cort'es, J. [2008] "Discontinuous dynamical systems: A tutorial on solutions, nonsmooth analysis, and stability," IEEE Contr. Syst. Mag. 28, 36–73 . Google Scholar
M.-F. Danca , Regul. Chaot. Dyn. 12 , 1 ( 2007 ) . Crossref, Web of Science, Google Scholar
M.-F. Danca and N. Lung , Appl. Math. Comput. 223 , 101 ( 2013 ) . Crossref, Web of Science, Google Scholar
M.-F. Danca , M. A. Aziz-Alaoui and M. Small , Chin. Phys. B 24 , ( 2015 ) . Crossref, Web of Science, Google Scholar
K. Diethelm , N. J. Ford and A. D. Freed , Nonlin. Dyn. 29 , 3 ( 2002 ) . Crossref, Web of Science, Google Scholar
K. Diethelm , Computing 71 , 305 ( 2003 ) . Crossref, Web of Science, Google Scholar
A. Dontchev and F. Lempio , SIAM Rev. 34 , 263 ( 1992 ) . Crossref, Web of Science, Google Scholar
Dorcak, L. [2002] "Numerical models for the simulation of the fractional-order control systems," [math.OC] , arXiv:math/0204108v1 . Google Scholar
K. Falconer , Fractal Geometry: Mathematical Foundations and Applications ( Wiley , Chichester , 1990 ) . Google Scholar
A. F. Filippov , Differential Equations with Discontinuous Right-Hand Sides ( Kluwer Academic , Dordrecht , 1988 ) . Crossref, Google Scholar
O. Hájek , J. Diff. Eqs. 32 , 149 ( 1979 ) . Crossref, Web of Science, Google Scholar
J. Henderson and O. Ouahab , Mediterr. J. Math. 9 , 453 ( 2012 ) . Crossref, Web of Science, Google Scholar
A. E. Kastner-Maresch , Computing 49 , 45 ( 1992 ) . Crossref, Web of Science, Google Scholar
A. Kaster-Maresch and F. Lempio , Numer. Funct. Anal. Optim. 14 , 555 ( 1993 ) . Crossref, Web of Science, Google Scholar
M. Kunze, Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, eds. M. Wiercigroch and B. de Kraker (World Scientific, Singapore, 2000) pp. 207–235. Link, Google Scholar
F. Lempio and V. Veliov , Bayreuth. Math. Schr. 54 , 149 ( 1998 ) . Google Scholar
C. Li and F. Zeng , Int. J. Bifurcation and Chaos 22 , 1230014-1 ( 2012 ) . Link, Web of Science, Google Scholar
K. B. Oldham and J. Spanier , The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order ( Academic Press , NY , 1974 ) . Google Scholar
I. Podlubny , J. Fract. Calc. Appl. Anal. 5 , 367 ( 2002 ) . Google Scholar
R. Scherer et al. , Comput. Math. Appl. 62 , 902 ( 2011 ) . Crossref, Web of Science, Google Scholar
M. Wiercigroch and B. de Kraker , Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities ( World Scientific , Singapore , 2000 ) . Link, Google Scholar