, Volume 81, Issue 1, pp 91–106 | Cite as

Numerical fixed grid methods for differential inclusions

  • W.-J. BeynEmail author
  • J. Rieger


Numerical methods for initial value problems for differential inclusions usually require a discretization of time as well as of the set valued right hand side. In this paper, two numerical fixed grid methods for the approximation of the full solution set are proposed and analyzed. Convergence results are proved which show the combined influence of time and (phase) space discretization.


differential inclusions numerical methods 

AMS Subject Classifications

34A60 49J53 65L20 


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  1. Aubin J.P. and Cellina A. (1984). Differential inclusions. Grundlehren der mathematischen Wissenschaften, vol. 264. Springer, Berlin Google Scholar
  2. Chahma I.A. (2003). Set-valued discrete approximation of state-constrained differential inclusions. Bayreuth Math Schr 67: 3–161 MathSciNetGoogle Scholar
  3. Dontchev A. and Farkhi E. (1988). Error estimates for discretized differential inclusions. Computing 41: 349–358 CrossRefMathSciNetGoogle Scholar
  4. Dontchev A. and Lempio F. (1992). Difference methods for differential inclusions: a survey. SIAM Rev 34(2): 263–294 zbMATHCrossRefMathSciNetGoogle Scholar
  5. Grammel G. (2003). Towards fully discretized differential inclusions. Set-Valued Anal 11: 1–8 zbMATHCrossRefMathSciNetGoogle Scholar
  6. Grüne L. (2002). Asymptotic behavior of dynamical systems and control systems under perturbation and discretization. Springer, Heidelberg zbMATHGoogle Scholar
  7. Grüne L. and Junge O. (2005). A set oriented approach to optimal feedback stabilization. Sys Control Lett 54: 169–180 zbMATHCrossRefGoogle Scholar
  8. Junge O. and Osinga H. (2004). A set oriented approach to global optimal control. ESAIM Control Optim Calc Var 10: 259–270 zbMATHCrossRefMathSciNetGoogle Scholar
  9. Komarov V.A. and Pevchikh K.E. (1991). A method of approximating attainability sets for differential inclusions with a prescribed accuracy. USSR Comput Math Math Phys 31(1): 109–112 zbMATHMathSciNetGoogle Scholar
  10. Lempio F. and Veliov V. (1998). Discrete approximations of differential inclusions. Bayreuth Math Schr 54: 149–232 zbMATHMathSciNetGoogle Scholar
  11. Murray J.D. (1989). Mathematical biology. Springer, Berlin zbMATHGoogle Scholar
  12. Szolnoki D. (2003). Set oriented methods for computing reachable sets and control sets. Discrete Contin Dyn Sys Ser B 3(3): 361–382 zbMATHMathSciNetCrossRefGoogle Scholar
  13. Warga J. (1972). Optimal control of differential and functional equations. Academic Press, New York zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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