Differential Equations with Fuzzy Parameters via Differential Inclusions

Abstract We give a definition of solutions of ordinary differential equations in R n containing parameters which are described by changing in time fuzzy sets. They are defined as fuzzy subsets of the space of absolutely continuous functions. We introduce a hypograph metric in the space of fuzzy sets and prove a theorem on continuous dependence of fuzzy solutions on parameters and initial conditions with respect to that metric.

[1]  M. Puri,et al.  Differentials of fuzzy functions , 1983 .

[2]  M. Kisielewicz.  Differential Inclusions and Optimal Control , 1991 .

[3]  Abraham Kandel,et al.  On the validity of the Peano theorem for fuzzy differential equations , 1997, Fuzzy Sets Syst..

[4]  Siegfried Gottwald,et al.  Applications of fuzzy sets to systems analysis , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[5]  Phil Diamond,et al.  Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations , 1999, IEEE Trans. Fuzzy Syst..

[6]  G. Beer.  Upper semicontinuous functions and the stone approximation theorem , 1982 .

[7]  Phil Diamond,et al.  Regularity of solution sets for differential inclusions quasi-concave in a parameter , 2000, Appl. Math. Lett..

[8]  Osmo Kaleva.  The Cauchy problem for fuzzy differential equations , 1990 .

[9]  Peter E. Kloeden.  Compact supported endographs and fuzzy sets , 1980 .

[10]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[11]  Jong Yeoul Park,et al.  Fuzzy differential equations , 2000, Fuzzy Sets Syst..

[12]  P. Kloeden,et al.  Metric Spaces of Fuzzy Sets: Theory and Applications , 1994 .

[13]  K. Deimling.  Multivalued Differential Equations , 1992 .

[14]  P. Kloeden.  Remarks on Peano-like theorems for fuzzy differential equations , 1991 .

[15]  A natural topology for upper semicontinuous functions and a Baire category dual for convergence in measure. , 1981 .

[16]  James J. Buckley,et al.  Solving fuzzy equations , 1992 .

[17]  P. Kloeden.  Fuzzy dynamical systems , 1982 .

[18]  Eyke Hüllermeier,et al.  An Approach to Modelling and Simulation of Uncertain Dynamical Systems , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[19]  S. Seikkala.  On the fuzzy initial value problem , 1987 .

[20]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[21]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

[22]  V. A. Baidosov.  FUZZY DIFFERENTIAL INCLUSIONS , 1990 .