In this work we consider the fuzzy dynamic programming problems. For this purpose, using the generalized Hukuhara differentiability for fuzzy functions, new concepts, for example: the fuzzy product, the fuzzy collocation and the Bellman’s principle, we have the neccesary and sufficient conditions.
AgarwalR.P., ReganD.O. and LakshmikanthamV., A stacking theorem approach for fuzzy differential equations, J Nonlinear Anal55 (2003), 299–312.
2.
AgarwalR.P., ÓReganD. and LakshmikanthamV., Viability theory and fuzzy differential equations, Fuzzys and Systems151 (2005), 536–580.
3.
AllahviranlooT., KianiN.A. and MotamediN., Solving fuzzy differential equations by differential transformation method, Information Sciences179 (2009), 956–966.
4.
AllahviranlooT., AmirteimooriA., KhezerlooM. and KhezerlooS., A new method for solving fuzzy volterra integro –differential equations, Australian Journal of Basic and Applied Sciences5(4) (2011), 154–164.
5.
AllahviranlooT., GhanbariM., HaghiE., HosseinzadehA. and NouraeiR., A note on “Fuzzy linear systems, Journal of Fuzzy Sets and Systems177 (2011), 87–92.
6.
AllahviranlooT., SalahshourS. and AbbasbandyS., Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation17 (2012), 1372–1381. doi: 10.1016/j.cnsns.2011.07.005
7.
AllahviranlooT., AbbasbandyS., SedaghatfarO. and DarabiP., A new method for solving fuzzy integro-differential equation under generalized differentiability, J Neural Computing and Applications21 (2012), 191–196. doi: 10.1007/s00521-011-0759-3
8.
BarrosL.C., BassaneziR.C. and TonelliP.A., Fuzzy modeling in population dynamics, J Ecological Modeling128 (2000), 27–33.
9.
BedeB. and GalS.G., Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, J Fuzzy Sets and Systems151 (2005), 581–599.
10.
BedeB., RudasI.J. and BencsikA.L., First order linear fuzzy differential equations under generalized differentiability, J Information Sciences177 (2007), 1648–1662.
11.
BedeB. and RudasI.J., Approximation properties of fuzzy transforms, J Fuzzy Sets and Systems180 (2011), 1648–1662.
12.
BedeB., Mathematics of Fuzzy Sets and Fuzzy Logic, Springer VerlagBerlin Heidelberg, 2013, p. 276.
13.
BedeB. and StefaniniL., Generalized differentiability of fuzzyvalued functions, Fuzzy Sets and Systems230 (2013), 119–141.
14.
BezdekJ.C., Partern Recognition with Fuzzy Objective Function Algoriths, Plenum Press, New York, 1981.
15.
Chalco-CanoY. and Román-FloresH., On new solutions of fuzzy differential equations, Chaos Solitons Fractals38 (2008), 112–119.
16.
ChangS.S.L. and ZadehL., On fuzzy mapping and control, IEEE Transactions on System, Man and Cybernetics2 (1972), 30–34.
17.
DuboisD. and PradeH., Operation on fuzzy numbers, Int J Syst Sci9 (1978), 613–626.
18.
DuboisD. and PradeH., Towards fuzzy differential calculus, Fuzzy Sets and Systems8 (1982), 225–233.
19.
FengY. and HuL., On the quasi-controllability of continuoustime dynamic fuzzy control systems, Chaos, Solitons and Fractals30 (2006), 177–188.
20.
KalevaO., Fuzzy differential equations, Fuzzy Sets and Systems24 (1987), 301–317.
21.
KhastanA., NietoJ.J. and RodrÖuez–LÃşezR., Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets Syst177 (2011), 20–33.
22.
HoaN.V. and PhuN.D., Fuzzy functional integro-differential equations under generalized H-differentiability, Journal of Intelligent and Fuzzy Systems26 (2014), 2073–2085.
23.
LakshmikanthamV. and LeelaS., Fuzzy differential systems and the new concept of stability, J Nonlinear Dynamics and Systems Theory1 (2001), 111–119.
24.
LakshmikanthamV. and Mohapatra, Theory of fuzzy differential equations and inclusionsTaylor Francis Inc, London, 2003.
25.
LupulescuV., On a class of fuzzy functional differential equations, Fuzzy Sets and Systems160 (2009), 1547–1562.
26.
MalinowskiM.T., Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Applied Mathematics Letters218 (2012), 9427–9437.
27.
NajariyanM. and FarahiM.H., Optimal control of fuzzy linear controlled systems with fuzzy initial conditions, Iranian Journal of Fuzzy Systems10 (2013), 21–35.
28.
NajariyanM. and FarahiM.H., A new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients, Journal of Computational and Applied Mathematics259 (2014), 682–694.
29.
NietoJ.J., The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems102 (1999), 259–262.
30.
NguyenH.T., A note on the extension principle for fuzzy sets, Journal Math Anal Aplications64 (1978), 369–380.
31.
Osuna-GomezR., Chalco-CanoY., Ruflian-LizanaA. and Hernadez JimenezB., Necessary and sufficient conditions for fuzzy optimality problems,20. In Press, Fuzzy Sets and Systems (2015), 20. In Press.
32.
PhuN.D., AnT.V., HoaN.V. and HienN.T., Interval-valued functional di_erential equations under dissipative conditions, J. Advances in Di_erence Equations, (2014), 2014, 198.
33.
PhuN.D. and DungL.Q., On the Stability and Contrllability of Fuzzy Set Control Di_erential Equations, International Journal of Reliability and Safety, Vol 5 Nos 3/4, 320-335, 2011.
34.
PuriM.L. and RalescuD., Di_erential for fuzzy function, Journal of Mathematical Analysis and Applications91 (1983), 552–558.
35.
RamikJ., Duality in fuzzy linear programming: Some new concepts and results, Journal Fuzzy Optimization and Dicision Making, Vol 4 (2005), pp. 25–39.
36.
SongS. and WuC., Existence and uniqueness of solutions to Cauchy problem of fuzzy di_erential equations, Fuzzy Sets and Systems110 (2000), 55–67.
37.
StefaniniS., A generalization of Hukuhara di_erence and division for interval and fuzzy arithmetic, Fuzzy sets syst161 (2010), 1564–1584.
38.
TriP.V., HoaN.V. and PhuN.D., Sheaf fuzzy problems for functional di_erential equations, J, Advanced in Di_erence Equations2014 (2014), 156.
39.
VuH., HoaN.V. and PhuN.D., The local existence of solutions for random fuzzy integro-diff_erential equations under generalized H-di_erentiability, Journal of Intelligent and Fuzzy Systems26 (2014), 2701–2717.
40.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.