The existence and uniqueness of a mild solution to nonlinear fuzzy differential equation constrained by initial value were proven. Initial value constraint was then replaced by delay function constraint and the existence of a solution to this type of problem was also proven. Furthermore, the existence of a solution to optimal control problem of the latter type of equation was proven.
References
C. Y. Shin and P. P. Wang, “Economic applications of fuzzy subset theory and fuzzy logic: a brief survey,” New Mathematics and Natural Computation, vol. 6, no. 3, pp. 301–320, 2010. MR2725258 10.1142/S1793005710001773 1231.91077 C. Y. Shin and P. P. Wang, “Economic applications of fuzzy subset theory and fuzzy logic: a brief survey,” New Mathematics and Natural Computation, vol. 6, no. 3, pp. 301–320, 2010. MR2725258 10.1142/S1793005710001773 1231.91077
N. H. Phuong and V. Kreinovich, “Fuzzy logic and its applications in medicine,” International Journal of Medical Informatics, vol. 62, no. 2-3, pp. 165–173, 2001. N. H. Phuong and V. Kreinovich, “Fuzzy logic and its applications in medicine,” International Journal of Medical Informatics, vol. 62, no. 2-3, pp. 165–173, 2001.
A. V. Plotnikov and T. A. Komleva, “Linear problems of optimal control of fuzzy maps,” Intelligent Information Management, vol. 1, no. 3, pp. 139–144, 2009. A. V. Plotnikov and T. A. Komleva, “Linear problems of optimal control of fuzzy maps,” Intelligent Information Management, vol. 1, no. 3, pp. 139–144, 2009.
C. Duraisamy and B. Usha, “Another approach to solution of fuzzy differential equations,” Applied Mathematical Sciences, vol. 4, no. 16, pp. 777–790, 2010. MR2595516 1191.65085 C. Duraisamy and B. Usha, “Another approach to solution of fuzzy differential equations,” Applied Mathematical Sciences, vol. 4, no. 16, pp. 777–790, 2010. MR2595516 1191.65085
D. Bauerova, “Fuzzy modelling of mortgage loans,” http://www.polytech.univ-savoie.fr/fileadmin/polytech$\_$autres$\_$sites/sites/listic/busefal/Papers/69.zip/69$\_$02.pdf. http://www D. Bauerova, “Fuzzy modelling of mortgage loans,” http://www.polytech.univ-savoie.fr/fileadmin/polytech$\_$autres$\_$sites/sites/listic/busefal/Papers/69.zip/69$\_$02.pdf. http://www
G. M. Schoen, “Stability and stabilization of time-delay systems,” Diss. ETH No. 11166, Swiss Federal Institute of Technology, Zurich, Switzerland, 1995. G. M. Schoen, “Stability and stabilization of time-delay systems,” Diss. ETH No. 11166, Swiss Federal Institute of Technology, Zurich, Switzerland, 1995.
H. T. Nguyen, “A note on the extension principle for fuzzy sets,” Journal of Mathematical Analysis and Applications, vol. 64, no. 2, pp. 369–380, 1978. MR0480044 0377.04004 10.1016/0022-247X(78)90045-8 H. T. Nguyen, “A note on the extension principle for fuzzy sets,” Journal of Mathematical Analysis and Applications, vol. 64, no. 2, pp. 369–380, 1978. MR0480044 0377.04004 10.1016/0022-247X(78)90045-8
J. Y. Park and H. K. Han, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 110, no. 1, pp. 69–77, 2000. MR1748109 10.1016/S0165-0114(98)00150-X 0946.34055 J. Y. Park and H. K. Han, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 110, no. 1, pp. 69–77, 2000. MR1748109 10.1016/S0165-0114(98)00150-X 0946.34055
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965. MR219427 0139.24606 10.1016/S0019-9958(65)90241-X L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965. MR219427 0139.24606 10.1016/S0019-9958(65)90241-X
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, Germany, 1971. MR0271512 0203.09001 J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, Germany, 1971. MR0271512 0203.09001
M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 552–558, 1983. MR690888 0528.54009 10.1016/0022-247X(83)90169-5 M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 552–558, 1983. MR690888 0528.54009 10.1016/0022-247X(83)90169-5
A. V. Plotnikov and T. A. Komleva, “Fuzzy quasi-differential equations in connection with the control problems,” International Journal of Open Problems in Computer Science and Mathematics, vol. 3, no. 4, 2010. A. V. Plotnikov and T. A. Komleva, “Fuzzy quasi-differential equations in connection with the control problems,” International Journal of Open Problems in Computer Science and Mathematics, vol. 3, no. 4, 2010.
Y.-K. Chang, W.-T. Li, and J. J. Nieto, “Controllability of evolution differential inclusions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 2, pp. 623–632, 2007. MR2317194 10.1016/j.na.2006.06.018 1128.93005 Y.-K. Chang, W.-T. Li, and J. J. Nieto, “Controllability of evolution differential inclusions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 2, pp. 623–632, 2007. MR2317194 10.1016/j.na.2006.06.018 1128.93005
O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987. MR919058 0646.34019 10.1016/0165-0114(87)90029-7 O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987. MR919058 0646.34019 10.1016/0165-0114(87)90029-7
S. S. L. Chang and L. A. Zadeh, “On fuzzy mapping and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 2, no. 1, pp. 30–34, 1972. MR0363548 0305.94001 10.1109/TSMC.1972.5408553 S. S. L. Chang and L. A. Zadeh, “On fuzzy mapping and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 2, no. 1, pp. 30–34, 1972. MR0363548 0305.94001 10.1109/TSMC.1972.5408553
B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005. \endinput MR2126175 10.1016/j.fss.2004.08.001 1061.26024 B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005. \endinput MR2126175 10.1016/j.fss.2004.08.001 1061.26024
