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**Linear Control Problems of the Fuzzy Maps** ()

In the present paper, we show the some properties of the fuzzy R-solution of the control linear fuzzy differential inclu-sions and research the optimal time problems for it.

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A. Plotnikov, T. Komleva and I. Molchanyuk, "Linear Control Problems of the Fuzzy Maps,"

*Journal of Software Engineering and Applications*, Vol. 3 No. 3, 2010, pp. 191-197. doi: 10.4236/jsea.2010.33024.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | A. Marchaud, “Sur les champs de demicones et equations differentielles du premier order,” Bulletin of Mathemati-cal Society, France, No. 62, pp. 1–38, 1934. |

[2] | S. C. Zaremba, “Sur une extension de la notion d’equation differentielle,” Comptes Rendus l’Académie des Sciences, Paris, No. 199, pp. 1278–1280, 1934. |

[3] | T. Wazewski, “Systemes de commande et equations au contingent,” Bulletin L’Académie Polonaise des Science, SSMAP, No. 9, pp. 151–155, 1961. |

[4] | T. Wazewski, “Sur une condition equivalente e l’equation au contingent,” Bulletin L’Académie Polonaise des Sci-ence, SSMAP, No. 9, pp. 865–867, 1961. |

[5] | A. F. Filippov, “Classical solutions of differential equa-tions with multi-valued right-hand side,” SIAM Journal of Control, No. 5, pp. 609–621, 1967. |

[6] | J.-P. Aubin and A. Cellina, “Differential inclusions. Set-valued maps and viability theory,” Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. |

[7] | N. Kikuchi, “On contingent equations,” Japan-United States Seminar on Ordinary Differential and Functional Equations, Lecture Notes in Mathematics, Springer, Ber-lin, Vol. 243, pp. 169–181, 1971. |

[8] | V. A. Plotnikov, A. V. Plotnikov, and A. N. Vityuk, “Differential equations with multivalued right-hand sides,” Asymptotics Methods, AstroPrint, Odessa, 1999. |

[9] | G. V. Smirnov, “Introduction to the theory of differential inclusions,” Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, Vol. 41, 2002. |

[10] | J.-P. Aubin and H. Frankovska, “Set-valued analysis,” Birk- hauser, Systems and Control: Fundations and Applications, 1990. |

[11] | F. S. de Blasi and F. IerVolino, “Equazioni differential- icon soluzioni a valore compatto convesso,” Bollettino della Unione Matematica Italiana, Vol. 2, No. 4–5, pp. 491–501, 1969. |

[12] | A. I. Panasyuk, “Dynamics of sets defined by differential inclusions,” Siberian Mathematical Journal, Vol. 27, No. 5, pp. 155–165, 1986. |

[13] | A. I. Panasyuk, “On the equation of an integral funnel and its applications,” Differential Equations, Vol. 24, No. 11, pp. 1263–1271, 1988. |

[14] | A. I. Panasyuk, “Equations of attainable set dynamics, part 1: Integral funnel equations,” Journal of Optimization Theory and Applications, Vol. 64, No. 2, pp. 349–366, 1990. “Equations of attainable set dynamics part 2: Partial differential equations,” Journal of Optimization Theory and Applications, Vol. 64, No. 2, pp. 367–377, 1990. |

[15] | A. I. Panasyuk and V. I. Panasyuk, “Asymptotic optimi- zation of nonlinear control systems,” Izdatel Belorussia Gosudarstvo University, Minsk, 1977. |

[16] | A. I. Panasjuk and V. I. Panasjuk, “An equation generated by a differential inclusion,” Matematicheskie Zametki, Vol. 27, No. 3, pp. 429–437, 1980. |

[17] | A. I. Panasyuk and V. I. Panasyuk, “Asymptotic turnpike optimization of control systems,” Nauka i Tekhnika, Minsk, 1986. |

[18] | A. A. Tolstonogov, “On an equation of an integral funnel of a differential inclusion,” Matematicheskie Zametki, Vol. 32, No. 6, pp. 841–852, 1982. |

[19] | A. I. Panasyuk, “Quasidifferential equations in a metric space,” Differentsial’nye Uravneniya, Vol. 21, No. 8, pp. 1344–1353, 1985. |

[20] | D. A. Ovsyannikov, “Mathematical methods for the control of beams,” Leningrad University, Leningrad, 1980. |

[21] | V. I. Zubov, “Dynamics of controlled systems,” Vyssh. Shkola, Moscow, 1982. |

[22] | V. I. Zubov, “Stability of motion: Lyapunov methods and their application,” Vyssh. Shkola, Moscow, 1984. |

[23] | S. Otakulov, “A minimax control problem for differential inclusions,” Soviet Doklady Mathematics, Vol. 36, No. 2, pp. 382–387, 1988. |

[24] | S. Otakulov, “Approximation of the optimal-time prob-lem for controlled differential inclusions,” Cybernetics Systems Analysis, Vol. 30, No. 3, pp. 458–462, 1994. |

[25] | A. V. Plotnikov, “Linear control systems with multivalued trajectories,” Kibernetika, Kiev, No. 4, pp. 130–131, 1987. |

[26] | A. V. Plotnikov, “Compactness of the attainability set of a nonlinear differential inclusion that contains a control,” Kibernetika, Kiev, No. 6, pp. 116–118, 1990. |

[27] | A. V. Plotnikov, “A problem on the control of pencils of trajectories,” Siberian Mathematical Journal, Vol. 33, No. 2, pp. 351–354, 1992. |

[28] | A. V. Plotnikov, “Two control problems under uncertainty conditions,” Cybernet Systems Analysis, Vol. 29, No. 4, pp. 567–573, 1993. |

[29] | A. V. Plotnikov, “Controlled quasi-differential equations and some of their properties,” Differential Equations, Vol. 34, No. 10, pp. 1332–1336, 1998. |

[30] | A. V. Plotnikov, “Necessary optimality conditions for a nonlinear problems of control of trajectory bundles,” Cy-bernetics and System Analysis, Vol. 36, No. 5, pp. 729–733, 2000. |

[31] | A. V. Plotnikov, “Linear problems of optimal control of multiple-valued trajectories,” Cybernetics and System Analysis, Vol. 38, No. 5, pp. 772–782, 2002. |

[32] | A. V. Plotnikov and T. A. Komleva, “Some properties of trajectory bunches of controlled bilinear inclusion,” Ukrainian Mathematical Journal, Vol. 56, No. 4, pp. 586– 600, 2004. |

[33] | A. V. Plotnikov and L. I. Plotnikova, “Two problems of encounter under conditions of uncertainty,” Journal of Applied Mathematics and Mechanics, Vol. 55, No. 5, pp. 618–625, 1991. |

[34] | V. A. Plotnikov and A. V. Plotnikov, “Multivalued dif-ferential equations and optimal control,” Applications of Mathematics in Engineering and Economics, Heron Press, Sofia, pp. 60–67, 2001. |

[35] | L. A. Zadeh, “Fuzzy sets,” Information and Control, No. 8, pp. 338–353, 1965. |

[36] | O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, Vol. 24, No. 3, pp. 301–317, 1987. |

[37] | O. Kaleva, “The Cauchy problem for fuzzy differential equations,” Fuzzy Sets and Systems, Vol. 35, No. 3, pp. 389–396, 1990. |

[38] | O. Kaleva, “The Peano theorem for fuzzy differential equations revisited,” Fuzzy Sets and Systems, Vol. 98, No. 1, pp. 147–148, 1998. |

[39] | O. Kaleva, “A note on fuzzy differential equations,” Nonlinear Analysis, Vol. 64, No. 5, pp. 895–900, 2006. |

[40] | T. A. Komleva, L. I. Plotnikova, and A. V. Plotnikov, “Averaging of the fuzzy differential equations,”Work of the Odessa Polytechnical University, Vol. 27, No. 1, pp. 185–190, 2007. |

[41] | T. A. Komleva, A. V. Plotnikov, and N. V. Skripnik, “Differential equations with set-valued solutions,” Ukrainian Mathematical Journal, Springer, New York, Vol. 60, No. 10, pp. 1540–1556, 2008. |

[42] | V. Lakshmikantham, T. G. Bhaskar, and D. J. Vasundhara, “Theory of set differential equations in metric spaces,” Cambridge Scientific Publishers, Cambridge, 2006. |

[43] | V. Lakshmikantham and R. N. Mohapatra, “Theory of fuzzy differential equations and inclusions,” Series in Mathematical Analysis and Applications, Taylor & Fran-cis Ltd., London, Vol. 6, 2003. |

[44] | J. Y. Park and H. K. Han, “Existence and uniqueness theorem for a solution of fuzzy differential equations,” International Journal of Mathematics and Mathematical Sciences, Vol. 22, No. 2, pp. 271–279, 1999. |

[45] | J. Y. Park and H. K. Han, “Fuzzy differential equations,” Fuzzy Sets and Systems, Vol. 110, No. 1, pp. 69–77, 2000. |

[46] | S. Seikkala, “On the fuzzy initial value problem,” Fuzzy Sets and Systems, Vol. 24, No. 3, pp. 319–330, 1987. |

[47] | D. Vorobiev and S. Seikkala, “Towards the theory of fuzzy differential equations,” Fuzzy Sets and Systems, Vol. 125, No. 2, pp. 231–237, 2002. |

[48] | J.-P. Aubin, “Mutational equations in metric spaces,” Set-Valued Analysis, Vol. 1, No. 1, pp. 3–46, 1993. |

[49] | J.-P. Aubin, “Fuzzy differential inclusions,” Problems of Control and Information Theory, Vol. 19, No. 1, pp. 55– 67, 1990. |

[50] | V. A. Baidosov, “Differential inclusions with fuzzy right- hand side,” Soviet Mathematics, Vol. 40, No. 3, pp. 567–569, 1990. |

[51] | V. A. Baidosov, “Fuzzy differential inclusions,” Journal of Applied Mathematics and Mechanics, Vol. 54, No. 1, pp. 8–13, 1990. |

[52] | E. Hullermeier, “An approach to modeling and simulation of uncertain dynamical systems,” International Journal of Uncertainty, Fuzziness Knowledge-Based Systems, Vol. 5, No. 2, pp. 117–137, 1997. |

[53] | N. D. Phu and T. T. Tung, “Some properties of sheaf- solutions of sheaf fuzzy control problems,” Electronic Journal of Differential Equations, No. 108, pp. 1–8, 2006. http://www.ejde.math.txstate.edu. |

[54] | N. D. Phu and T. T. Tung, “Some results on sheaf-solu-tions of sheaf set control problems,” Nonlinear Analysis, Vol. 67, No. 5, pp. 1309–1315, 2007. |

[55] | N. D. Phu and T. T. Tung, “Existence of solutions of fuzzy control differential equations,” Journal of Sci-Tech Development, Vol. 10, No. 5, pp. 5–12, 2007. |

[56] | I. V. Molchanyuk and A. V. Plotnikov, “Linear control systems with a fuzzy parameter,” Nonlinear Oscillator, Vol. 9, No. 1, pp. 59–64, 2006. |

[57] | V. S. Vasil’kovskaya and A. V. Plotnikov, “Integro- differential systems with fuzzy noise,” Ukrainian Mathe-matical Journal, Vol. 59, No. 10, pp. 1482–1492, 2007. |

[58] | C. V. Negoito and D. A. Ralescu, “Applications of fuzzy sets to systems analysis,” A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont., 1975. |

[59] | M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” Journal of Mathematical Analysis and Applications, No. 114, pp. 409–422, 1986. |

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