Linear Control Problems of the Fuzzy Maps

Affiliation(s)

Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine..

Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine.

Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine..

Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine.

ABSTRACT

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.

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.

Cite this paper

nullA. 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.

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

References

[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.

[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.