Pontryagin's Minimum Principle for Fuzzy Optimal Control Problems

Document Type: Research Paper

Author

Department of Mathematics, Quchan University of Advanced Tech- nologies, Iran

Abstract

The objective of this article is to derive the necessary optimality conditions, known as Pontryagin's minimum principle, for fuzzy optimal control problems based on the concepts of differentiability and integrability of a fuzzy mapping that may be parameterized by the left and right-hand functions of its $\alpha$-level sets.

Keywords


bibitem{Ath} {M. Athans and P. L. Falb, textit{Optimal control: an introduction to the theory and its applications}, New York, McGraw-Hill, Inc.,} 1966.

%bibitem{Bv} {Brunt B. V., The calculus of variations, Springer-Verlag Heidelberg,} (2004).

bibitem{BF} {J. J. Buckley and T. Feuring, textit{Introduction to
fuzzy partial differential equations}, Fuzzy Sets and Systems,} {bf 105} (1999), 241-248.

%bibitem{Da} {Dacorogna B., Introduction to the calculus of variations, Imperial College Press,} (2004).

bibitem{DP} {D. Dubois and H. Prade, textit{Operation on fuzzy numbers}, Internat. J. Systems Sci.,} {bf 9} (1978), 613-626.

%bibitem{Em} {Emamizadeh B., Decreasing rearrangement and a fuzzy variational problem, Applied Math. Letters,} {bf 18,} (2005), 171-178.

bibitem{far} {B. Farhadinia, textit{Necessary optimality conditions for fuzzy variational problems}, Information Sciences,} {bf 181} (2011), 1348-1357.

bibitem{GV}{R. Goetschel and W. Voxman, textit{Elementary fuzzy calculus}, Fuzzy Sets and
Systems,} {bf 18} (1986), 31-43.

bibitem{huk} {M. Hukuhara, textit{Integration des applications measurables dont la valeur est un compact convexe}, Funkcialaj. Ekvacioj,} {bf 10} (1967), 205-223.

bibitem{hyu}{C. H. Hyun, C. W. Park and S. Kim, textit{Takagi-Sugeno fuzzy model based indirect adaptive fuzzy observer
and controller design}, Information Sciences,} {bf 180} (2010), 2314-2327.

%bibitem{K} {Kirk D. E., Optimal control theory, an introduction, Dover Publications,} (2004).

bibitem{Kl} {G. J. Klir and B. Yuan, textit{Fuzzy sets and fuzzy logic-theory and applications}, Prentice-Hall Inc.,} 1995.

bibitem{ku}{C. C. Ku, P. H. Huang and W. J. Chang, textit{Passive fuzzy controller design for nonlinear systems
with multiplicative noises}, J. Franklin Institute,} {bf 347} (2010), 732-750.

bibitem{kwu}{Y. C. Kwun, J. S. Kim, M. J. Park and J. H.Park, textit{Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space}, Advances in Difference Equations,} doi:10.1155/2009/734090, {bf 2009}.

bibitem{liu}{B. Liu, textit{Fuzzy process, hybrid process and uncertain process}, J. Un-
certain Systems,} {bf 2} (2008), 3-16.

bibitem{liu2}{B. Liu, textit{Uncertainty theory}, 2nd Ed., Springer-Verlag, Berlin,} 2007.

bibitem{Mo} {B. S. Mordukhovich and L. Wang, textit{Optimal control of constrained delay-differential
inclusions with multivalued initial conditions}, J. Control Cybernet.,} {bf 32} (2003), 585-609.

bibitem{phu} {N. D. Phu and T. T. Tung, textit{Existence of solutions of fuzzy control differential equations}, J. Science and Technology
Development,} {bf 10} (2007), 5-12.

bibitem{tak} {T. Takagi and M. Sugeno, textit{Fuzzy identification of systems and its applications to modeling and control}, IEEE Trans. on SMC,} {bf 15} (1985), 116-132.

bibitem{yan} {D. Yang and K. Y. Cai, textit{Finite-time quantized guaranteed cost fuzzy control for continuous-time
nonlinear systems}, Expert Systems with Applications,} doi:10.1016/j.eswa.2010.03.024, 2010.

bibitem{Zd} {L. A. Zadeh, textit{Fuzzy sets}, Information and Control,} {bf 8} (1965), 338-353.