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##### Strict Stability of Fuzzy Differential Equations by Lyapunov Functions

Authors: Mustafa Bayram Gücen, Coşkun Yakar

Abstract:

In this study, we have investigated the strict stability of fuzzy differential systems and we compare the classical notion of strict stability criteria of ordinary differential equations and the notion of strict stability of fuzzy differential systems. In addition that, we present definitions of stability and strict stability of fuzzy differential equations and also we have some theorems and comparison results. Strict Stability is a different stability definition and this stability type can give us an information about the rate of decay of the solutions. Lyapunov’s second method is a standard technique used in the study of the qualitative behavior of fuzzy differential systems along with a comparison result that allows the prediction of behavior of a fuzzy differential system when the behavior of the null solution of a fuzzy comparison system is known. This method is a usefull for investigating strict stability of fuzzy systems. First of all, we present definitions and necessary background material. Secondly, we discuss and compare the differences between the classical notion of stability and the recent notion of strict stability. And then, we have a comparison result in which the stability properties of the null solution of the comparison system imply the corresponding stability properties of the fuzzy differential system. Consequently, we give the strict stability results and a comparison theorem. We have used Lyapunov second method and we have proved a comparison result with scalar differential equations.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1316718

References:

 Aumann, R.J. Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12.
 Bernfeld, S. and Lakshmikantham, V. An Introduction to Nonlinear Boundary V alue Problems. Academic Press, New York, 1974.
 Bede, B. and Gal, S. G. Generalizations of the differentiability of fuzzy-number valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, Vol. 151, No. 3, (2005) 581-589.
 Bede, B., Rudas, I. J. and Bencsik, A. L. First order linear fuzzy differential equations under generalized differentiability,Information Sciences, vol. 177, no. 7, (2007) pp. 1648-1662.
 Bede, B., Stefanini, L. Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, Vol. 230, November, (2013) pp. 119–141.
 Buckley, J.J. and Feuring, T.H. Fuzzy differential equations,Fuzzy Sets and Systems,Vol. 110 , (2000) pp. 43–54.
 Gomes, L. T., Barros, L. C., Bede, B. Fuzzy Differential Equations in Various Approaches, Springer International Publishing (2015).
 Buckley, J. J. and Feuring, T. H. Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 43 -54.
 Chen, Z., Fu, X., The variational Lyapunov function and strict stability theory for differential systems, Nonlinear Analysis 64, 1931 – 1938, (2006).
 Ding, Z., Ming, M. and Kandel, A. Existence of solutions of Fuzzy differential equations, Inform. Sci., 99 (1997), 205 - 217.
 Dubois, D. and Prade, H. Towards fuzzy differential calculus, Part I, Part II, Part III, Fuzzy Sets and Systems, 8 (1982), 1 - 17, 105 - 116, 225 - 234.
 Kaleva, O. Fuzzy differential equations. Fuzzy Sets and Systems 24 (1987) 301–317.
 Kaleva, O. On the calculus of fuzzy valued mappings, Appl. Math. Lett., 3 (1990), 55 - 59.
 Kaleva, O. The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems 35 (1990) 389–396.
 Lakshmikantham, V. and Leela, S. Differential and Integral Inequalities, Vol. I. Academic Press, New York, 1969.
 Lakshmikantham, V. and Leela, S. Fuzzy differential systems and the new concept of stability. Nonlinear Dynamics and Systems Theory, 1 (2) (2001), 111-119
 Lakshmikantham, V. and Leela, S. A new concept unifying Lyapunov and orbital stabilities. Communications in Applied Analysis, (2002), 6 (2).
 Lakshmikantham, V. and Leela, S. Stability theory of fuzzy differential equations via differential inequalities. Math. Inequalities and Appl. 2 (1999) 551–559.
 Lakshmikantham, V., Leela, S. and Martynyuk, A.A. Practical Stability of Nonlinear System. World Scientific Publishing, NJ, 1990.
 Lakshmikantham, V., Leela, S. and Martynyuk, A. A. Stability Analysis of Nonlinear System. Marcel Dekker, New York, 1989.
 Lakshmikantham, V. and Mohapatra, R. Basic properties of solutions of fuzzy differential equations. Nonlinear Studies 8 (2001) 113–124.
 Lakshmikantham, V. and Mohapatra, R. N. : Strict Stability of Differential Equations, Nonlinear Analysis, Volume 46, Issue 7, Pages 915-921 (2001)
 Lakshmikantham, V. and Mohapatra, R. N. Theory of Fuzzy Differential Equations. Taylor and Francis Inc. New York, 2003.
 Lakshmikantham, V. and Vatsala, A.S., Differential inequalities with time difference and application, Journal of Inequalities and Applications 3, (1999) 233-244.
 Li, A., Feng, E. and Li, S., Stability and boundedness criteria for nonlinear differential systems relative to initial time difference and applications. Nonlinear Analysis: Real World Applications 10(2009) 1073–1080.
 Liu, K., Yang, G., Strict Stability Criteria for Impulsive Functional Differential Systems, Journal of Inequalities and Applications , 2008:243863 doi:10.1155/2008/243863, (2008).
 Lyapunov, A. Sur les fonctions-vecteurs completement additives. Bull. Acad. Sci. URSS, Ser. Math 4 (1940) 465-478.
 Massera, J. L. The meaning of stability. Bol. Fac. Ing. Montevideo 8 (1964) 405–429.
 Nieto, J. J. The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems, (102 (1999), 259 - 262.
 Park, J. Y. and Hyo, K. H. Existence and uniqueness theorem for a solution of Fuzzy differential equations, Inter. J. Math.and Math. Sci, 22 (1999), 271-279.
 Puri, M. L. D. and Ralescu, A. Differential of Fuzzy functions, J. Math. Anal. Appl, 91 (1983), 552 - 558.
 Rojas K., Gomez D., Monteroa J., Rodrigueza J. Tinguaro, Strictly stable families of aggregation operators, Fuzzy Sets and Systems, 228, 44–63 (2013).
 Rojas K., Gomez D., Monteroa J., Rodrigueza J. Tinguaro, Valdivia A., Paiva F., Development of child’s home environment indexes based on consistent families of aggregation operators with prioritized hierarchical information, Fuzzy Sets and Systems 241, 41–60, (2014).
 Shaw, M. D. and Yakar, C., Generalized variation of parameters with initial time difference and a comparison result in term Lyapunov-like functions, International Journal of Non-linear Differential Equations-Theory-Methods and Applications 5, (1999). 86-108.
 Shaw, M. D. and Yakar, C., Stability criteria and slowly growing motions with initial time difference, Problems of Nonlinear Analysis in Engineering Systems 1, (2000) 50-66.
 Song, S. J., Guo, L. and Feng, C. H. Global existence of solutions of Fuzzy differential equations, Fuzzy Sets and Systems, 115 (2000), 371 - 376.
 Song, S.J. and Wu, C. Existence and Uniqueness of solutions to Cauchy problem of Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 55 - 67.
 Yakar, C. Boundedness criteria with initial time difference in terms of two measures, Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 14, supplement 2, (2007) 270–274, .
 Yakar C. and C¸ ic¸ek M. and G¨ucen, M.B. ”Boundedness and Lagrange stability of fractional order perturbed system related to unperturbed systems with initial time difference in Caputo’s sense.” Advances in difference Equations (2011):54. doi:10.1186/1687-1847-2011-54. ISSN: 1687-1847.
 Yakar C., C¸ ic¸ek M. and G¨ucen M. B. “Practical Stability in Terms of Two Measures for Fractional Order Dynamic Systems in Caputo’s Sense with Initial Time Difference” Journal of the Franklin Institute.PII: S0016-0032(13)00377-3 DOI: http://dx.doi.org/10.1016/j.jfranklin.2013.10.009. Ref.: FI1903. (2013). (SCI).
 Yakar C., C¸ ic¸ek M. and G¨ucen M. B. “Practical Stability, Boundedness Criteria and Lagrange Stability of Fuzzy Differential Systems” Journal of Computers and Mathematics with Applications. 64 (2012) 2118-2127. Doi: 10.1016/j.camwa.2012.04.008. (2012).
 Yakar, C. Strict stability criteria of perturbed systems with respect to unperturbed systems in terms of initial time difference. Complex Analysis and Potential Theory, World Scientific, Hackensack, NJ, USA (2007) 239–248.
 Yakar, C. and Shaw, M. D. A comparison result and Lyapunov stability criteria with initial time difference. Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 12, no. 6, (2005) 731–737.
 Yakar, C. and Shaw, M. D. Initial time difference stability in terms of two measures and a variational comparison result. Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 3, (2008) 417–425, .
 Yakar, C. and Shaw, M. D. Practical stability in terms of two measures with initial time difference. Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, (2009) e781–e785.
 Yoshizawa, T. Stability Theory by Lyapunov’s second Method, The Mathematical Society of Japan, Tokyo, 1966.
 Zadeh, L. A. Fuzzy Sets, Inform. Control., 8 (1965), 338 - 353.
 Zhang, Y. Criteria for boundedness of Fuzzy differential equations, Math. Ineq. Appl., 3 (2000), 399 -410.
 Zhang, Y., Sun J., Strict stability of impulsive functional differential equations, J. Math. Anal. Appl., 301, 237–248 (2005).