Stability of Fuzzy Differential Equations With the Second Type of Hukuhara Derivative

In this paper, we consider fuzzy differential equations with the second type of Hukuhara derivative. With the help of Lyapunov-like functions, we establish stability, uniform stability, and uniformly asymptotical stability theorems, respectively, for a class of fuzzy differential equations with the second type of Hukuhara derivative. Besides, a very useful comparison principle is derived, which connects the stability relationships between fuzzy differential equations with the second type of Hukuhara derivative and scalar differential equations. In addition, some examples are shown to illustrate the effectiveness of the results.

[1]  Xiaoping Xue,et al.  Impulsive functional differential inclusions and fuzzy population models , 2003, Fuzzy Sets Syst..

[2]  Jinxian Li,et al.  The Cauchy problem of fuzzy differential equations under generalized differentiability , 2012, Fuzzy Sets Syst..

[3]  M. T. Malinowski.  Random fuzzy differential equations under generalized Lipschitz condition , 2012 .

[4]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[5]  Marek T. Malinowski,et al.  Interval differential equations with a second type Hukuhara derivative , 2011, Appl. Math. Lett..

[6]  Junping Shi,et al.  Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks , 2011 .

[7]  Y Chalco Cano,et al.  ON NEW SOLUTIONS OF FUZZY DIFFERENTIAL EQUATIONS , 2008 .

[8]  Abraham Kandel,et al.  The Cauchy problem for complex fuzzy differential equations , 2014, Fuzzy Sets Syst..

[9]  Marek T. Malinowski,et al.  Some properties of strong solutions to stochastic fuzzy differential equations , 2013, Inf. Sci..

[10]  Phil Diamond,et al.  Stability and periodicity in fuzzy differential equations , 2000, IEEE Trans. Fuzzy Syst..

[11]  V. Lakshmikantham,et al.  Differential and integral inequalities : theory and applications , 1969 .

[12]  Juan J. Nieto,et al.  The Cauchy problem for continuous fuzzy differential equations , 1999, Fuzzy Sets Syst..

[13]  V. Lakshmikantham,et al.  Theory of Fuzzy Differential Equations and Inclusions , 2003 .

[14]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[15]  Sue Ann Campbell,et al.  Stability and Bifurcations of Equilibria in a Multiple-Delayed Differential Equation , 1994, SIAM J. Appl. Math..

[16]  Marek T. Malinowski,et al.  Itô type stochastic fuzzy differential equations with delay , 2012, Syst. Control. Lett..

[17]  Yi Shen,et al.  Stability and boundedness of nonlinear hybrid stochastic differential delay equations , 2013, Syst. Control. Lett..

[18]  Marek T. Malinowski,et al.  Interval Cauchy problem with a second type Hukuhara derivative , 2012, Inf. Sci..

[19]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[20]  Marek T. Malinowski,et al.  On random fuzzy differential equations , 2009, Fuzzy Sets Syst..

[21]  Mustafa Bayram Gücen,et al.  Practical stability, boundedness criteria and Lagrange stability of fuzzy differential systems , 2012, Comput. Math. Appl..

[22]  Feiqi Deng,et al.  New-type stability theorem for stochastic functional differential equations with application to SFDSs with distributed delays , 2014, Int. J. Syst. Sci..

[23]  M. Puri,et al.  Differentials of fuzzy functions , 1983 .

[24]  Imre J. Rudas,et al.  First order linear fuzzy differential equations under generalized differentiability , 2007, Inf. Sci..

[25]  Jitao Sun,et al.  Stabilization of discrete-time Markovian jump linear systems via time-delayed and impulsive controllers , 2008, Autom..

[26]  D. Aeyels,et al.  A new asymptotic stability criterion for nonlinear time-variant differential equations , 1998, IEEE Trans. Autom. Control..

[27]  M. T. Malinowski.  On a new set-valued stochastic integral with respect to semimartingales and its applications , 2013 .