Periodic behavior of semi-linear uncertain dynamical systems

On generalized solutions for discontinuous fuzzy differential equations and strong fuzzy Henstock integrals

“…This led Seikkala [27] to introduce the notion of fuzzy derivative as an extension of the Hukuhara derivative and the fuzzy integral, which was the same as that proposed by Dubois and Prade [11,12]. Naturally, the investigation of fuzzy differential and integral equations, existence and uniqueness theorems for the solutions of fuzzy initial value problems, drew upon the interest of many researchers of the fuzzy domain (see [1,3,4,7,8,14,18,19,22,24,29]). In 2002, Xue and Fu [33] established solutions to fuzzy differential equations with right-hand side functions satisfying Caratheoedory conditions on a class of Lipschitz fuzzy sets.…”

The Cauchy problems for discontinuous fuzzy systems under generalized differentiability

“…Because the metric space (E n , D) has a linear structure, it can be imbedded isomorphically as a cone in a Banach space. It is worth mentioning that Chen et al [4][5][6] studied the initial value problems of fuzzy differential equations by using the parametric representation of fuzzy numbers and the new framework of calculus for fuzzy number valued functions established in [7]. One can see that their method was more convenient than the original method to calculate derivatives, integrals and compute numerical solutions, etc.…”

Periodic problem of first order nonlinear uncertain dynamic systems

“…(2) Zadeh's Extension Principle (see [19,20]); (3) differential inclusions (see [6][7][8][13][14][15]). …”

“…In addition, we know that the theories of fuzzy differential equations are different under these three senses (see [1,7,15]). Usually, when one studies practical problems such as uncertain periodic control systems and neural networks with uncertainty, one often needs to consider the following periodic problem of fuzzy differential equation: x = f(t, x), x(0) = x(T ), (1.1) where f : [0, T ] × E 1 → E 1 has at least one fuzzy (non-real) value, i.e., there exists t 0 ∈ [0, T ] such that f(t 0 , x(t 0 )) ∈ E 1 \ R.…”

“…From the results of [4,7,15], we know that there are no periodic solution to fuzzy differential equations in the sense of H-derivatives, so fuzzy differential equations in the senses of H-derivatives cannot be used to describe periodic phenomena in the real world. In this paper, following with [7] and [8], we apply the methods of differential inclusion, functional analysis, set-valued analysis and Kakutani fixed point theorem to consider a periodic problem of fuzzy differential equation in the sense of differential inclusion, i.e., the formula (1.1) holds under the conditions f : [0, T ] × E c → E c and E c ⊂ E 1 , and the periodic behavior of first order uncertain dynamical system can be seen as:…”