Advanced
Exact Controllability for Fuzzy Differential Equations in Credibility Space
Exact Controllability for Fuzzy Differential Equations in Credibility Space
International Journal of Fuzzy Logic and Intelligent Systems. 2014. Jun, 14(2): 145-153
Copyright © 2014, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : September 23, 2013
  • Accepted : May 09, 2014
  • Published : June 25, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
Bu Young Lee
Hae Eun Youm
Jeong Soon Kim

Abstract
With reasonable control selections on the space of functions, various application models can take the shape of a well-defined control system on mathematics. In the credibility space, controlability management of fuzzy differential equation is as much important issue as stability. This paper addresses exact controllability for fuzzy differential equations in the credibility space in the perspective of Liu process. This is an extension of the controllability results of Park et al. (Controllability for the semilinear fuzzy integro-differential equations with nonlocal conditions) to fuzzy differential equations driven by Liu process.
Keywords
1. Introduction
The concept of fuzzy set was initiated by Zadeh via membership function in 1965. Fuzzy differential equations are a field of increasing interest, due to their applicability to the analysis of phenomena where imprecision in inherent. Kwun et al. [1 - 4] and Lee et al. [5] have studied the existence and uniqueness for solutions of fuzzy equations.
The theory of controlled processes is one of the most recent mathematical concepts to enable very important applications in modern engineering. However, actual systems subject to control do not admit a strictly deterministic analysis in view of various random factors that influence their behavior. The theory of controlled processes takes the random nature of a systems behavior into account. Many researchers have studied controlled processes. With regard to fuzzy systems, Kwun and Park [6] proved controllability for the impulsive semilinear fuzzy differential equation in n-dimension fuzzy vector space. Park et al. [7] studied the controllability of semilinear fuzzy integrodifferential equations with nonlocal conditions. Park et al. [8] demonstrated the controllability of impulsive semilinear fuzzy integrodifferential equations, while Phu and Dung [9] studied the stability and controllability of fuzzy control set differential equations. Lee et al. [10] examined the controllability of a nonlinear fuzzy control system with nonlocal initial conditions in n -dimensional fuzzy vector space EnN .
In terms of the controllability of stochastic systems, P. Balasubramaniam [11] studied quasilinear stochastic evolution equations in Hilbert spaces, and the controllability of stochastic control systems with time-variant coefficients was proved by Yuhu [12] . Arapostathis et al. [13] studied the controllability properties of stochastic differential systems that are characterized by a linear controlled diffusion perturbed by a smooth, bounded, uniformly Lipschitz nonlinearity.
Stochastic differential equations driven by Brownian motion have been studied for a long time, and are a mature branch of modern mathematics. A new kind of fuzzy differential equation driven by a Liu process was defined as follows by Liu [14] dXt = f(Xt, t)dt + g(Xt, t)dCt where C t is a standard Liu process, and f , g are some given functions. The solution of such equation is a fuzzy process. You [15] discussed the solutions of some special fuzzy differential equations, and derived an existence and uniqueness theorem for homogeneous fuzzy differential equations. Chen [16] for fuzzy differential equations. Liu [17] studied an analytic method for solving uncertain differential equations. In this paper, we extend the result of Liu [17] to fuzzy differential equations driven by a Liu process within a controlled system.
We study the exact controllability of abstract fuzzy differential equations in a credibility space:
PPT Slide
Lager Image
where the state x ( t , θ ) takes values in X (
PPT Slide
Lager Image
E N ) and another bounded space Y (
PPT Slide
Lager Image
E N ). We use the following notation: E N is the set of all upper semi-continuously convex fuzzy numbers on R , (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, C r ) is the credibility space, A is a fuzzy coefficient, the state function x : [0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, C r )
PPT Slide
Lager Image
X is a fuzzy process, f : [0, T] × X
PPT Slide
Lager Image
X is a fuzzy function, u : [0, T ]×(
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr )
PPT Slide
Lager Image
Y is a control function, B is a linear bounded operator from Y to X , C t is a standard Liu process and x 0
PPT Slide
Lager Image
E N is an initial value.
In Section 2, we discuss some basic concepts related to fuzzy sets and Liu processes.
In Section 3, we show the existence of solutions to the free fuzzy differential equation (1)( u ≡ 0).
Finally, in Section 4, we prove the exact controllability of the fuzzy differential Eq. (1).
2. Preliminaries
In this section, we give some basic definitions, terminology, notation, and Lemmas that are relevant to our investigation and are needed in latter sections. All undefined concepts and notions used here are standard.
We consider E N to be the space of one-dimensional fuzzy numbers u : R
PPT Slide
Lager Image
[0, 1], satisfying the following properties:
  • (1)uis normal, i.e., there exists anu0Rsuch thatu(to) = 1;
  • (2)uis fuzzy convex, i.e.,u(λt+(1╶λ)s) ≥ min{u(t);u(s)} for anyt,sR, 0 ≤ λ ≤ 1;
  • (3)u(t) is upper semi-continuous, i.e.,for anytkR(k= 0, 1, 2, ․ ․ ․ ),tkt0;
  • (4) [u]0is compact.
The level sets of u , [ u ] α = { t
PPT Slide
Lager Image
R : u ( t ) ≥ α }, α
PPT Slide
Lager Image
(0, 1], and [ u ] 0 are nonempty compact convex sets in R [8] .
Definition 2.1 [19] We define a complete metric D L on E N by for any u , v
PPT Slide
Lager Image
E N , which satisfies D L ( u + w , v + w ) = D L ( u , v ) for each w
PPT Slide
Lager Image
E N , and
PPT Slide
Lager Image
for every α
PPT Slide
Lager Image
[0, 1] where
PPT Slide
Lager Image
,
PPT Slide
Lager Image
PPT Slide
Lager Image
R with
PPT Slide
Lager Image
PPT Slide
Lager Image
.
Definition 2.2 [20] For any u , v
PPT Slide
Lager Image
C ([0, T ], E N ), the metric H 1 ( u , v ) on C ([0, T ], E N ) is defined by
Let
PPT Slide
Lager Image
be a nonempty set, and let
PPT Slide
Lager Image
be the power set of
PPT Slide
Lager Image
. Each element in
PPT Slide
Lager Image
is called an event. To present an axiomatic definition of credibility, it is necessary to assign a number Cr { A } to each event A indicating the credibility that A will occur. To ensure that the number Cr { A } has certain mathematical properties that we intuitively expect, we accept the following four axioms:
  • (1) (Normality)Cr{} = 1.
  • (2) (Monotonicity)Cr{A} ≤Cr{B} whenever AB.
  • (3) (Self−Duality)Cr{A}+Cr{Ac} = 1 for any eventA.
  • (4) (Maximality)Cr{∪iAi} = supiCr{Ai} for any events {Ai} with supiCr{Ai} < 0.5.
Definition 2.5 [21] Let
PPT Slide
Lager Image
be a nonempty set,
PPT Slide
Lager Image
be the power set of
PPT Slide
Lager Image
, and Cr be a credibility measure. Then the triplet (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) is called a credibility space.
Definition 2.6 [14] A fuzzy variable is a function from a credibility space (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) to the set of real numbers.
Definition 2.7 [14] Let T be an index set and (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) be a credibility space. A fuzzy process is a function from T × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) to the set of real numbers.
That is, a fuzzy process x ( t , θ ) is a function of two variables such that the function x ( t *, θ ) is a fuzzy variable for each t *. For each fixed θ *, the function x ( t , θ *) is called a sample path of the fuzzy process. A fuzzy process x ( t , θ ) is said to be sample-continuous if the sample ping is continuous for almost all θ . Instead of writing x ( t , θ ), we sometimes we use the symbol x t .
Definition 2.8 Let (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) be a credibility space. For fuzzy random variable x t in credibility space, for each α
PPT Slide
Lager Image
[0, 1], the α -level set
PPT Slide
Lager Image
is defined by where
PPT Slide
Lager Image
with
PPT Slide
Lager Image
when α
PPT Slide
Lager Image
[0, 1].
Definition 2.9 [22] Let 𝜉 be a fuzzy variable and r is real number. Then the expected value of 𝜉 is defined by provided that at least one of the integrals is finite.
Lemma 2.1 [22] Let 𝜉 be a fuzzy vector. The expected value operator E has the following properties:
  • (i) iff≤g, thenE[f(𝜉)] ≤E[g(𝜉)],
  • (ii)E[−f(𝜉)] = −E[f(𝜉)],
  • (iii) if functionsfandgare comonotonic, then for any nonnegative real numbersaandb, we haveE[af(𝜉) +bg(𝜉)] =aE[f(𝜉)] +bE[g(𝜉)].
Where f (𝜉) and g (𝜉) are fuzzy variables.
Definition 2.10 [?] A fuzzy process Ct is said to be a Liu process if
  • (i)C0= 0,
  • (ii)Cthas stationary and independent increments,
  • (iii) every incrementCt+s―Csis a normally distributed fuzzy variable with expected value et and varianceσ2t2, whose membership function is,xR.
The parameters e and σ are called the drift and diffusion coefficients, respectively. Liu process is said to be standard if e = 0 and σ = 1.
Definition 2.11 [23] Let x t be a fuzzy process and let C t be a standard Liu process. For any partition of closed interval [ c , d ] with c = t 0 < · · · < t n = d , the mesh is written as
PPT Slide
Lager Image
. Then the fuzzy integral of x t with respect to C t is provided that the limit exists almost surely and is a fuzzy variable.
Lemma 2.2 [23] Let C t be a standard Liu process. For any given θ with Cr { θ } > 0, the path C t is Lipschitz continuous, that is, the following inequality holds |Ct1Ct2 | < K(θ)|t1t2|, where K is a fuzzy variable called the Lipschitz constant of a Liu process with and E [ K p ] < ∞, ∀ p > 0.
Lemma 2.3 [23] Let C t be a standard Liu process, and let h ( t ; c ) be a continuously differentiable function. Define x t = h ( t ; C t ). Then we have the following chain rule
Lemma 2.4 [23] Let f ( t ) be continuous fuzzy process, the following inequality of fuzzy integral holds where K = K ( θ ) is defined in Lemma 2.2.
3. Existence of Solutions for Abstract Fuzzy Differential Equations
In this section, by Definition 2.7, instead of longer notation x ( t , θ ), sometimes we use the symbol x t . We consider the existence and uniquencess of solutions for the fuzzy differential Eq (1)( u ≡ 0).
PPT Slide
Lager Image
where the state x t takes values in X (
PPT Slide
Lager Image
E N ). E N is the set of all upper semi-continuously convex fuzzy numbers on R , (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) is credibility space, A is fuzzy coefficient, the state function x : [0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr )
PPT Slide
Lager Image
X is a fuzzy process, f : [0, T ] × X
PPT Slide
Lager Image
X is regular fuzzy function, C t is a standard Liu process, x 0
PPT Slide
Lager Image
E N is initial value.
Lemma 3.1 [19] Let ɡ be a function of two variables and let at be an integrable uncertain process. Then a given uncertain differential equation by dXt = atXtdt + ɡ(t,Xt)dCt has a solution where and Z t is the solution of uncertain differential equation dZt = Ytɡ(t, Y −1Zt)dCt with initial value Z 0 = X 0 .
Using Lemma 3.1, we show that, for fuzzy coefficient A , the Eq. (2) have a solution.
Lemma 3.2 For x (0) = x 0 , if x t is solution of the Eq. (2), then the solution x t is given by where S ( t ) is continuous with S (0) = I , | S ( t )| ≤ c , c > 0, for all t
PPT Slide
Lager Image
[0, T ].
Proof For fuzzy coefficient A , the following define inverse of S ( t ) S −1(t) = eAt. Then it follows that dS −1(t) = −AeAtdt = −AS −1(t)dt. Applying the integration by parts to the above equation provides That is, d(S −1(t)xt) = S −1(t)f(t, xt)dCt. Defining zt = S −1 ( t ) xt , we obtain xt = S ( t ) zt and dzt = S −1(t)f(t, S(t)zt)dCt. Furthermore, we get in virtue of S (0) = I , and z 0 = x 0 , Therefore the Eq. (2) has the following solution where we write S ( t s ) instead of S ( t ) S −1 ( s ).
Assume the following statements:
(H1) For xt , yt C ([0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ), t ∈ [0, T ], there exists positive number m such that dL([f(t, xt)]α, [f(t, yt)]α) ≤ mdL([xt]α, [yt]α) and f (0, X {0} (0)) ≡ 0.
(H2) 2 cm KT ≤ 1.
By Lemma 3.2, we know that the Eq. (2) have a solution xt . Thus in Theorem 3.1, we show that uniqueness of solution for Eq. (2).
Theorem 3.1 For every x 0 EN , if hypotheses (H1), (H2) are hold, then the eEq. (2) have a unique solution xt C ([0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ).
Proof For each 𝜉 t C ([0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ), t ∈ [0, T ] define Thus, one can show that
PPT Slide
Lager Image
𝜉 : [0, T ]× (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr )
PPT Slide
Lager Image
C ([0, T ]× (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ) is continuous, then : C([0, T] × (,,Cr),X) C([0, T] × (,,Cr),X).
It is also obvious that a fixed point of
PPT Slide
Lager Image
is solution for the Eq. (2). For
PPT Slide
Lager Image
, ηt C ([0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ), by Lemma 2.4 and hypothesis (H1), we have Therefore, we obtain that Hence, for a.s. θ
PPT Slide
Lager Image
, by Lemma 2.1, By hypotheses (H2),
PPT Slide
Lager Image
is a contraction mapping. By the Banach fixed point theorem, Eq. (2) have a unique fixed point xt C ([0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ).
4. Exact Controllability for Abstract Fuzzy Differential Equations
In this section, we study exact controllability for abstract fuzzy differential Eq. (1).
We consider solution for the Eq. (1), for each u in Y (⊂ EN ).
PPT Slide
Lager Image
where S ( t ) is continuous with S (0) = I , | S ( t )| ≤ c , c > 0, for all t ∈ [0, T ].
We define the controllability concept for abstract fuzzy differential equations.
Definition 4.1 The Eq. (1) are said to be controllable on [0,T], if for every x 0 EN there exists a control ut Y such that the solution x of (1) satisfies xT = x 1 X , a.s. θ (i.e., [ xT ] α = [ x 1 ] α ).
Define the fuzzy mapping
PPT Slide
Lager Image
:
PPT Slide
Lager Image
( R )
PPT Slide
Lager Image
X where
PPT Slide
Lager Image
( R ) is a nonempty fuzzy subset of R and
PPT Slide
Lager Image
is closure of support u . Then there exists
PPT Slide
Lager Image
( i = l , r ) such that We assume that
PPT Slide
Lager Image
,
PPT Slide
Lager Image
are bijective mappings.
We can introduce α -level set of us defined by
Then substitute this expression into the Eq. (3) yields α -level of xT . Hence this control ut satisfis xT = x 1 , a.s. θ .
We now set where the fuzzy mappings
PPT Slide
Lager Image
satisfies above statements.
(H3) Assume that the linear system of Eq. (1) (f ≡ 0) is controllable.
Theorem 4.1 If Lemma 2.4 and the hypotheses (H1), (H2) and (H3) are satisfied, then the Eq. (1) are controllable on [0, T ].
Proof We can easily check that Φ is continuous from C ([0, T ]× (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ) to itself. By Lemma 2.4 and hypotheses (H1) and (H2), for any given θ with Cr { θ } > 0, xt , yt C ([0, T ]× (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ), X ), we have
Therefore by Lemma 2.1,
We take sufficiently small T , (2 cm KT) < 1. Hence Φ is a contraction mapping. We now apply the Banach fixed point theorem to show that the Eq. (3) have a unique fixed point.
Consequently, the Eq. (1) are controllable on [0, T ].
Example 4.1 We consider the following abstract fuzzy differential equations in credibility space where the state xt takes values in X (⊂ EN ) and another bounded space Y (⊂ EN ). EN is the set of all upper semicontinuously convex fuzzy numbers on R , (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) is credibility space, A is a fuzzy coefficient, the state function x : [0, T ]×(
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) → X is a fuzzy process, f : [0, T X X is a regular fuzzy function, u : [0, T ] × (
PPT Slide
Lager Image
,
PPT Slide
Lager Image
, Cr ) → Y is a control function, B is a linear bounded operator from Y to X . Ct is a standard Liu process, x 0 EN is an initial value.
Let f ( t , xt ) =
PPT Slide
Lager Image
txt ,
PPT Slide
Lager Image
, defining zt = S −1 ( t ) xt , then the balance equations become
PPT Slide
Lager Image
Therefore Lemma 3.2 is satisfy.
The α -level set of fuzzy number
PPT Slide
Lager Image
is [2] α = [ α + 1, 3 − α ] for all α ∈ [0, 1]. Then α -level sets of f ( t , xt ) is [ f ( t , xt )] α =
PPT Slide
Lager Image
Further, we have where m = 3 T satisfies the inequality in hypothesis (H1), (H2). Then all the conditions stated in Theorem 3.1 are satisfied.
Let an initial value x 0 is
PPT Slide
Lager Image
. Target set is x 1 =
PPT Slide
Lager Image
The α -level set of fuzzy number
PPT Slide
Lager Image
is [
PPT Slide
Lager Image
] = [ α − 1, 1 − α ], α ∈ (0, 1]. We introduce the α -level set of us of Eq. (4). Then substituting this expression into the Eq. (5) yields α -level of xT . Then all the conditions stated in Theorem 4.1 are satisfied. So the Eq. (4) are controllable on [0, T ].
5. Conclusions
If there is an exact controllability encouraged for the abstract fuzzy differential equations, it can provide a benchmark for an approach to handle controllability about the equations such as fuzzy semilinear integrodifferential equations, fuzzy delay integrodifferential equations on the credibility space. Therefore, the theoretical result of this study can be used to make stochastic extension on the credibility space.
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This work was supported by the research grant of Dong-A University in 2013.
BIO
Bu Young Lee received the Ph. D. degree from Gyeongsang National University in 1990. He is a professor at the Department of Mathematics, Dong-A University since 1983. His research interests include general topology, fuzzy mathematics, and fuzzy topology.
E-mail: bylee@dau.ac.kr
Hae Eun Youm received the Ph. D. degree from Dong-A University in 2013. She is a lecturer in Kyungnam high school. Her research interest is in fuzzy mathematics.
E-mail: cara4303@hanmail.net
Jeong Soon Kim received the Ph. D. degree from Dong-A University in 2012. She is an assistant professor at the Department of Mathematics, Dong-A University since 2013. Her principal research area is fuzzy mathematics.
E-mail: jeskim@dau.ac.kr
References
Kwun Y. C. , Kim W. H. , Nakagiri S. , Park J. H. 2009 “Existence and uniqueness of solutions for the fuzzy differential equations in n-dimension fuzzy vector space,” International Journal of Fuzzy Logic and Intelligent Systems http://dx.doi.org/10.5391/IJFIS.2009.9.1.016 9 (1) 16 - 19    DOI : 10.5391/IJFIS.2009.9.1.016
Kwun Y. C. , Kim J. S. , Hwang J. , Park J. H. 2010 “Existence of periodic solutions for fuzzy differential equations,” International Journal of Fuzzy Logic and Intelligent Systems http://dx.doi.org/10.5391/IJFIS.2010.10.3.184 10 (3) 184 - 193    DOI : 10.5391/IJFIS.2010.10.3.184
Kwun Y. C. , Han C. W. , Kim S. Y. , Park J. S. 2004 “Existence and uniqueness of fuzzy solutions for the nonlinear fuzzy integro-differential equation EnN,” International Journal of Fuzzy Logic and Intelligent Systems http://dx.doi.org/10.5391/IJFIS.2004.4.1.040 4 (1) 40 - 44    DOI : 10.5391/IJFIS.2004.4.1.040
Kwun Y. C. , Kim J. S. , Hwang J. , Park J. H. 2011 “Existence of solutions for the impulsive semilinear fuzzy intergrodifferential equations with nonlocal conditions and forcing term with memory in n-dimensional fuzzy vector space(EnN, d),” International Journal of Fuzzy Logic and Intelligent Systems http://dx.doi.org/10.5391/IJFIS.2011.11.1.025 11 (1) 25 - 32    DOI : 10.5391/IJFIS.2011.11.1.025
Lee B. Y. , Kwun Y. C. , Ahn Y. C. , Park J. H. 2009 “The existence and uniqueness of fuzzy solutions for semilinear fuzzy integrodifferential equations using integral contractor,” International Journal of Fuzzy Logic and Intelligent Systems http://dx.doi.org/10.5391/IJFIS.2009.9.4.339 9 (4) 339 - 342    DOI : 10.5391/IJFIS.2009.9.4.339
Kwun Y. C. , Park J. H. 2009 “Controllability for the impulsive semilinear fuzzy differential equation in n-dimensional fuzzy vector space,” Proceedings of the 10th International Symposium on Advanced Intelligent Systems 269 - 271
Park J. , Park J. , Kwun Y. , Wang L. , Jiao L. , Shi G. , Li X. , Liu J. 2006 “Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions,” Springer Berlin Heidelberg, Germany 221 - 230
Park J. , Park J. , Ahn Y. , Kwun Y. , Cao B. Y. 2007 “Controllability for the impulsive semilinear fuzzy integrodifferential equations,” Springer Berlin Heidelberg, Germany 704 - 713
Phu N. D. , Dung L. Q. 2011 “On the stability and controllability of fuzzy control set differential equations,” International Journal of Reliability and Safety http://dx.doi.org/10.1504/IJRS.2011.041183 5 (3-4) 320 - 335    DOI : 10.1504/IJRS.2011.041183
Lee B. Y. , Park D. G. , Choi G. T. , Kwun Y. C. 2006 “Controllability for the nonlinear fuzzy control system with nonlocal initial condition in EnN,” International Journal of Fuzzy Logic and Intelligent Systems 6 (1) 15 - 20    DOI : 10.5391/IJFIS.2006.6.1.015
Balasubramaniam P. 2001 “Controllability of quasilinear stochastic evolution equations in Hilbert spaces,” Journal of Applied Mathematics and Stochastic Analysis http://dx.doi.org/10.1155/S1048953301000119 14 (2) 151 - 159    DOI : 10.1155/S1048953301000119
Yuhu F. 1999 “Convergence theorems for fuzzy random variables and fuzzy martingales,” Fuzzy Sets and Systems http://dx.doi.org/10.1016/S0165-0114(97)00180-2 103 (3) 435 - 441    DOI : 10.1016/S0165-0114(97)00180-2
Arapostathis A. , George R. K. , Ghosh M. K. 2001 “On the controllability of a class of nonlinear stochastic systems,” Systems & Control Letters http://dx.doi.org/10.1016/S0167-6911(01)00123-2 44 (1) 25 - 34    DOI : 10.1016/S0167-6911(01)00123-2
Liu B. 2008 “Fuzzy process, hybrid process and uncertain process,” Journal of Uncertain Systems 2 (1) 3 - 16
You C. “Existence and uniqueness theorems for fuzzy differential equations,” http://www.orsc.edu.cn/process/080316.pdf
Chen X. “A new existence and uniquenss theorem for fuzzy differential equations,” http://orsc.edu.cn/process/080929.pdf
Liu Y. 2012 “An analytic method for solving uncertain dierential equations,” Journal of Uncertain Systems 6 (4) 244 - 249
Diamond P. , Kloeden P. E. 1994 Metric Spaces of Fuzzy Sets: Theory and Applications World Scientific River Edge, NJ
Wang G. , Li Y. , Wen C. 2007 “On fuzzy-cell numbers and -dimension fuzzy vectors,” Fuzzy Sets and Systems http://dx.doi.org/10.1016/j.fss.2006.09.006 158 (1) 71 - 84    DOI : 10.1016/j.fss.2006.09.006
Kwun Y. , Kim J. , Park M. , Park J. 2009 “Nonlocal controllability for the semilinear fuzzy integrodifferential equations in-dimensional fuzzy vector space,” Advances in Difference Equations http://dx.doi.org/10.1155/2009/734090 2009 (1) 734090 -    DOI : 10.1155/2009/734090
Liu B. 2006 “A survey of credibility theory,” Fuzzy Optimization and Decision Making http://dx.doi.org/10.1007/s10700-006-0016-x 5 (4) 387 - 408    DOI : 10.1007/s10700-006-0016-x
Liu B. , Liu Y. K. 2002 “Expected value of fuzzy variable and fuzzy expected value models,” IEEE Transactions on Fuzzy Systems http://dx.doi.org/10.1109/TFUZZ.2002.800692 10 (4) 445 - 450    DOI : 10.1109/TFUZZ.2002.800692
Fei W. “Uniqueness of solutions to fuzzy differential equations driven by Liu’s process with non-lipschitz coefficients,” Proceedings of the 6th International Conference on Fuzzy Systems and Knowledge Discovery Tianjin, China August 14-16, 2009 http://dx.doi.org/10.1109/FSKD.2009.603 565 - 569    DOI : 10.1109/FSKD.2009.603