Exact Controllability for Fuzzy Differential Equations in Credibility Space
 Author: Lee Bu Young, Youm Hae Eun, Kim Jeong Soon
 Organization: Lee Bu Young; Youm Hae Eun; Kim Jeong Soon
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 14, Issue2, p145~153, 25 June 2014

ABSTRACT
With reasonable control selections on the space of functions, various application models can take the shape of a welldefined 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 integrodifferential equations with nonlocal conditions) to fuzzy differential equations driven by Liu process.

KEYWORD
fuzzy differential equations , Credibility space , Liu process , Fuzzy process

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. [14] 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 ndimension 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 spaceE^{n}_{N} .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 timevariant 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, andf ,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:
where the state
x (t ,θ ) takes values inX (E _{N}) and another bounded spaceY (E _{N}). We use the following notation:E _{N} is the set of all upper semicontinuously convex fuzzy numbers onR , (,,C r ) is the credibility space,A is a fuzzy coefficient, the state functionx : [0,T ] × (,,C r )X is a fuzzy process,f : [0, T] ×X X is a fuzzy function,u : [0,T ]×(,,Cr )Y is a control function,B is a linear bounded operator fromY toX ,C _{t} is a standard Liu process andx _{0}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 onedimensional fuzzy numbersu :R [0, 1], satisfying the following properties: (1) u is normal, i.e., there exists an u0 R such that u(to) = 1; (2) u is fuzzy convex, i.e., u(λt+(1╶λ)s) ≥ min{u(t); u(s)} for any t, s R, 0 ≤ λ ≤ 1; (3) u(t) is upper semicontinuous, i.e., for any tk R (k = 0, 1, 2, ․ ․ ․ ), tk t0; (4) [u]0 is compact. The level sets ofu , [u ]^{α} = {t R :u (t ) ≥α },α (0, 1], and [u ]^{0} are nonempty compact convex sets inR [8].Definition 2.1 [19] We define a complete metricD _{L} onE _{N} by for anyu ,v E _{N}, which satisfiesD _{L}(u +w ,v +w ) =D _{L}(u ,v ) for eachw E _{N}, and for everyα [0, 1] where ,R with ≤ .Definition 2.2 [20] For anyu ,v C ([0,T ],E _{N}), the metricH _{1}(u ,v ) onC ([0,T ],E _{N}) is defined byLet be a nonempty set, and let be the power set of . Each element in is called an event. To present an axiomatic definition of credibility, it is necessary to assign a number
Cr {A } to each eventA indicating the credibility thatA will occur. To ensure that the numberCr {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 A B. (3) (Self − Duality) Cr{A}+Cr{Ac} = 1 for any event A. (4) (Maximality) Cr{∪iAi} = supi Cr{Ai} for any events {Ai} with supi Cr{Ai} < 0.5.
Definition 2.5 [21] Let be a nonempty set, be the power set of , andCr be a credibility measure. Then the triplet (,,Cr ) is called a credibility space.Definition 2.6 [14] A fuzzy variable is a function from a credibility space (,,C_{r} ) to the set of real numbers.Definition 2.7 [14] Let T be an index set and (,,Cr ) be a credibility space. A fuzzy process is a function fromT × (,,Cr ) to the set of real numbers.That is, a fuzzy process
x (t ,θ ) is a function of two variables such that the functionx (t *,θ ) is a fuzzy variable for eacht *. For each fixedθ *, the functionx (t ,θ *) is called a sample path of the fuzzy process. A fuzzy processx (t ,θ ) is said to be samplecontinuous if the sample ping is continuous for almost allθ . Instead of writingx (t ,θ ), we sometimes we use the symbolx _{t}.Definition 2.8 Let (,,Cr ) be a credibility space. For fuzzy random variablex _{t} in credibility space, for eachα [0, 1], theα level set is defined by where with whenα [0, 1].Definition 2.9 [22] Let 𝜉 be a fuzzy variable andr 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 operatorE has the following properties: (i) if f ≤ g, then E[f(𝜉)] ≤ E[g(𝜉)], (ii) E[− f(𝜉)] = − E[f(𝜉)], (iii) if functions f and g are comonotonic, then for any nonnegative real numbers a and b, we haveE[af(𝜉) + bg(𝜉)] = aE[f(𝜉)] + bE[g(𝜉)]. Wheref (𝜉) andg (𝜉) are fuzzy variables.Definition 2.10 [?] A fuzzy processC_{t} is said to be a Liu process if (i) C0 = 0, (ii) Ct has stationary and independent increments, (iii) every increment Ct+s ― Cs is a normally distributed fuzzy variable with expected value et and variance σ2t2, whose membership function is , x R. The parameterse andσ are called thedrift anddiffusion coefficients, respectively. Liu process is said to be standard ife = 0 andσ = 1.Definition 2.11 [23] Letx _{t} be a fuzzy process and letC _{t} be a standard Liu process. For any partition of closed interval [c ,d ] withc =t _{0} < · · · <t _{n} =d , the mesh is written as . Then the fuzzy integral ofx _{t} with respect toC _{t} is provided that the limit exists almost surely and is a fuzzy variable.Lemma 2.2 [23] LetC _{t} be a standard Liu process. For any givenθ withCr {θ } > 0, the pathC _{t }is Lipschitz continuous, that is, the following inequality holds Ct1 − Ct2  < K(θ)t1 − t2, whereK is a fuzzy variable called the Lipschitz constant of a Liu process with andE [K ^{p}] < ∞, ∀_{p} > 0.Lemma 2.3 [23] LetC _{t} be a standard Liu process, and leth (t ;c ) be a continuously differentiable function. Definex _{t} =h (t ;C _{t}). Then we have the following chain ruleLemma 2.4 [23] Letf (t ) be continuous fuzzy process, the following inequality of fuzzy integral holds whereK =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 symbolx _{t}. We consider the existence and uniquencess of solutions for the fuzzy differential Eq (1)(u ≡ 0).where the state
x _{t} takes values inX (E _{N}).E _{N} is the set of all upper semicontinuously convex fuzzy numbers onR , (,,Cr ) is credibility space,A is fuzzy coefficient, the state functionx : [0,T ] × (,,Cr )X is a fuzzy process,f : [0,T ] ×X X is regular fuzzy function,C _{t} is a standard Liu process,x _{0}E _{N} is initial value.Lemma 3.1 [19] Letɡ 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 andZ _{t} is the solution of uncertain differential equation dZt = Ytɡ(t, Y −1Zt)dCt with initial valueZ _{0} =X _{0}.Using Lemma 3.1, we show that, for fuzzy coefficient
A , the Eq. (2) have a solution.Lemma 3.2 Forx (0) =x _{0}, ifx _{t} is solution of the Eq. (2), then the solutionx _{t} is given bywhereS (t ) is continuous withS (0) =I , S (t ) ≤c ,c > 0, for allt [0,T ].Proof For fuzzy coefficientA , the following define inverse ofS (t ) S −1(t) = e −At. Then it follows that dS −1(t) = −Ae −Atdt = −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. Definingz_{t} =S ^{ −1}(t )x_{t} , we obtainx_{t} =S (t )z_{t} and dzt = S −1(t)f(t, S(t)zt)dCt. Furthermore, we get in virtue ofS (0) =I , andz _{0} =x _{0}, Therefore the Eq. (2) has the following solution where we writeS (t −s ) instead ofS (t )S ^{ −1}(s ).Assume the following statements:
(H1) For
x_{t} ,y_{t} ∈C ([0,T ] × (,,C_{r} ),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) andf (0,X _{{0}}(0)) ≡ 0.(H2) 2
cm KT ≤ 1.By Lemma 3.2, we know that the Eq. (2) have a solution
x_{t} . Thus in Theorem 3.1, we show that uniqueness of solution for Eq. (2).Theorem 3.1 For everyx _{0} ∈E_{N} , if hypotheses (H1), (H2) are hold, then the eEq. (2) have a unique solutionx_{t} ∈C ([0,T ] × (,,C_{r} ),X ).Proof For each 𝜉_{t} ∈C ([0,T ] × (,,C_{r} ),X ),t ∈ [0,T ] define Thus, one can show that 𝜉 : [0,T ]× (,,C_{r} )C ([0,T ]× (,,C_{r} ),X ) is continuous, then : C([0, T] × (,,Cr),X) C([0, T] × (,,Cr),X).It is also obvious that a fixed point of is solution for the Eq. (2). For ,
η_{t} ∈C ([0,T ] × (,,C_{r} ),X ), by Lemma 2.4 and hypothesis (H1), we have Therefore, we obtain that Hence, for a.s.θ ∈ , by Lemma 2.1, By hypotheses (H2), is a contraction mapping. By the Banach fixed point theorem, Eq. (2) have a unique fixed pointx_{t} ∈C ([0,T ] × (,,C_{r} ),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 inY (⊂E_{N} ).where
S (t ) is continuous withS (0) =I , S (t ) ≤c ,c > 0, for allt ∈ [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 everyx _{0} ∈E_{N} there exists a controlu_{t} ∈Y such that the solution x of (1) satisfiesx_{T} =x ^{1} ∈X , a.s.θ (i.e., [x_{T} ]^{α} = [x ^{1}]^{α}).Define the fuzzy mapping : (
R )X where (R ) is a nonempty fuzzy subset ofR and is closure of supportu . Then there exists (i =l ,r ) such that We assume that , are bijective mappings.We can introduce
α level set ofu_{s} defined byThen substitute this expression into the Eq. (3) yields
α level ofx_{T} . Hence this controlu_{t} satisfisx_{T} =x ^{1}, a.s.θ .We now set where the fuzzy mappings 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 fromC ([0,T ]× (,,C_{r} ),X ) to itself. By Lemma 2.4 and hypotheses (H1) and (H2), for any givenθ withC_{r} {θ } > 0,x_{t} ,y_{t} ∈C ([0,T ]× (,,C_{r} ),X ), we haveTherefore by Lemma 2.1,
We take sufficiently small
T , (2cm 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 statex_{t} takes values inX (⊂E_{N} ) and another bounded spaceY (⊂E_{N} ).E_{N} is the set of all upper semicontinuously convex fuzzy numbers onR , (,,C_{r} ) is credibility space,A is a fuzzy coefficient, the state functionx : [0,T ]×(,,C_{r} ) →X is a fuzzy process,f : [0,T ]×X →X is a regular fuzzy function,u : [0,T ] × (,,C_{r} ) →Y is a control function,B is a linear bounded operator fromY toX .C_{t} is a standard Liu process,x _{0} ∈E_{N} is an initial value.Let
f (t ,x_{t} ) =tx_{t} , , definingz_{t} =S ^{ −1}(t )x_{t} , then the balance equations becomeTherefore Lemma 3.2 is satisfy.
The
α level set of fuzzy number is [2]^{α} = [α + 1, 3 −α ] for allα ∈ [0, 1]. Thenα level sets off (t ,x_{t} ) is [f (t ,x_{t} )]^{α} = Further, we have wherem = 3T 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 . Target set isx ^{1} = Theα level set of fuzzy number is [] = [α − 1, 1 −α ],α ∈ (0, 1]. We introduce theα level set of us of Eq. (4). Then substituting this expression into the Eq. (5) yieldsα level ofx_{T} . 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.

4. 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 ndimensional fuzzy vector space(EnN, d),” [International Journal of Fuzzy Logic and Intelligent Systems] Vol.11 P.2532

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