Stability in Distribution for Backward Uncertain Differential Equation

Transcription

1 Sabiliy in Diribuion for Backward Uncerain Differenial Equaion Yuhong Sheng 1, Dan A. Ralecu 2 1. College of Mahemaical and Syem Science, Xinjiang Univeriy, Urumqi 8346, China heng-yh12@mail.inghua.edu.cn 2. Deparmen of Mahemaical Science, Univeriy of Cincinnai, OH , USA dan.ralecu@uc.edu Abrac Backward uncerain differenial equaion i a ype of differenial equaion driven by a Liu proce. So far, concep of abiliy in meaure, abiliy in mean, abiliy in p-h momen, almo ure abiliy and ph momen exponenial of backward uncerain differenial equaion have been propoed. A a upplemen, hi paper give a concep of abiliy in diribuion of backward uncerain differenial equaion. Some ufficien condiion for a backward uncerain differenial equaion being able in diribuion are provided. In addiion, hi paper furher dicue heir relaionhip among abiliy in diribuion, abiliy in meaure, abiliy in mean and abiliy in momen. La, hi paper dicue ome example o illurae he heoreical conideraion. Keyword: Backward uncerain differenial equaion; Sabiliy in diribuion; Uncerain proce; 1 Inroducion We know ha a premie of applying probabiliy heory i ha he obained diribuion i cloe enough o he real frequency. However, omeime we can no obain daa o eimae a probabiliy diribuion. In hi cae, we have o invie ome exper o evaluae heir belief degree abou he poible even. In order o deal wih he belief degree, Liu [5] inroduced an uncerainy heory in 27 and perfeced in 29 [7]. For decribe he evoluion of an uncerain phenomenon, Liu [6] propoed an uncerain proce ha i i eenially a equence of uncerain variable indexed by ime, ha i, i i an uncerain variable a each ime. In order o deal wih whie noie, Liu [7] preened a Liu proce which i almo all ample pah are Lipchiz coninuou, and i i a aionary and independen incremen proce whoe incremen are normal uncerain variable. Meanwhile, Liu [7] founded uncerain calculu o deal wih he inegral and differenial of an uncerain proce wih repec o Liu proce, Chen and Ralecu [2] propoed an uncerain inegral wih repec o general Liu proce. Then Liu and Yao [9] exended uncerain inegral from ingle Liu proce o muliple one. Correponding auhor: dan.ralecu@uc.edu 1

2 In 28, Liu [6] fir preened a ype of uncerain differenial equaion driven by Liu proce. The exience and uniquene heorem of oluion of uncerain differenial equaion wa fir proved by Chen and Liu [1] under linear growh condiion and Lipchiz coninuou condiion. Furhermore, hi heorem wa verified by Gao [3] under local linear growh condiion and local Lipchiz coninuou condiion. Sabiliy of uncerain differenial equaion ha recenly received a lo of aenion. The concep of abiliy in meaure of uncerain differenial equaion wa preened by Liu [7], and Yao e al. [17] proved ome abiliy heorem of uncerain differenial equaion. Following ha, Yao e al. [18] dicued abiliy in mean, Sheng and Wang [11] conidered abiliy in p-h momen, Sheng and Gao [12] conidered an exponenial abiliy of uncerain differenial equaion, Liu e al. [1] inveigaed almo ure abiliy and Yang e al. [19] inveigaed abiliy in diribuion. The backward uncerain differenial equaion wa inroduced by Ge and Zhu [4]. A o he oluion of a backward uncerain differenial equaion, an exience and uniquene heorem wa given by Ge and Zhu [4]. For he abiliy analyi of he oluion for a backward uncerain differenial equaion, he concep of abiliy in meaure, abiliy in mean and abiliy in p-h momen for a backward uncerain differenial equaion were propoed by Wang and Ning [13]. Meanwhile, hey derived ome ufficien condiion for a backward uncerain differenial equaion being able in meaure, in mean and in p-h momen. Beide, Wang and Ning [14] dicued he almo ure abiliy and momen exponenial abiliy for backward uncerain differenial equaion. A an exenion of he previou work on abiliy of backward uncerain differenial equaion he objecive of hi paper i o develop he abiliy in diribuion of backward uncerain differenial equaion. The re of hi paper i organized a follow. In Secion 2, hi paper review ome baic concep abou uncerain meaure, uncerain diribuion, uncerain proce and uncerain calculu. Secion 3 preen he concep of abiliy in diribuion for backward uncerain differenial equaion, and ome ufficien condiion will be given. Secion 4 dicue relaionhip among abiliy in diribuion, abiliy in meaure, abiliy in mean and abiliy in momen. Finally, ome concluion are given in Secion 5. 2 Preliminarie In hi ecion, we review ome baic concep in uncerainy heory including uncerain variable, uncerain proce and uncerain calculu. For modelling belief degree, an uncerain meaure wa defined by Liu [5] according o he following hree axiom: Axiom 1. Normaliy Axiom) MΓ} = 1 for he univeral e Γ. Axiom 2. Dualiy Axiom) MΛ} + MΛ c } = 1 for any even Λ. Axiom 3. Subaddiiviy Axiom) For every counable equence of even Λ 1, Λ 2,, we have } M Λ i MΛ i }. i=1 i=1 Beide, Liu [7] defined a produc uncerain meaure by he fourh axiom: Axiom 4. Produc Axiom) Le Γ k, L k, M k ) be uncerainy pace for k = 1, 2,. The produc uncerain meaure M i an uncerain meaure aifying } M Λ k = M k Λ k } k=1 k=1 2

3 where Λ k are arbirarily choen even from L k for k = 1, 2,, repecively. Definiion 2.1. Liu [5]) Le Γ be a nonempy e, le L be a σ-algebra over Γ. Then he riple Γ, L, M) i called an uncerainy pace. Definiion 2.2. Liu [5]) An uncerain variable ξ i a funcion from an uncerainy pace Γ, L, M) o he e of real number uch ha for any Borel e B of real number, he e i an even. ξ B} = γ Γ ξγ) B} An uncerain variable i eenially a meaurable funcion from an uncerainy pace o he e of real number. follow. In order o decribe an uncerain variable, a concep of uncerainy diribuion i defined a Definiion 2.3. Liu [5]) The uncerainy diribuion Φ of an uncerain variable ξ i defined by for any real number x. Φx) = Mξ x} Definiion 2.4. Liu [5]) The uncerain variable ξ 1, ξ 2,, ξ n are aid o be independen if n } n M ξ i B i ) = Mξ i B i } i=1 for any Borel e B 1, B 2,, B n of real number. Definiion 2.5. Liu [8]) Le ξ be an uncerain variable wih regular uncerainy diribuion Φx). Then he invere funcion Φ 1 α) i called he invere uncerainy diribuion of ξ. Theorem 2.1. Liu [8]) Le ξ 1, ξ 2,, ξ n be independen uncerain variable wih regular uncerainy diribuion Φ 1, Φ 2,, Φ n, repecively. If f i a ricly increaing funcion for ξ 1, ξ 2,, ξ m and f i a ricly decreaing funcion for ξ m+1, ξ m+2,, ξ n, hen ξ = fξ 1, ξ 2,, ξ n ) i an uncerain variable wih invere uncerainy diribuion Ψ 1 α) = fφ 1 1 α), Φ 1α),, Φ 1 m α), Φ 1 2 m+1 i=1 1 α), Φ 1 m+2 1 α),, Φ 11 α)). An uncerain variable ξ i aid o be normal if i ha a normal uncerainy diribuion )) 1 πe x) Φx) = 1 + exp, x R 3σ denoed by Ne, σ) where e and σ are real number wih σ >. In hi cae, he uncerain variable expξ) ha a lognormal uncerainy diribuion )) 1 πe ln x) Ψx) = 1 + exp, x R + 3σ denoed by LOGNe, σ). An uncerain proce i eenially a equence of uncerain variable indexed by ime. A formal definiion of uncerain proce i a follow. n 3

4 Definiion 2.6. Liu [6]) Le T be a oally ordered e e.g. ime) and le Γ, L, M) be an uncerainy pace. An uncerain proce i a funcion X γ) from T Γ, L, M) o he e of real number uch ha X B} i an even for any Borel e B of real number a each ime. Definiion 2.7. Liu [6]) An uncerain proce X i aid o have independen incremen if X, X 1 X, X 2 X 1,, X k X k 1 are independen uncerain variable where, 1,, k 1 are any ime wih < 1 < < k 1. An uncerain proce X i aid o have aionary incremen if, for any given >, he incremen X + X are idenically diribued uncerain variable for all >. Definiion 2.8. Liu [6]) An uncerain proce i aid o be a aionary independen incremen proce if i ha aionary and independen incremen. Definiion 2.9. Liu [7]) An uncerain proce C i aid o be a Liu proce if i) C = and almo all ample pah are Lipchiz coninuou, ii) C ha aionary and independen incremen, iii) every incremen C + C i a normal uncerain variable wih an uncerainy diribuion Φ x) = 1 + exp πx )) 1, x R. 3 Definiion 2.1. Liu [7]) Le X be an uncerain proce and le C be a Liu proce. For any pariion of cloed inerval [a, b] wih a = 1 < 2 < < k+1 = b, he meh i wrien a Then Liu inegral of X wih repec o C i b a = max 1 i k i+1 i. X dc = lim i=1 k X i C i+1 C i ), provided ha he limi exi almo urely and i finie. In hi cae, he uncerain proce X i aid o be inegrable. Definiion Liu [6]) Suppoe C i a Liu proce, and f and g are wo given funcion. Then dx = f, X )d + g, X )dc 1) i called an uncerain differenial equaion. A oluion i an uncerain proce ha aifie 1) idenically in. Theorem 2.2. Yao-Chen Formula [15]) Le X and X α be he oluion and α-pah of he uncerain differenial equaion dx = f, X )d + g, X )dc. Then MX X α, } = α, MX > X α, } = 1 α. 4

5 Theorem 2.3. Yao e al. [16]) Le C be a Liu proce on an uncerainy pace Γ, L, M). Then here exi an uncerain variable K uch ha Kγ) i a Lipchiz conan of he ample pah C γ) for each γ, and Definiion Yang e al. )) 1 πx Mγ Kγ) x} exp 1. x [19]) Le ξ, ξ 1, ξ 2, be an uncerain equence wih regular uncerainy diribuion Φ, Φ 1, Φ 2,, repecively. We ay ha ξ n } converge in invere diribuion o ξ if for all α, 1). lim n Φ 1 n α) = Φ 1 α) Theorem 2.4. Yang e al. [19]) Le ξ, ξ 1, ξ 2, be an uncerain equence wih regular uncerainy diribuion Φ, Φ 1, Φ 2,, repecively. Then ξ n } converge in invere diribuion o ξ if and only if i converge in diribuion o ξ. Definiion Ge and Zhu [4]) Le C be a Liu proce, and give wo real valued funcion f and g. dx = f, X )d + g, X )dc, [, T ) X = x T 2) i called a backward uncerain differenial equaion wih final value X = x T. The backward uncerain differenial equaion 2) and he backward uncerain inegral equaion 3) are equivalen. X = x T f, X )d g, X )dc, [, T ]. 3) Theorem 2.5. Ge and Zhu [4]) The backward uncerain differenial equaion 2) ha a unqiue oluion X wih given final value on [, T ] if he coefficien f, x) and g, x) aify he Lipchiz condiion and he linear growh condiion where L i a poiive conan. f, x) f, y) + g, x) g, y) L x y, x, y R, [, T ] 4) f, x) + g, x) L1 + x ), x R, [, T ], 5) Definiion Wang and Ning [13]) The backward uncerain differenial equaion 2) i aid o be able in meaure if for any wo oluion X and Y wih differen final value, we have for any given number ɛ >. lim M X Y > ɛ} =, [, T ] x T x T Definiion Wang and Ning [13]) The backward uncerain differenial equaion 2) i aid o be able in mean if for any wo oluion X and Y wih differen final value, we have lim E[ X Y ] =, [, T ]. x T x T 5

6 Definiion Wang and Ning [13]) The backward uncerain differenial equaion 2) i aid o be able in p-h momen < p < + ) if for any wo oluion X and Y wih differen final value, we have lim E[ X Y ) p ] =, [, T ]. x T x T Theorem 2.6. Wang and Ning [13]) If he backward uncerain differenial equaion 2) i able in mean, hen i i able in meaure. Theorem 2.7. Wang and Ning [13]) If he backward uncerain differenial equaion 2) i able in p-h momen < p < + ), hen i i able in meaure. 3 Sabiliy in diribuion In hi ecion, we inveigae he abiliy in diribuion for backward uncerainy differenial equaion, and prove a ufficien condiion for a backward uncerain differenial equaion being able in diribuion. Definiion 3.1. Le X and Y be wo oluion of backward uncerain differenial equaion 2) wih differen final value. Aume heir uncerainy diribuion of X and Y are Φ x) and Ψ y), repecively. Then he backward uncerain differenial equaion 2) i aid o be able in diribuion if for all x a which Φ and Ψ are coninuou. lim Φ x) Ψ x) =, [, T ] X T Y T Theorem 3.1. Aume he backward uncerain differenial equaion 2) ha a unique oluion for each given final value. Then i i able in diribuion if he coefficien f, x) and g, x) aify he rong Lipchiz condiion f, x 1 ) f, x 2 ) + g, x 1 ) g, x 2 ) L x 1 x 2, x 1, x 2 R, [, T ], where L i a poiive funcion aifying L d < +. Proof: Aume ha X and Y are wo oluion of backward uncerain differenial equaion 2) wih differen final value x T and y T [, T ]), repecively. Tha i, dx = f, X )d + g, X )dc, [, T ) X = x T, = T and dy = f, Y )d + g, Y )dc, [, T ) Y = y T, = T. Then for a Lipchiz coninuou ample C γ), we have X γ) = x T f, X γ))d g, X γ))dc γ), [, T ) X = x T, = T and Y γ) = y T f, Y γ))d g, Y γ))dc γ), [, T ) Y = y T, = T. 6

7 By Theorem 2.2, he invere uncerainy diribuion Φ 1 α) and Ψ 1 α) of X and Y aify he ordinary delay differenial equaion repecively, where dφ 1 dψ 1 Then for all α, 1), we obain Φ 1 α) = f, Φ 1 α))d + g, Φ 1 α)) Υ 1 α)d, α) = f, Ψ 1 α))d + g, Ψ 1 α)) Υ 1 α)d, α) Ψ 1 α) = x T y T Υ 1 α) = f, Φ 1 x T y T + + Υ 1 α) x T y T + + Υ 1 α) x T y T + + Υ 1 α) By he Grönwall inequaliy, we have Since Φ 1 f, Ψ 1 3 π ln α 1 α. α)) + g, Φ 1 f, Φ 1 g, Φ 1 f, Φ 1 g, Φ 1 L Φ 1 L Φ 1 x T y T Υ 1 α) ) x T y T Υ 1 α) ) α)) + g, Ψ 1 α))d f, Ψ 1 α)) g, Ψ 1 α)) Υ 1 α) ) d ) α)) Υ 1 α)d )) α)) ) d α)) ) d α))d f, Ψ 1 α)) d α)) g, Ψ 1 α)) d α) Ψ 1 α) d α) Ψ 1 α) d α) Ψ 1 α) x T y T exp here exi a real number N > uch ha exp 1 + Υ 1 α) ) L Φ 1 L Φ 1 L d < +, α) Ψ 1 α) d α) Ψ 1 α) d. 1 + Υ 1 α) ) L d for any [, T ]. Thu, for any given ɛ >, we e δ = ɛ/n uch ha Φ 1 α) Ψ 1 α) x T y T exp ) < N 1 + Υ 1 α) ) L d L d ) ). < δn = ɛ 7

8 for any provided x T y T < δ. Then we have By Theorem 2.4, we obain lim x T y T Φ 1 α) Ψ 1 α) =, α, 1). lim Φ x) Ψ x) =, [, T ], x R. x T y T Therefore, he backward uncerain differenial equaion 2) i able in diribuion. Thu he heorem i compleed. Remark 3.1. Theorem 3.1 give a ufficien condiion bu no a neceary condiion for a backward uncerain differenial equaion being able in diribuion. Example 3.1: Conider a backward uncerain differenial equaion dx = X d + σdc, [, T ) X = x T, = T. 6) and The oluion of equaion 6) wih wo final value x T and y T are X = Y = x T σ y T σ whoe invere uncerainy diribuion are and repecively. Then Therefore, which i equivalen o expt )dc ) exp T ) expt )dc ) exp T ), Φ 1 α) = x T exp T ) + σexp T ) 1) 3 π ln α 1 α Ψ 1 α) = y T exp T ) + σexp T ) 1) 3 π ln α 1 α lim x T y T Φ 1 Φ 1 α) Ψ 1 α) = x y ) exp T ) α) Ψ 1 α) =, [, T ], α, 1) lim Φ x) Ψ x) =, [, T ], x R. x T y T By Definiion 3.1, he backward uncerain differenial equaion 6) i able in diribuion. Example 3.2: Conider a backward uncerain differenial equaion dx = exp 2 )X d + exp X 2 )dc, [, T ) X = x T, = T. 7) Noe ha f, x) = exp 2 )x, g, x) = exp x 2 ) aify he ong Lipchiz condiion f, x 1 ) f, x 2 ) + g, x 1 ) g, x 2 ) exp 2 ) + exp )) x 1 x 2, x 1, x 2 R, [, T ], 8

9 wih exp 2 ) + exp ))d < +. From Theorem 3.1, he backward uncerain differenial equaion 7) i able in diribuion. Corollary 3.1. The linear backward uncerain differenial equaion dx = a 1 X + b 1 )d + a 2 X + b 2 )dc, [, T ) X = x T, = T. 8) i able in diribuion if he coefficien a 1, b 1, a 2 and b 2 are real valued bounded funcion, and a 1 and a 2 are i inegrable on [, T ]. f, x 1 ) f, x 2 ) + g, x 1 ) g, x 2 ) L x 1 x 2, x 1, x 2 R, [, T ], where L i a poiive funcion aifying L d < +. Proof: Take f, x) = a 1 x + b 1 and g, x) = a 2 x + b 2, and le R denoe he maxinum value of upper bound of a 1, b 1, a 2 and b 2. Since he experion f, x) + g, x) = a 1 x + b 1 + a 2 x + b 2 2R1 + x ) hold, he linear growh condiion i aified. Beide, we have f, x 1 ) f, x 2 ) + g, x 1 ) g, x 2 ) = a 1 + a 2 ) x 1 x 2, x 1, x 2 R, [, T ], Leing L = a 1 + a 2 ), we have L d = a 1 d + a 2 d < +. The rong Lipchiz condiion i aified. By uing Theorem 3.1, linear backward uncerain differenial equaion 8) i able in diribuion. Thu he corollary i compleed. Example 3.3: Conider a linear backward uncerain differenial equaion dx = X + 2 )d + in)x co))dc, [, T ) X = x T, = T. 9) Noe ha a 1 =, b 1 = 2, b 1 = in) and b 2 = co) aify condiion of Corollary 3.1, i.e., a 1 =, b 1 = 2, b 1 = in) and b 2 = co) are bounded funcion on [, T ], and and d = T < +. in)d = cot ) 1 < +. So he backward uncerain differenial equaion 9) i able in diribuion. 9

10 4 Some relaionhip of abiliy In hi ecion, we furher dicu heir relaionhip among abiliy in diribuion, abiliy in meaure, abiliy in mean and abiliy in momen. Theorem 4.1. If he backward uncerain differenial equaion 2) i able in meaure, hen i i able in diribuion. Proof: Aume ha X and Y are wo oluion of backward uncerain differenial equaion 2) wih differen final value x T and y T, repecively. According o Definiion 2.14, we have lim M X Y > ɛ} =, [, T ] x T y T for any given number ɛ >. Suppoe x i a given real number. On he one hand, for any y > x, i i clear ha X x} = X x, Y y} X x, Y > y} Y y} X Y y x}. I follow from he monooniciy heorem and ubaddiiviy axiom ha Thu we have Φ x) Ψ y) M X Y y x}. Φ x) Ψ y) lim M X Y y x}. x T y T I i clear ha M X Y y x} a x T y T. Then we have for any y > x. Leing y x +, we have On he oher hand, for any z < x, i i clear ha lim Φ x) Ψ y) x T y T lim Φ x) Ψ x). 1) x T y T Y z} = X x, Y z} X > x, Y z} X x} X Y x z}. I follow from he monooniciy heorem and ubaddiiviy axiom ha Thu we have Ψ z) Φ x) M X Y x z}. Ψ z) Φ x) lim M X Y x z}. x T y T I i clear ha M X Y x z} a x T y T. Then we have for any z < x. Leing z x, we have lim Ψ z) Φ x) x T y T lim Ψ x) Φ x). 11) x T y T 1

11 By equaion 1) and 11), we obain lim Φ x) Ψ x) =. x T y T According o Definiion 3.1, he backward uncerain differenial equaion 2) i able in diribuion. Thu he heorem i compleed. Theorem 4.2. If he backward uncerain differenial equaion 2) i able in mean, hen i i able in diribuion. Proof: If he backward uncerain differenial equaion 2) i able in mean, by Theorem 2.6, he backward uncerain differenial equaion 2) i able in meaure. And baed on Theorem 4.1, we obain he backward uncerain differenial equaion 2) i able in diribuion. Remark 4.1. Theorem 4.2 give a ufficien condiion bu no a neceary condiion for a backward uncerain differenial equaion being able in diribuion. Theorem 4.3. If he backward uncerain differenial equaion 2) i able in p-h momen < p < + ), hen i i able in diribuion. Proof: If he backward uncerain differenial equaion 2) i able in p-h momen < p < + ), by Theorem 2.7, he backward uncerain differenial equaion 2) i able in meaure. And baed on Theorem 4.1, we obain he backward uncerain differenial equaion 2) i able in diribuion. Remark 4.2. Theorem 4.3 give a ufficien condiion bu no a neceary condiion for a backward uncerain differenial equaion being able in diribuion. 5 Concluion Thi paper furher dicued abiliy in diribuion for backward uncerain differenial equaion, and proved a ufficien condiion for being able in diribuion. Moreover, hi paper alo dicued heir relaionhip among abiliy in diribuion, abiliy in meaure, abiliy in mean and abiliy in momen. Some example were dicued o illurae he heoreical conideraion. Acknowledgmen Thi work wa uppored by Naional Naural Science Foundaion of China Gran No ), Join Key Program of Naional Naural Science Foundaion of China Gran No. U173262), Innovaion Team Reearch Program of Univeriie in Xinjiang Uyghur Auonomou Region Gran No. XJEDU217T1) and hi work wa graned by China Scholarhip Council. Thi work wa compleed during he fir auhor viiing in he Deparmen of Mahemaical Science a Univeriy of Cincinnai, OH, USA. He i graeful for he hopialiy of Prof. Dan A. Ralecu. 11

12 Reference [1] X. Chen and B. Liu, Exience and uniquene heorem for uncerain differenial equaion, Fuzzy Opim. Deci. Making, 9 1) 21) pp [2] X. Chen and D. Ralecu, Liu proce and uncerain calculu, J. Uncerain. Anal. Appl ) Aricle 2. [3] Y. Gao, Exience and uniquene heorem on uncerain differenial equaion wih local Lipchiz condiion, J. Uncerain Sy. 6 3) 212) pp [4] X. Ge and Y. Zhu, A neceary condiion of opimaliy for uncerain opimal conrol problem, Fuzzy Opim. Deci. Making 12 1) 213) pp [5] B. Liu, Uncerainy Theory, 2nd ed., Springer-Verlag, Berlin, 27. [6] B. Liu, Fuzzy proce, hybrid proce and uncerain proce, J. Uncerain Sy. 2 1) 28) pp [7] B. Liu, Some reearch problem in uncerainy heory, J. Uncerain Sy. 3 1) 29) pp [8] B. Liu, Uncerainy Theory: A Branch of Mahemaic for Modeling Human Uncerainy, Springer-Verlag, Berlin, 21. [9] B. Liu and K. Yao, Uncerain inegral wih repec o muliple canonical procee, J. Uncerain Sy. 6 4) 212) pp [1] H. Liu, H. Ke and W. Fei, Almo ure abiliy for uncerain differenial equaion, Fuzzy Opim. Deci. Making, 13 4) 214) pp [11] Y. Sheng and C. Wang, Sabiliy in p-h momen for uncerain differenial equaion, J. In. Fuzz. Sy. 26 3) 214) pp [12] Y. Sheng and J. Gao, Exponenial abiliy for uncerain differenial equaion, Sof Compu. 2 9) 216) pp [13] X. Wang and Y. Sabiliy analyi of backward uncerain differenial equaion, under review. [14] X. Wang and Y. Ning, Almo ure and ph momen exponenial abiliy of backward uncerain differenial equaion, J. In. Fuzz. Sy., ) pp [15] K. Yao and X. Chen, A numerical mehod for olving uncerain differenial equaion, J. In. Fuzz. Sy. 25 3) 213) pp [16] K. Yao, J. Gao, and Y. Gao, Some abiliy heorem of uncerain differenial equaion, Fuzzy Opim. Deci. Making, 12 1) 213) pp [17] K. Yao, J. Gao and Y. Gao, Some abiliy heorem of uncerain differenial equaion, Fuzzy Opim. Deci. Making, 12 1) 213) pp [18] K. Yao, H. Ke. and Y. Sheng, Sabiliy in mean for uncerain differenial equaion, Fuzzy Opim. Deci. Making, 14 3) 215) pp [19] X. Yang, Y. Ni, and Y. Zhang, Sabiliy in invere diribuion for uncerain differenial equaion, J. In. Fuzz. Sy. 32 3) 217) pp