Stability and attractivity in optimistic value for dynamical systems with uncertainty

International Journal of General Systems ISSN: 38-179 (Print 1563-514 (Online Journal homepage: http://www.tandfonline.com/loi/ggen2 Stability and attractivity in optimistic value for dynamical systems with uncertainty Nana Tao & Yuanguo Zhu To cite this article: Nana Tao & Yuanguo Zhu (216 Stability and attractivity in optimistic value for dynamical systems with uncertainty, International Journal of General Systems, 45:4, 418-433, DOI: 1.18/38179.215.172522 To link to this article: http://dx.doi.org/1.18/38179.215.172522 Published online: 26 Oct 215. Submit your article to this journal Article views: 19 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalinformation?journalcode=ggen2 Download by: [Nanjing University Science & Technology] Date: 4 May 216, At: :32

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS, 216 VOL. 45, NO. 4, 418 433 http://dx.doi.org/1.18/38179.215.172522 Stability and attractivity in optimistic value for dynamical systems with uncertainty Nana Tao a,b and Yuanguo Zhu a a Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, China; b Department of Mathematics, Kaifeng University, Kaifeng, China Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 ABSTRACT Stability analysis is an important aspect to investigate the qualitative analysis of solutions of dynamical systems with uncertainty. And the optimistic value of uncertain variable is a critical value for handing optimization problems under uncertain environments. In this paper, we introduce two concepts of stability and attractivity in optimistic value for dynamical systems with uncertainty. Then, we present a sufficient condition of stability and a sufficient and necessary condition of attractivity for linear dynamical systems with uncertainty. Furthermore, we discuss the relationship between stability in measure (in mean, in distribution and stability in optimistic value, and the relationship between attractivity in measure (in mean, in distribution and attractivity in optimistic value. Last, a population model is considered. 1. Introduction ARTICLE HISTORY Received 19 June 214 Accepted 1 June 215 KEYWORDS Stability; attractivity; dynamical systems; uncertainty; optimistic value; population model Uncertainty theory was founded by Liu (27 and refined by Liu (21. Different from randomness and fuzziness, uncertainty theory provides a new mathematical model for uncertain phenomena, and becomes a branch of mathematics based on the normality axiom, duality axiom, subadditivity axiom and product axiom. For describing the evolution of uncertain phenomenon, a concept of uncertain process was presented by Liu (28, which is essentially a sequence of uncertain variable indexed by time or space. A canonical process designed by Liu (29 is a special type of uncertain process, and it is an independent and stationary process with normal uncertain increments. Uncertain dynamical system (UDS is a type of differential equation driven by the canonical Liu process and has been applied in many areas. For example, UDS has been applied to uncertain financial market by Liu (29 and uncertain optimal controlby Zhu (21. With many applications of UDS, the study on properties of the solutions also developed well (see Peng and Chen 214; Yao 213. The definition of stability was given by Liu (29. Following that many researchers did a lot of work (see Chen and Liu 21; Deng and Zhu 212; Gao 212; Ge and Zhu 212; Yao, Ke, and Sheng 215; Zhang 213 on CONTACT Yuanguo Zhu ygzhu@njust.edu.cn 215 Informa UK Limited, trading as Taylor & Francis Group

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 419 Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 UDS. Then, Tao and Zhu (215 gave the concept of attractivity of UDS and presented some sufficient and necessary conditions for linear UDS. In a pioneering study, optimism found by Scheier and Carver (1985affectsthe manner in which subjective judgments are constructed, and it is a fundamental construct that reflects how people respond to their perceived environments as well as how they explain and predict life events. Based on the uncertain measure, the optimistic value of an uncertain variable was introduced by Liu (29 for handling optimization problems under uncertain environments.sheng and Zhu (213 studied optimistic value model of uncertain optimal control. In this paper, we will introduce two concepts of stability and attractivity in optimistic value for UDS. For linear UDS, we will give a sufficient condition of stability and a sufficient and necessary condition of attractivity. Then for nonlinear UDS, we will discuss the relationship between stability in measure (in mean, in distribution and stability in optimistic value, and the relationship between attractivity in measure (in mean, in distribution and attractivity in optimistic value. Last, a population model will be given to illustrate our main results. The rest of this paper is organized as follows. In Section 2, we will review some concepts about uncertainty theory. In Section 3, we will introduce some concepts of stability and attractivity in optimistic value for UDS. In Section 4, we will propose a sufficient condition of stability in optimistic value for linear UDS and discuss the relationship between stability in measure (in mean, in distribution and stability in optimistic value for nonlinear UDS. In Section 5, We will give a sufficient and necessary condition of attractivity in optimistic value for linear UDS. Furthermore, we will discuss the relationship between attractivity in measure (in mean, in distribution and attractivity in optimistic value for nonlinear UDS. In Section 6, a population model will be given to illustrate our main results. 2. Preliminary We first review some notations and concepts in Liu (27. Let Ɣ be a nonempty set, and L a σ -algebra over Ɣ. Each element L is called an event. A set function M defined on L is called an uncertain measure if it satisfies that M{Ɣ} =1 for the universal set Ɣ; M{ }+M{ c }=1for any event ; { } M i M{ i } i=1 for i L. Then, the triplet (Ɣ, L, M is said to be an uncertainty space. To obtain an uncertain measure of a compound event, a product uncertain measure is defined. Let (Ɣ k, L k, M k be uncertainty spaces for k = 1, 2,. Then, the product uncertain measure M is an uncertain measure on the product σ -algebra L 1 L 2 satisfying i=1 { } M k = M k { k }. k=1 An uncertain variable is a measurable function ξ from an uncertainty space (Ɣ, L, Mto the set R of real numbers, i.e. for any Borel set of real numbers, the set {ξ B} ={γ k=1

42 N. TAO AND Y. ZHU Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 Ɣ ξ(γ B} is an event. The distribution :R [, 1] of an uncertain variable ξ is defined by (x = M{γ Ɣ ξ(γ x} for x R. The expected value of an uncertain variable ξ is defined by E[ξ] = M{ξ r}dr M{ξ r}dr provided that at least one of the two integrals is finite, and the variance of ξ is V[ξ] =E[(ξ E[ξ] 2 ]. Definition 1: Liu (28 Let (Ɣ, L, M be an uncertainty space and let T be a totally ordered set (e.g. time. An uncertain process is a function X t (γ from T (Ɣ, L, M to the set of real numbers such that {X t B} is an event for any Borel set B at each time t. Definition 2: Liu (29 An uncertain process C t is said to be a canonical process if (i C = and almost all sample paths are Lipschitz continuous, (ii C t has stationary and independent increments, (iii every increment C s+t C s is a normal uncertain variable N (, t with expected value and variance t 2, whose uncertainty distribution is t (x = ( ( 1 + exp πx 1, x R. 3t Definition 3: Liu (27 Letξ be an uncertain variable, and α (, 1].Then ξ sup (α = sup{r M{ξ r} α} is called the α-optimistic value to ξ. This means that the uncertain variable ξ will reach upwards of the α-optimistic value ξ sup (α with the α degree of belief. Definition 4: Liu (29 Let X t be an uncertain process and let C t be a canonical process. For any partition of closed interval [a, b] with a = t 1 <t 2 <t k+1 = b, the mesh is written as = max t i+1 t i. 1 i k Then the uncertain integral of X t with respect to C t is b a X t dc t = lim k X ti (C ti+1 C ti i=1 provided that the limit exists almost surely and is finite. In this case, the uncertain process X t is said to be integrable. For example, let f (t be an integrable function with respect to t. Then the uncertain integral s f (tdc t is a normal uncertain variable at each time s,and s ( f (tdc t N, s f (t dt.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 421 Definition 5: Liu (28 Suppose C t is a canonical process, and f and g are two given functions. Then dx t = f (t, X t dt + g(t, X t dc t (1 is called an uncertain differential equation (or UDS. A solution of (1 isanuncertain process X t that satisfies X t = X + f (s, X sds + g(s, X sdc s. Definition 6: Liu (29 TheUDS(1 is said to be stable (i.e. stable in measure if for any two solutions X t and Y t with initial values X and Y,respectively,wehave lim X Y M { X t Y t > ε} =, t > Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 for any given number ε >. Definition 7: Yao, Ke, and Sheng (215 The UDS (1 issaidtobestableinmeanif for any two solutions X t and Y t with initial values X and Y,respectively,wehave lim X Y E [ X t Y t ] =, t >. Definition 8: Zhang (213 LetX t and Y t be two solutions of the UDS (1withinitial values X and Y, respectively. Assume that the uncertainty distributions of X t and Y t are t (x and t (x, respectively. Then, the UDS (1 is said to be stable in distribution if lim X Y t(x t (x =, t >, x R. Definition 9: Tao and Zhu (215 LetX t and Y t be two solutions of the UDS (1with initial values X and Y, respectively. Then the UDS (1 is said to be attractive in measure if for any given ε >, there exists σ >, such that when X Y < σ,wehave lim t + M { X t Y t > ε} =. Definition 1: Tao and Zhu (215 LetX t and Y t be two solutions of the UDS (1with initial values X and Y, respectively. Then the UDS (1 is said to be attractive in mean if there exists σ >, such that when X Y < σ,wehave lim t + E[ X t Y t ] =. Definition 11: Tao and Zhu (215 Let X t and Y t be two solutions of the UDS (1 with initial values X and Y, respectively. Then the UDS (1 is said to be attractive in distribution if there exists σ >, such that when X Y < σ,wehave lim ϒ t(x t (x =, x R, t + where ϒ t (x and t (x are uncertainty distributions of X t and Y t, respectively. Theorem 1: (Liu 27, Markov Inequality Let ξ be an uncertain variable. Then for any given numbers t>andp>, we have M{ ξ t} E[ ξ p ] t p.

422 N. TAO AND Y. ZHU Theorem 2: Yao, Ke, and Sheng (215 Suppose that a(t, b(t, c(t and d(t are real functions [, +. Then the linear UDS is stable in mean if and only if dx t = (a(tx t + b(tdt + (c(tx t + d(tdc t (2 sup a(sds <+, t c(s ds < π 3. Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 Theorem 3: Tao andzhu (215 Suppose thata(t, b(t, c(t and d(t are continuous functions on [, +, and c(s ds < π. Then the linear UDS (2 is attractive in 3 mean if and only if a(sds =. 3. Concepts of stability and attractivity We will introduce concepts of stability and attractivity in optimistic value of UDS (1. 3.1. Concept of stability in optimistic value Definition 12: Let X t and Y t be two solutions of the UDS (1 with initial values X and Y, respectively. Then the UDS (1 is said to be stable in optimistic value if for α (, 1], we have Example 1: lim X t Y t sup (α = lim sup{r M{ X t Y t r} α} =. X Y X Y Consider the UDS dx t = μdt + σ dc t. Let X t and Y t be two solutions of UDS with initial values X and Y, respectively. Then X t = X + μt + σ C t, Y t = Y + μt + σ C t. For any given ε >, let δ = ε 2. When X Y < δ, one can obtain sup {r M{ X t Y t r} α} = sup {r M{ X Y r} α} < sup {r M{ ε } 2 r} α ε 2 < ε, where α (, 1]. Then the UDS dx t = μdt + σ dc t is stable in optimistic value.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 423 Example 2: Consider the UDS dx t = X t dt + σ dc t. Let X t and Y t be two solutions of UDS with initial values X and Y, respectively. Then Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 X t = exp (tx + σ exp (t Y t = exp (ty + σ exp (t exp ( sdc s, exp ( sdc s. Let ε = 1. For any δ > and< X Y < δ, there exists t>ln 2 X Y such that sup {r M{ X t Y t r} α} = sup { r M{exp (t X Y r} α } > 1 = ε, where α (, 1]. Thus the UDS dx t = X t dt + σ dc t is unstable in optimistic value. 3.2. Concept of attractivity in optimistic value Definition 13: Let X t and Y t be two solutions of the UDS (1 with initial values X and Y, respectively. Then the UDS (1 is said to be attractive in optimistic value if there exists σ >, such that when X Y < σ,wehave lim X t Y t sup (α = lim sup{r M{ X t Y t r} α} =, t + t + where α (, 1]. Example 3: Consider the UDS dx t = X t dt + σ dc t. Let X t and Y t be two solutions of the UDS with initial values X and Y, respectively. We have dx t = X t dt + σ dc t, dy t = Y t dt + σ dc t. It is easy to see that d ( X t Y t = ( Xt Y t dt. Thus X t Y t = ( X Y exp ( t.

424 N. TAO AND Y. ZHU For any given ε >, let σ = ε. When X Y < σ and t>ln 3, it follows that sup {r M{ X t Y t r} α} = sup { r M{ X Y exp ( t r} α } < sup {r M{ ε } 3 r} α ε 3 < ε, Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 where α (, 1]. Then the UDS dx t = X t dt + σ dc t is attractive in optimistic value. Example 4: Consider the UDS dx t = X t dt + σ dc t. Let X t and Y t be two solutions of UDS with initial values X and Y, respectively. Then X t = exp (tx + σ exp (t exp ( sdc s, Y t = exp (ty + σ exp (t exp ( sdc s. Let ε = 1. For any σ > and X Y < σ, such that when t>ln 2 X Y,wehave sup {r M{ X t Y t r} α} = sup { r M{exp (t X Y r} α } > 1 = ε, where α (, 1]. It follows that the UDS dx t = X t dt + σ dc t is not attractive in optimistic value. 4. Stability analysis 4.1. Stability analysis for linear UDS We will give a sufficient condition of stability for linear UDS dx t = (a(tx t + b(tdt + (c(tx t + d(tdc t (3 Theorem 4: Suppose that a(t, b(t, c(t and d(t are continuous functions [,+. Then the linear UDS (3 is stable in optimistic value if sup a(sds <+, t and c(s ds <+. Proof: Suppose X t and Y t are two solutions of the linear UDS (3 with initial values X and Y, respectively. It follows that d ( X t Y t = a(t(xt Y t dt + c(t(x t Y t dc t.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 425 Then, we have ( X t Y t = (X Y exp a(sds + c(sdc s. Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 Note that c(sdc s N (, c(s ds and c(s ds <+, it is easy to see that 3 Ft 1 (1 α = c(s ds ln 1 α < +, π α where F t (x is the uncertainty distribution of c(sdc s and α (, 1].Since it follows that { ( M{ X t Y t r} =M X Y exp a(sds + { r = M c(sdc s > ln X Y = 1 F t (ln r X Y a(sds, } c(sdc s r } a(sds lim X t Y t sup (α = lim sup{r M{ X t Y t r} α} X Y X Y ( = lim sup r t } {r 1 F t ln X Y X Y a(sds α { ( } = lim sup r r X Y exp Ft 1 (1 α + a(sds X Y = with Ft 1 (1 α < + and sup a(sds <+. t Therefore, the linear UDS (3 is stable in optimistic value. Example 5: Consider the linear UDS dx t = cos tx t dt + exp ( 2tX t dc t. Since a(t = cos t and c(t = exp ( 2t, we immediately have sup a(sds = 1 < + t and c(s ds = 1 2 < +. Thus, the linear UDS dx t = cos tx t dt + exp ( 2tX t dc t is stable in optimistic value.

426 N. TAO AND Y. ZHU 4.2. Stability analysis for nonlinear UDS We will discuss the relationship between stability in measure (in mean, in distribution and stability in optimistic value for UDS dx t = f (t, X t dt + g(t, X t dc t (4 Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 Theorem 5: The UDS (4 is stable in measure if and only if it is stable in optimistic value. Proof: Suppose that X t and Y t are two solutions of the UDS (4 with initial values X and Y,respectively.IftheUDS(4 is stable in measure, i.e. for any given number ɛ > and ε >, there exists δ >, such that when X Y < δ,wehave for any t>. But if M { X t Y t > ɛ} < ε, lim X t Y t sup (α = lim sup{r M{ X t Y t r} α } =, X Y X Y where α (, 1], there exists ε >, such that when X Y < δ, sup{r M{ X t Y t r} α } > ε, it follows that M{ X t Y t > ε } α >. It contradicts with the concept of stability in measure. Thus, the UDS (4 is stable in optimistic value. Conversely,suppose that the UDS (4 is stable in optimistic value, i.e. lim X t Y t sup (α = lim sup{r M{ X t Y t r} α} =. X Y X Y If the UDS (4 is unstable in measure, there exist ɛ > andε >, such that for any δ > with X Y < δ, M { X t Y t > ɛ } > ε, it follows that sup{r M { X t Y t r} ε } ɛ. It contradicts with the concept of stability in optimistic value. Thus, the UDS (4 is stable in measure. The theorem is proved. Theorem 6: Suppose that the UDS (4 is stable in mean. Then it is stable in optimistic value. Proof: Suppose that X t and Y t are two solutions of the UDS (4 with initial values X and Y, respectively, and the UDS (4 is stable in mean. It follows from Theorem 1 Markov inequality that for any given ε >, we have M { X t Y t > ε} E[ X t Y t ] ε

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 427 as X Y. Thus the UDS (4 is stable in measure. By Theorem 5,itiseasytosee that theuds (4 is stable in optimistic value. Example 6: Consider the linear UDS dx t = exp (tx t dt + exp ( 2tX t dc t. Since a(t = exp (t and c(t = exp ( 2t, we immediately have a(sds = and c(s ds = 1 2 < π 3. Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 By Theorem 2,thelinearUDS dx t = exp (tx t dt+exp ( 2tX t dc t is stable in mean. Then, it is stable in optimistic value. Remark 1: Generally, stability in optimistic value does not imply stability in mean. Consider the linear UDS dx t = cos tx t dt + 2exp( tx t dc t. Since a(t = cos t and c(t = 2exp( t, we immediately have sup a(sds = 1 < + and c(s ds = 2 > π. t 3 By Theorem 4,thelinearUDS dx t = cos tx t dt + 2exp( tx t dc t is stable in optimistic value. But it is unstable in mean from Theorem 2. Theorem 7: Suppose that the UDS (4 is stable in optimistic value. Then it is stable in distribution. Proof: Suppose that the UDS (4 is stable in optimistic value. By Theorem 5, it is stable in measure. Then it is stable in distribution. Example 7: Consider the UDS dx t = 1 1 + t 2 X tdt + exp ( tx t dc t. It is easy to see that and 1 sup t 1 + t 2 dt = π 2 < +, exp ( tdt = 1 < +.

428 N. TAO AND Y. ZHU By Theorem 4,theUDS dx t = 1 1 + t 2 X tdt + exp ( tx t dc t Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 is stable in optimistic value. Then it is stable in distribution. Remark 2: We know that optimal control theory has been applied to many fields. Our work provides a good method for solving problems of the optimistic value controls from the relationship among stability concepts. stability in measure stability = = stability in mean stability in optimistic value in distribution 5. Attractivity analysis 5.1. Attractivity analysis for linear UDS We will give a sufficient and necessary condition of attractivity for linear UDS dx t = (a(tx t + b(tdt + (c(tx t + d(tdc t (5 Theorem 8: Suppose that a(t, b(t, c(t and d(t are continuous functions [, +, and c(s ds <+. Then, the linear UDS (5 is attractive in optimistic value if and only if a(sds =. Proof: Suppose X t and Y t are two solutions of the linear UDS (5 with initial values X and Y, respectively. Then, we have ( X t Y t = (X Y exp a(sds + c(sdc s. Note that c(sdc s N (, c(s ds and c(s ds <+. It is easy to see that 3 Ft 1 (1 α = c(s ds ln 1 α π α < +. where F t (x is the uncertainty distribution of c(sdc s and α (, 1].Since

it follows that { ( M{ X t Y t r} =M X Y exp { = M c(sdc s > ln = 1 F t (ln INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 429 r X Y a(sds + } c(sdc s r } r X Y a(sds a(sds, Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 lim X t Y t sup (α = lim sup{r M{ X t Y t r} α} t + t + ( = lim {r sup r t } 1 F t ln t + X Y a(sds α { ( } = lim sup r r X Y exp F 1 t + t (1 α + a(sds. Since Ft 1 (1 α < +, weknowthat lim X t Y t sup (α = if and only if t + a(sds =. Thus, the theorem is proved. Example 8: Consider the linear UDS dx t = 1 t + 1 X tdt + exp ( tx t dc t. Since a(t = t+1 1 and c(t = exp ( t,weimmediatelyhave a(sds = and c(s ds = 1 < +. Thus the linear UDS dx t = 1 t+1 X tdt + exp ( tdc t is attractive in optimistic value. 5.2. Attractivity analysis for nonlinear UDS We will discuss the relationship between attractivity in measure (in mean, in distribution and attractivity in optimistic value for UDS dx t = f (t, X t dt + g(t, X t dc t (6 Theorem 9: The UDS (6 is attractive in measure if and only if it is attractive in optimistic value. Proof: Suppose that X t and Y t are two solutions of (6 with initial values X and Y, respectively. If (6 is attractive in measure, i.e. for any given number ɛ > andε >, there exists σ > andt>such that when X Y < σ and t>t,wehave

43 N. TAO AND Y. ZHU But if M { X t Y t > ɛ} < ε. lim X t Y t sup (α = lim sup{r M{ X t Y t r} α } =, t + t + Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 where α (, 1], there exist ε > such that for any σ > andt>, when X Y < δ and t>t,wehave sup{r M{ X t Y t r} α } ε, it follows that M{ X t Y t ε } α > which contradicts with the concept of attractive in measure. Thus, the UDS (6 is attractive in optimistic value. Conversely,suppose that the UDS (6 is attractive in optimistic value, i.e. lim X t Y t sup (α = lim sup{r M{ X t Y t r} α} =. t + t + If the UDS (6 is not attractive in measure, there exist ɛ > andε > such that for any δ > andt>, when X Y < δ and t>t,wehave It follows that M { X t Y t > ɛ } > ε. sup{r M { X t Y t r} ε } ɛ which contradicts with the concept of attractivity in optimistic value. Thus, the UDS (6is attractive in measure. The theorem is proved. Theorem 1: Suppose that the UDS (6 is attractive in mean. Then it is attractive in optimistic value. Proof: Suppose that X t and Y t are two solutions of (6 with initial values X and Y, respectively, and (6 is attractive in mean. It follows from Theorem 1 Markov inequality that for any given ε >, we have M { X t Y t > ε} E[ X t Y t ] ε as t +. Thus, the UDS (6 is attractive in measure. By Theorem 9, itiseasytosee that theuds (6 is attractive in optimistic value. Example 9: Consider the linear UDS dx t = exp (tx t dt + exp ( 2tX t dc t. Since a(t = exp (t and c(t = exp ( 2t, we immediately have a(sds = and c(s ds = 1 2 < π 3.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 431 By Theorem 3,thelinearUDS dx t = exp (tx t dt + exp ( 2tX t dc t Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 is attractive in mean. Then it is attractive in optimistic value. Theorem 11: Suppose that the UDS (6 is attractive in optimistic value. Then it is attractive in distribution. Proof: Suppose that the UDS (6 is attractive in optimistic value. By Theorem 9, itis attractive in measure. Then it is attractive in distribution. Example 1: Consider the linear UDS dx t = exp (2tX t dt + exp ( tx t dc t. Since a(t = exp (2t and c(t = exp ( t, we immediately have c(s ds = 1 < + and lim a(sds =. t + Then dx t = exp (2tX t dt + exp ( tx t dc t is attractive in optimistic value by Theorem 8. And it is easy to see that dx t = exp (2tX t dt + exp ( tx t dc t is attractive in distribution by Theorem 11. Remark 3: Relationship among attractivity concepts attractivity in mean 6. Application attractivity in measure = = attractivity in optimistic value attractivity in distribution Let X t denote the population number at time t in a district. Generally a dynamical system is used to model the behavior of X t : dx t = μ ( K Xt where μ > is the growing rate parameter, and K is a constant which means the balance level for the number of population in the district. In fact, it is impossible that the population number X t exactly follows that dynamical system. It is reasonable to say that a dynamical system with some noises is suitable for the population number. Malinowski and Michta (211 studied population dynamics with random noises. However, there is no evidence to show that the noises upon the population dynamical system are indeed random. Now we will suppose that the noises affecting the population dynamics are uncertain. That is, we propose the following UDS as a mathematical model of population growth: dx t = μ K ( K Xt K dt dt + σ dc t,

432 N. TAO AND Y. ZHU where σ >, and C t is a canonical process. For a(t = μ K and c(t =, we immediately have c(s ds = < + and ( K Xt lim t + a(sds =. Then dx t = μ K dt + σ dc t is attractive in optimistic value by Theorem 8. Let X <K. The above result shows that the population will come back to a balanced level in a long run despite that the population at the initial stage may have an explosive growth. Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 7. Conclusion In this paper, we introduced stability and attractivity concepts for UDS: stability in optimistic value and attractivity in optimistic value. For linear UDS, we presented a sufficient condition of stability and a sufficient and necessary condition of attractivity. Besides, for nonlinear UDS, we deduced stability (attractivity in measure can be equivalent to the stability (attractivity in optimistic value. And stability (attractivity in mean implies stability (attractivity in optimistic value, and then stability (attractivity in distribution. A population model was given to illustrate our main results. Funding This work is supported by the National Natural Science Foundation of China [grant number 612739]. Notes on contributors Nana Tao is a PhD candidate in Nanjing University of Science and Technology. Yuanguo Zhu is a professor of Mathematics in Nanjing University of Science and Technology located at No. 2 Xiaolingwei Street, Nanjing 2194, Jiangsu, China. His research interests include uncertain systems, optimal control, optimization and intelligent computing.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 433 References Downloaded by [Nanjing University Science & Technology] at :32 4 May 216 Chen, X., and B. Liu. 21. Existence and Uniqueness Theorem for Uncertain Differential Equations. Fuzzy Optimization and Decision Making 9 (1:69 81. Deng, L., and Y. Zhu. 212. Existence and Uniqueness Theorem of Solution for Uncertain Differential Equations with Jump. ICIC Express Letters 6 (1:2693 2698. Liu, B. 27. Uncertainty Theory. 2nd ed. Berlin: Springer-Verlag. Liu, B. 28. Fuzzy Process, Hybrid Process and Uncertain Process. Journal of Uncertain Systems 2 (1: 3 16. Liu, B. 29. Some Research Problems in Uncertainty Theory. Journal of Uncertain Systems 3(1: 3 1. Liu, B. 21. Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin: Springer-Verlag. Gao, Y. 212. Existence and Uniqueness Theorems of Uncertain Differential Equations with Local Lipschitz Condition. Journal of Uncertain Systems 6 (3: 223 232. Ge, X., and Y. Zhu. 212. Existence and Uniqueness Theorem for Uncertain Delay Differential Equations. Journal of Computational Information Systems 8 (2:8341 8447. Malinowski, M., and M. Michta. 211. Stochastic Fuzzy Differential Equations with an Application. Kybernetika 47: 123 143. Peng, Z., and X. Chen. 214. Uncertain Systems are Universal Approximators. Journal of Uncertainty Analysis and Aplications 2: Article 13. Scheier, M. F., and C. Carver. 1985. Optimism, Coping and Health: Assessment and Ipllications of Generalized Expectancies. Health Psychology 4 (3: 217 247. Sheng, L., and Y. Zhu. 213. Optimistic Value Model of Uncertain Optimal Control. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21 (1: 75 83. Tao, N., and Y. Zhu. 215. Attractivity and Stability Analysis of Uncertain Differential Systems. International Journal of Bifurcation and Chaos 25 (2: 15522-1 15522-1. Yao, K. 213. A Type of Nonlinear Uncertain Differential Equations with Analytic Solution. Journal of Uncertainty Analysis and Applications 1: Article 8. Yao, K., H. Ke, and Y. Sheng. 215. Stability in Mean for Uncertain Differential Equation. Fuzzy Optimization and Decision Making.doi:1.17/s17-14-924-2. Zhang, Y. 213. Stability in Distribution for Uncertain Differential Equation. http://orsc.edu.cn/ online/1353.pdf. Zhu, Y. 21. Uncertain Optimal Control with Application to a Portfolio Selection Model. Cybernetics and Systems: An International Journal 41 (7: 535 547.