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2012 9th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2012)

Existence and Uniqueness of Solutionsto Uncertain Functional Differential Equations

Hongjian Liu, Weiyin FeiSchool of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, Anhui, P.R. China

Abstract: The canonical process is a Lipschitz continuousuncertain process with stationary and independent incre-ments, and uncertain functional differential equationsdriven by the canonical process give a mathematicalformulation for dynamic systems. This paper proves anexistence and uniqueness theorem of solutions for uncer-tain functional differential equations under the uniformLipschitz condition and the linear growth condition.

1. Introduction

Randomness is a basic type of objective uncertainty, andprobability theory is a branch of mathematics for studyingthe behavior of random phenomena. The study of probabilitytheory was started by Pascal and Fermat (1654), and anaxiomatic foundation of probability theory was given byKolmogoroff (1933).

The concept of fuzzy set was initiated by Zadeh [13] viamembership function in 1965. In order to measure a fuzzyevent, Zadeh [14] introduces the theory of possibility. Someinformation and knowledge in practice are usually repre-sented by human language like “about 100km”, “roughly80kg”, “low speed”, “middle age”, and “big size”. A lotof surveys show that the real life phenomenon imprecisequantities behave neither like randomness nor like fuzziness.In order to deal with this problem, uncertainty theory,founded by Liu [7] in 2007 to study the behavior of uncertainphenomena, is a branch of mathematics based on normal-ity, monotonicity, self-duality, countable subadditivity, andproduct measure axioms. It is a new tool to study subjectiveuncertainty.

Differential equations have been widely applied inphysics, engineering, biology, economics and other fields.With the development of science and technology, practi-cal problems require more and more accurate description.A wide range of uncertainties are added to the differen-tial equation system, thus producing stochastic differentialequations, fuzzy differential equations and fuzzy stochas-tic differential equations, e.g., see Fei [4] and Fei et al.[6]. Furthermore, uncertain differential equation, a typeof differential equations driven by canonical process, wasdefined by Liu [7] in 2007. Chen and Liu [1] present anexistence and uniqueness theorem of solution for uncertain

differential equation under Lipschitz condition and lineargrowth condition.

In many applications, one assumes that the system underconsideration is governed by a principle of causality; thatis, the future state of the system is independent of the paststates and is determined solely by the present. However,under closer scrutiny, it becomes apparent that the principleof causality is often only a first approximation to the truesituation and that a more realistic model would includesome of the past states of the system. Uncertain functionaldifferential equations give a mathematical formulation forsuch system.

The simplest type of past dependence in a differentialequation is that in which the past dependence is through thestate variable but not the derivative of the state variable. LordCherwell (see Wright [12]) has encountered the differentialdifference equation

x(t) = −αx(t− 1)[1 + x(t)]

in his study of the distribution of primes. Dunkel [2] suggeststhe more general equation

x(t) = −α

[ ∫ 0

−1

x(t+ θ)dη(θ)

][1 + x(t)]

for the growth of a single species. The equation

x(t) = −∫ t

t−τ

a(t− θ)g(x(θ))dθ

is encountered by Ergen [3] in the theory of a circulatingfuel nuclear reactor. Taking into account the transmissiontime in the triode oscillator, Rubanik [11] has studied thevan der Pol equation

x(t) + αx(t)− f(x(t− τ))x(t− τ) + x(t) = 0

with the delayed argument τ . All these equations are specialcases of the general functional differential equation

x(t) = f(xt, t),

where xt = (x(t + θ) : −τ ≤ θ ≤ 0) is the past history ofthe state. Taking into account the environmental noise weare led to the uncertain functional differential equation

dx(t) = f(xt, t)dt+ g(xt, t)dC(t).

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In this paper, we study the existence and uniqueness of thesolution to the uncertain functional differential equationswith a given initial-value.

The rest of the paper is organized as follows. Somepreliminary concepts of uncertainty theory are recalled inSect. 2. An existence and uniqueness theorem is proved inSect. 3. Finally, a brief summary is given in Sect. 4.

2. Preliminaries

Let Γ be a nonempty set, and L a σ-algebra over Γ. EachΛ ∈ L is called an event.

Definition 1 (Liu[7]) A set function M is called an uncer-tain measure if it satisfies the following four axioms:Axiom 1 (Normality) M{Λ} = 1;Axiom 2 (Monotonicity) M{Λ1} ≤ M{Λ2}, wheneverΛ1 ⊂ Λ2;Axiom 3 (Self -Duality) M{Λ} + M{Λc} = 1, for anyevent Λ;Axiom 4 (Countable Subadditivity) For every countablesequence of events {Λi}, we have

M{ ∞∪

i=1

Λi

}≤

∞∑i=1

M{Λi}.

The following is the definition of uncertain variable.

Definition 2 (Liu[7]) An uncertain variable is a measurablefunction from an uncertainty space (Γ,L,M) to the set ofreal numbers, i.e., for any Borel set B of real numbers, theset

M{ξ ∈ B} = {γ ∈ Γ|ξ(γ) ∈ B}

is an event.Let T be an index set and let (Γ,L,M) be an uncertaintyspace. An uncertain process is a measurable function fromT×(Γ,L,M) to the set of real numbers, i.e., for each t ∈ Tand any Borel set B of real numbers, the set

{xt ∈ B} = {γ ∈ Γ|xt(γ) ∈ B}

is an event.

Now, let us introduce a special uncertain process calledthe canonical process, that plays the role of counterpart ofBrownian motion.

Definition 3 (Liu[9]) An uncertain process Ct = (C(t), t ∈[0,∞)) is said to be a canonical process if(i) it starts at zero: C(0) = 0 and almost all sample pathsare Lipschitz continuous,(ii) it has stationary and independent increments,(iii) every increment C(t+ s)−C(s) is a normal uncertainvariable with expected value 0 and variance t2.

The uncertain process dCt = (C(t+ dt)−C(t), t ∈ [0,∞))has the properties that E[dCt] = 0 and dt2/2 ≤ E[dC2

t ] ≤dt2. Then dCt and dt are infinitesimals of the same order.Liu [7] has been proved that dCt/dt is a normal uncertainvariable with expected value 0 and variance 1.

Definition 4 (Liu[9]) Let xt be an uncertain process andlet Ct be a canonical process. For any partition of closedinterval [a, b] with a = t1 < t2 < · · · < tk+1 = b, the meshsize is written as

∆ = max1≤i≤k

|ti+1 − ti|.

Then the uncertain integral of x(t) with respect to C(t) is∫ b

a

x(t)dC(t) = lim∆→0

k∑i=1

x(ti) · (C(ti+1)− C(ti))

provided that the limit exists almost surely and is an uncer-tain variable.

Let h(t, c) be a continuously differentiable function. Thenxt = h(t, Ct) is an uncertain process. Liu [9] proved thefollowing chain rule

dx(t) =∂h

∂t(t, C(t))dt+

∂h

∂c(t, C(t))dC(t).

Definition 5 A m-dimensional process {Ct =(C1

t , · · · , Cmt )}{t≥0} is called a m-dimensional canonical

process if every {Cit} is a one-dimensional canonical

process, and {C1t }, · · · , {Cm

t } are independent.

3. Existence and uniqueness theorem

In the following, let (Γ,L,M) be an uncertain space withthe filtration {Lt}t≥0, and Ct is the given m-dimensionalcanonical process defined on the space. Let τ > 0 and denoteby C([−τ, 0];Rd) the family of continuous functions φ from[−τ, 0] to Rd with the norm ∥ φ ∥= sup−τ≤θ≤0 |φ(θ)|. Let0 ≤ t0 < T < ∞. Let f : C([−τ, 0];Rd) × [t0, T ] →Rd and g : C([−τ, 0];Rd) × [t0, T ] → Rd×m be Borelmeasurable. Consider the d-dimensional uncertain functionaldifferential equation

dx(t) = f(xt, t)dt+ g(xt, t)dC(t), t ∈ [t0, T ], (1)

where xt = {x(t + θ) : −τ ≤ θ ≤ 0} is regarded asa C([−τ, 0];Rd)-value uncertain process. We impose theinitial data:

xt0 = β = {β(θ) : −τ ≤ θ ≤ 0} is anLt0−measurable C([−τ, 0];Rd)− valueuncertain variable such that E ∥ β ∥2< ∞.

(2)

The initial-value problem for equation (1) is now to find thesolution of equation (1) satisfying the initial data (2). But,what is the solution?

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Definition 6 An Rd-value uncertain process xt on t0− τ ≤t ≤ T is called a solution to equation (1) with initial data(2) if it has the following properties:(i) it is continuous and {xt}t0≤t≤T is Lt-adapted;(ii)

∫ T

t0|f(x, t)|dt < ∞ and

∫ T

t0|g(x, t)|dt < ∞;

(iii) xt0 = β and, for every t0 ≤ t ≤ T ,

x(t) = β(0) +

∫ t

t0

f(xs, s)ds+

∫ t

t0

g(xs, s)dC(s) a.s.

A solution x(t) is said to be unique if any other solutionx(t) is indistinguishable from it, that is

M{x(t) = x(t) for all t0 − τ ≤ t ≤ T} = 1.

Let us now we begin to establish the theorem of theexistence and uniqueness of the solution for the uncertainfunctional differential equation (1) with initial data (2).

Theorem 1 Assume that there exists two positive constantsK and K such that(i) (uniform Lipschitz condition) for all t ∈ [t0, T ] andφ, ϕ ∈ C([−τ, 0];Rd)

|f(φ, t)− f(ϕ, t)| ∨ |g(φ, t)− g(ϕ, t)| ≤ K ∥ φ−ϕ ∥; (3)

(ii) (linear growth condition)for all (φ, t) ∈ C([−τ, 0];Rd)× [t0, T ],

|f(φ, t)| ∨ |g(φ, t)| ≤ K(1+ ∥ φ ∥). (4)

Then there exists a unique solution x(t) to equation (1) withinitial data (2).

We prepare following lemmas in order to prove thistheorem.

Lemma 1 Let g(t) be a continuous uncertain process, thefollowing inequality of uncertain integral holds∣∣∣∣ ∫ T

t0

g(t)dC(t)

∣∣∣∣ ≤ ζ

∫ T

t0

|g(t)|dt, (5)

where ζ is a uncertain variable called the Lipschitz constantof a canonical process with

ζ(γ) =

{sup0≤s<t

|C(t,γ)−C(s,γ)|t−s , if M{γ} > 0,

∞, othewise,

and E[ζp] < ∞,∀p > 0.

Proof. Similar to the proof of Lemma 2.3 in Fei [5], we canshow Lemma 1. ♯

Lemma 2 Let the linear growth condition (4) hold. If x(t) isa solution to equation (1) with initial data (2), then

E

(sup

t0−τ≤t≤T|x(t)|

)≤

(1+2E ∥ β ∥

)eK(T−t0)(T−t0+ζ).

(6)

Proof. For every integer n ≥ 1, define the stopping time

τn = T ∧ inf{t ∈ [t0, T ] :∥ xt ∥≥ n}.

Clearly, τn ↑ T a.s. Set xn(t) = x(t∧τn) for t ∈ [t0−τ, T ].Then, for t0 ≤ t ≤ T ,

xn(t) = β(0) +∫ t

t0f(xn

s , s)I[[t0,τn]](s)ds

+∫ t

t0g(xn

s , s)I[[t0,τn]](s)dC(s).

By Lemma 1, and the linear growth condition, we show that

E

(supt0−τ≤t≤T |xn(s)|

)≤ E|β(0)|+K(1 + ζ)

∫ t

t0(1 + E ∥ xn

s ∥)ds.

Noting that

supt0−τ≤s≤t

|xn(s)| ≤∥ β ∥ + supt0≤s≤t

|xn(s)|,

we obtain

1+ E

(sup

t0−τ≤s≤t|xn(s)|

)≤ 1 + 2E ∥ β ∥

+K(1 + ζ)∫ t

t0

[1 + E

(sup

t0−τ≤r≤s|xn(r)|

)]ds.

Now the Gronwall inequality yields that

1+E

(sup

t0−τ≤t≤τn

|x(t)|)

≤(1+2E ∥ β ∥

)eK(T−t0)(1+ζ).

Consequently,

E

(sup

t0−τ≤t≤τn

|x(t)|)

≤(1 + 2E ∥ β ∥

)eK(T−t0)(1+ζ).

Finally the required inequality (6) follows by letting n →∞. ♯

Proof of Theorem 1 Uniqueness. Let x(t) and x(t) be thetwo solutions. By Lemma 2, noting

x(t)− x(t) =∫ t

t0[f(xs, s)− f(xs, s)]ds

+∫ t

t0[g(xs, s)− g(xs, s)]dC(s),

we can easily show that

E

(sup

t0≤s≤t|x(s)− x(s)|

)≤ K(1 + ζ)

∫ t

t0E ∥ xs − xs ∥ ds

= K(1 + ζ)∫ t

t0E

(sup

t0≤r≤s|x(r)− x(r)|

)ds.

The Gronwall inequality then yields that

E(

supt0≤t≤T

|x(t)− x(t)|)= 0,

which implies that x(t) = x(t) for t0 ≤ t ≤ T , hence forall t0 − τ ≤ t ≤ T , almost surely. Thus, the uniqueness hasbeen proved.

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Existence. Define x0t0 = β and x0(t) = β(0) for t0 ≤ t ≤

T . For each n = 1, 2, · · ·, set xnt0 = β and define, by the

Picard iterations,

xn(t) = β(0) +

∫ t

t0

f(xn−1s , s)ds+

∫ t

t0

g(xn−1s , s)dC(s)

(7)for t ∈ [t0, T ]. We claim that for all n ≥ 0 and t ∈ [t0, T ]

E

(sup

t0≤s≤t|xn+1(t)− xn(s)|

)≤ H[M(t− t0)]

n

n!, (8)

where M = K(1 + ζ) and H will be defined below. Firstwe compute

E

(sup

t0≤t≤T|x1(t)− x0(t)|

)≤ K

∫ T

t0(1 + E∥x0

s∥)ds+Kζ∫ T

t0(1 + E∥x0

s∥)ds≤ K(1 + ζ)(T − t0)(1 + E∥β∥) := H.

So (8) holds for n = 0. Next, assume (8) holds for somen ≥ 0. Then

E(

supt0≤s≤t

|xn+2(s)− xn+1(s)|)

≤ K(t− t0 + ζ)E∫ t

t0∥xn+1

s − xns ∥ds

≤ M∫ t

t0E

(sup

t0≤r≤s|xn+1(r)− xn(r)|

)ds

≤ M∫ t

t0

H[M(s−t0)]n

n! ds = H[M(t−t0)]n+1

(n+1)! .

That is, (8) holds for n+1. Hence, by induction, (8) holdsfor all n ≥ 0. From (8), we can then show that xn(·)converges to x(t) in the sense of L1 as well as probability1, and the x(t) is a solution to equation (1) satisfying theinitial condition (2). The existence has also been proved.Thus, the proof of the theorem is complete. ♯

In the proof above we have shown that the Picard itera-tions xn(t) converge to the unique solution x(t) of equation(1).

4. Conclusions

The theory of uncertain functional differential equationsis an important tool to deal with dynamic systems includingthe past states in uncertain environments. The contributionof this paper to the uncertain functional differential equationtheory is to provide an existence and uniqueness theoremunder the uniform Lipschitz condition and the linear growthcondition.

Acknowledgment

This paper is supported by National Natural ScienceFoundation of China (No. 71171003), Anhui Natural ScienceFoundation (No. 090416225), Anhui Natural Science Foun-dation of Universities (No. KJ2010A037) and Anhui Poly-technic University young teachers and scientific researchfund projects (No. 2009YQ032) .

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