# Nonlinear Fuzzy Differential Equation with Time Delay and Optimal Control Problem

Authors: Wichai Witayakiattilerd
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

The existence and uniqueness of a mild solution to nonlinear fuzzy differential equation constrained by initial value were proven. Initial value constraint was then replaced by delay function constraint and the existence of a solution to this type of problem was also proven. Furthermore, the existence of a solution to optimal control problem of the latter type of equation was proven.


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## Body

## 1. Introduction

Fuzzy logic is originated by Zadeh in 1965. It is primarily based on the fact that “all things happening in real world are unstable and unpredictable.” This idea was put forward and successfully applied to many fields of research—such as medicine, computer science, engineering, and economics—owing to its remarkable effectiveness at solving problems that could not be solved by traditional logic; see [1–3] and references therein. In particular, fuzzy logic has long been applied to dynamic systems expressed in differential equations; see [4–15] and references therein. Moreover, dynamic system with time delay can be advantageously applied to many important problems such as determining the current position of a particle from the history of its past movement; see [16–19] and references therein. In this study, fuzzy differential equation of dynamic system constrained by time delay was investigated. The objectives of this investigation were to delineate the definitions of and theorems on fuzzy control system with time delay and to find the necessary conditions for the existence of a solution to this type of system by functional analysis.


## 2. Preliminaries

This section discusses the definitions and theorems pertaining to this research.Definition 1.
LetRF be a family of fuzzy subset of R, called afuzzy number space. It satisfies the following conditions, for each u∈RF:(1)
u is normal; that is, there exists x0∈R such that u(x0)=1.(2)
u is a convex fuzzy set; that is, u(tx+(1-t)y)≥min⁡{u(x),u(y)} for all t∈[0,1] and x,y∈R.(3)
u is upper semicontinuous on R; that is, for each x∈R and for all sequences xn∈R, if xn→x then limn→∞⁡sup⁡u(xn)≤u(x).(4)
{x∈R∣u(x)>0}¯ is compact; that is, for all sequences xn∈{x∈R∣u(x)>0}¯, there is a subsequence xkn such that xkn→y∈{x∈R∣u(x)>0}¯.Notice that R=χx∣xisreal and R⊂RF.Definition 2.
Letu∈RF and 0≤r≤1. The set of r-cut of u, denoted as [u]r, is defined by (1)ur=x∈R∣ux≥rfor0<r≤1,u0=x∈R∣ux>0¯.According to Definition1, conditions (1)–(4) imply that [u]r for all 0≤r≤1 is a compact set. Hence, we can denote [u]r by a closed interval [u_(r),u-(r)], where u_,u-:[0,1]→R is a function satisfying the following conditions:(1)
u_ is a bounded, left continuous, and nondecreasing function on [0,1].(2)
u- is a bounded, right continuous, and nonincreasing function on [0,1].(3)
u_(r)≤u-(r) for all r∈[0,1].Next, we define addition and scalar multiplication for the set in the sense of Minkowski.Definition 3.
LetA and B be any nonempty subsets of R and λ∈R; addition between A and B denoted by A+B is defined by(2)A+B=a+b∣a∈A,b∈B.Multiplication of A by a scalar λ denoted by λA is defined by(3)λA=λa∣a∈A,where, for λ>0, summation of A+(-λB) is denoted by A-λB; that is, (4)A-λB=A+-λB.Definition 4 (see [18]).
LetA and B be any nonempty subsets of R. A nonempty subset C is called a Hukuhara difference between A and B if A=B+C. A Hukuhara difference between A and B is denoted by A⊖B.Note the following: (1)A⊖B may not exist even when A-B definitely exists, so, for any A and B, A⊖B≠A-B and (2) A⊖A={0}.Definition 5.
Letg:R×R→R. Zadeh’s extension of g is the function g:RF×RF→RF (again, labeled as g) defined by (5)gu,vz=supgx,y=zmin⁡ux,vy∀z∈R.Theorem 6.
Letg:RF×RF→RF be Zadeh’s extension of g. Then, the set of r-cut of g(u,v) is of the form(6)gu,vr=gur,vrfor all u,v∈RF and 0≤r≤1.Definition5 together with Theorem 6 is called Zadeh’s extension principle. Following Zadeh’s extension principle and Minkowski’s definition, addition and scalar multiplication can be defined by the next definition.Definition 7.
Letu,v∈RF and λ∈R; addition between u and v, denoted by u⊕v, is defined by(7)u⊕vz=supx+y=zmin⁡ux,vy∀z∈R,and multiplication of u by a scalar λ, denoted by λ⊙u, is defined by(8)λ⊙uz=uzλ,λ≠00-,λ=0.Theorem 8.
Letu,v∈RF and λ∈R. Then, one has(9)u⊕vr=ur+vr=u_r+v_r,u-r+v-r,λ⊙ur=λur=λu_r,λu-r,λ>0λu-r,λu_r,λ<00,λ=0,where 0-=χ{0} is the addition identity on RF.Theorem 9 (see [20]).
Under addition⊕ and multiplication ⊙, one has the following:(1)
No element ofRF∖R, except  0-, has an inverse under ⊕.(2)
For alla,b∈R such that both a and b≤0 or ≥0 and, for all u∈RF,(10)a+b⊙u=a⊙u⊕b⊙u.(3)
For allλ∈R and for all u,v∈RF,(11)λ⊙u⊕v=λ⊙u⊕λ⊙v.(4)
For allλ,β∈R and for all u∈RF,(12)λ+β⊙u=λ⊙u⊕β⊙u.Definition 10.
Letu,v∈RF. If there exists w∈RF such that u=v⊕w, then w is called aHukuhara difference (fuzzy) between u and v, denoted by w=u⊖v.Next, we define the distance between any two elements inRF. R0+ denotes [0,∞).Definition 11 (see [15]).
Letu,v∈RF. The distance (Hausdorff distance) between u and v is defined by dH:RF×RF→R0+, where(13)dHu,v=supr∈0,1⁡max⁡u_r-v_r,u-r-v-r.
Hence, according to the property of the distancedH, (RF,dH) is a complete metric space.Definition 12.
A functionf:[a,b]→RF is calledfuzzy function and the r-cut of f(t) for all t∈[a,b] is denoted by [f(t)]r=[f_(t)(r),f-(t)(r)] for all r∈[0,1].Definition 13.
Letf:[a,b]→RF be a fuzzy function. f is calledfuzzy continuous on [a,b] if for all t0∈[a,b] and for all ϵ>0 there exists δ>0 such that for all t∈[a,b] if t-t0<δ, then dH(f(t),f(t0))<ϵ.Definition 14.
Letf:[a,b]→RF be a fuzzy function. f is calledfuzzy uniform continuous on [a,b] if for all ϵ>0 there exists δ>0 such that for all s,t∈[a,b] if s-t<δ, then dH(f(s),f(t))<ϵ.Definition 15.
Letf:[a,b]→RF be a fuzzy function. One says that f isbounded on [a,b] if there is M>0 such that dH(f(t),0-)≤M for all t∈[a,b].Definition 16 (see [18]).
Letf:[a,b]→RF be a fuzzy function. One says that f isfuzzy differentiable at x0∈(a,b) if there is y∈RF and δ>0 such that for all h<δ if f(x0+h)⊖f(x0) and f(x0)⊖f(x0-h) exist, then(14)limh→0+⁡dHfx+h⊖fx0h,y=0=limh→0+⁡dHfx0⊖fx-hh,y.The fuzzy number y∈RF is calledfuzzy derivative of f at x0 and is denoted by f′(x0) or df/dx(x0); that is,(15)f′x0=limh→0+⁡fx+h⊖fx0h=limh→0+⁡fx0⊖fx-hh.For the extremes of the interval [a,b], the fuzzy derivative of f at a is f′(a)=limh→0+f(x+h)⊖f(a)/h if limh→0+f(x+h)⊖f(a)/h exists, and the fuzzy derivative of f at b is f′(b)=limh→0+f(b)⊖f(x-h)/h if limh→0+f(x+h)⊖f(a)/h exists (the multiplier 1/h denotes a scalar fuzzy multiple).Theorem 17 (see [15]).
Let the following be true:λ∈R; α:[a,b]→R is differentiable; and f,g:[a,b]→RF is fuzzy differentiable on [a,b]; then the following is true:(1)
f⊕g′(t)=f′t⊕g′t.(2)
λ⊙f′(t)=λ⊙f′t.(3)
α⊙g′(t)=α(t)⊙f′t⊕a′t⊙f(t).Definition 18.
For eachf:[a,b]→RF, one says that f isintegrable if there exists a fuzzy function F:[a,b]→RF such that F′t=f(t) for all t∈[a,b]. The fuzzy function F is calledfuzzy antiderivative of f and is denoted by F(t)=∫atf(s)ds.Theorem 19 (see [15]).
Letf:[a,b]→RF be fuzzy differentiable; then(16)ft=fa⊕∫atf′sds.Theorem 20 (see [15]).
Letf:[a,b]→RF be fuzzy integrable and c∈[a,b]; then(17)∫abfsds=∫acfsds⊕∫cbfsds.Theorem 21 (see [15]).
Iff:[a,b]→RF is fuzzy differentiable, then f is fuzzy continuous.Definition 22.
A fuzzy sequence is a function fromN to RF. A fuzzy sequence f (where f(n)=un∈R for all n∈R) is denoted by un or more briefly by un.Definition 23.
Letun be a fuzzy sequence and u∈RF. One says that un converges to u if and only if for all ϵ>0 there exists n0∈N such that dH(un,u)<ϵ for all n≥n0, denoted by limn→∞⁡un=u.Definition 24.
Letun be a fuzzy sequence. One says that un is bounded if there is N>0 such that dH(un,0-)<N for all n∈N.


## 3. Fuzzy Initial Value Problem

In this section, we discuss initial value problem of fuzzy differential equation, give the definition of a solution and sufficient conditions for its existence, and prove the relevant theorems and lemmas and then the existence of the solution by using the method of successive approximation.In this paper,C([0,T],RF) denotes {f:[0,T]→RF∣fisfuzzycontinuous} with a weighted metric defined by dC(u,v)=supt∈[0,T]⁡e-λtdH(u(t),v(t)), where λ≥0 (which can be any given value). Since (RF,dH) is complete, the space (C([0,T],RF),dC) is also complete; see [15]. For convenience, we denote C([0,T],RF) as C0.


### 3.1. Fuzzy Differential Equation

Consider an initial value problem of a fuzzy differential equation:(18)x′t=at⊙xt⊕ft,xt,t∈0,T,x0=x0,where x is a fuzzy state function of time variable t, f(t,x) is a fuzzy input function of variable t and x, x′ is the fuzzy derivative of x, x(0)=x0 is a fuzzy number, and a:[0,T]→R is a continuous function. Throughout this paper, we denote x by [x_,x-] and its r-cut by [x(t)]r=[x_(t)(r),x-(t)(r)] for all 0<r≤1.The fuzzy functionf(t,x) denotes [f_(t,x),f¯(t,x)] with(19)f_t,x=min⁡ft,u∣u∈xtr,f¯t,x=max⁡ft,u∣u∈xtr.The r-cut of f(t,x) for t∈[0,T] is given by(20)ft,xtr=f_t,xtr,f¯t,xtr∀0<r≤1.Consider the fuzzy derivative ofS(s,t)⊙x(s) for all s∈[0,t], where S(s,t)=e∫sta(τ)dτ. If x is fuzzy differentiable, that is, a solution to (18), using Theorem 19, we get(21)ddsSs,t⊙xs=Ss,t⊙x′s⊕dSs,tds⊙xs=Ss,t⊙as⊙xs⊕fs,xs⊕-Ss,tas⊙xs=Ss,tas⊙xs⊕Ss,t⊙fs,xs⊕-Ss,tas⊙xs=Ss,t⊙fs,xs.Using Theorem 20 and an initial value x(0)=x0, we obtain(22)xt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds.Hence, (22) is a fuzzy integral that corresponds to the fuzzy differential equation (18). Solution to (22) is a type of solutions to (18) that we define next.Definition 25.
Letx∈C([0,T],RF). x is calledfuzzy mild solution of the fuzzy differential equation (18) if x satisfies the fuzzy integral equation(23)xt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds,where S(s,t)=e∫sta(τ)dτ.In the next section, we prove the existence of a fuzzy mild solution of (18) under the following assumption.Assumption H.
It declares that iff(t,x)=[f_(t,x),f¯(t,x)] is a fuzzy function with f_(t,x)=min⁡ft,u∣u∈xtr=F(t,x_,x-), f¯(t,x)=max⁡ft,u∣u∈xtr=G(t,x_,x-), where F,G:[0,T]×C([0,T],C([0,1],RF))2→C([0,1],RF), then there is l>0 such that lMT<1 and F(t1,x1(t1)(r),y1(t1)(r))-F(t2,x2(t2)(r),y2(t2)(r))(r), G(t1,x1(t1)(r),y1(t1)(r))-G(t2,x2(t2)(r),y2(t2)(r))(r)≤l(t1-t2 + max⁡x1t1r-x2t2r,y1t1r-y2t2r), for all (t1,x1,y1),(t2,x2,y2)∈[0,T]×C([0,T],C([0,1],R))×C([0,T],C([0,1],R)), where M=sups,t∈[0,T]⁡S(s,t) and S(s,t)=e∫sta(τ)dτ.


### 3.2. Existence of a Solution

In this subsection, we prove the existence of a mild fuzzy solution to system (18) under Assumption H by using the method of successive approximation. Let us begin by defining a sequence of function xn for an initial value x0∈RF as(24)xnt=S0,t⊙x0⊕∫0tSs,t⊙fs,xn-1sds∀t∈0,T,where x0∈C0 is a given initial function. For any x0∈RF and t∈[0,T], we have xn:[0,T]→RF. Next, we show that the sequence xn has the following properties:(1)
xn∈C0 for all n∈N.(2)
xn is a Cauchy sequence in C0.Property 1.
We show thatxn∈C0 for all n∈N by referring to the following statements.Lemma 26.
Letf be a fuzzy function that satisfies Assumption H. Then, for each t, where t0∈[0,T], there exists l>0 such that(25)dHft,xt,ft0,xt0≤lt-t0+dHxt,xt0.Proof.
Lett,t0∈[0,T]. Since(26)dHft,xt,ft0,xt0=supr∈0,1⁡max⁡Ft,x_tr,x-trr-Ft0,x_t0r,x-t0rr,Gt,x_tr,x-trr-Gt0,x_t0r,x-t0rr,by Assumption H, there is l>0 such that(27)max⁡Ft,x_tr,x-trr-Ft0,x_t0r,x-t0rr,Gt,x_tr,x-trr-Gt0,x_t0r,x-t0rr≤lt-t0+max⁡x_tr-x_t0r,x-tr-x-t0r.Hence,(28)dHft,xt,ft0,xt0≤lt-t0+supr∈0,1⁡max⁡x_tr-x_t0r,x-tr-x-t0r=lt-t0+dHxt,xt0.Lemma 27.
Letf be a fuzzy function that satisfies Assumption H. Then, for each x∈C0, the map t↦f(t,x(t)) is fuzzy continuous.Proof.
Lett,t0∈[0,T]. By Lemma 26, there is l>0 such that(29)dHft,xt,ft0,xt0≤lt-t0+dHxt,xt0.Given any ε>0, by the fuzzy continuity of x, there exists δ1>0 such that, for all t∈[0,T], if t-t0<δ1, then dH(x(t),x(t0))<ε/2l. Choose δ=min⁡ε/2l,δ1. Then, for each t∈[0,T] such that t-t0<δ, we have(30)dHft,xt,ft0,xt0≤lt-t0+dHxt,xt0<lε2l+ε2l=ε.Therefore, the map t↦f(t,x(t)) is fuzzy continuous.Lemma 28.
Ify∈C0 and β∈C([0,T],R), then β⊙y∈C0.Proof.
Lett0∈[0,T]. Because β∈C([0,T],R) and y∈C0, there exist M1 and M2>0 such that βt≤M1 and dH(y(t),0-)≤M2 for all t∈[0,T]. Set B=max⁡{M1,M2}. Given any ε>0, by the continuity of y and β, there is δ1>0 for each t∈[0,T]. If t-t0<δ1, then(31)βt-βt0<ε2B,dHyt,yt0<ε2B.Choose δ=min⁡ε/2B,δ1. Then, for each t∈[0,T] such that t-t0<δ, we have(32)dHβt⊙yt,βt0⊙yt0=dHβt⊙yt+βt⊙yt0,βt0⊙yt0+βt⊙yt0≤dHβt⊙yt,βt⊙yt0+dHβt⊙yt0,βt0⊙yt0≤βtdHyt,yt0+βt-βt0dHyt0,0-<Bε2B+Bε2B=ε.Therefore, we can conclude that the map t↦β(t)⊙y(t) is fuzzy continuous.Lemma 29.
Ify1,y2∈C0, then y1⊕y2∈C0.Proof.
Lett0∈[0,T]. Given any ε>0, by the continuity of y1 and y2, there is δ1>0 for all t∈[0,T] such that t-t0<δ1, and so we obtain dH(y1(t),y1(t0)),dH(y2(t),y2(t0))<ε/2.
Chooseδ=min⁡ε/2,δ1. Then, for each t∈[0,T] such that t-t0<δ, we have(33)dHy1t⊕y2t,y1t0⊕y2t0≤dHy1t,y1t0+dHy2t,y2t0<ε2+ε2=ε.Hence, y1⊕y2∈C0.Lemma 30.
Ify∈C0, then the map t↦∫0ty(s)ds is fuzzy continuous.Proof.
Sincey∈C0, there is M>0 such that dH(y(s),0-)≤M for all s∈[0,T].
Lett0∈[0,T]. Given any ε>0, choose δ=ε/M. Then, for each t∈[0,T] such that t-t0<δ and t>t0, by Theorem 20, we have(34)dH∫0tysds,∫0t0ysds=dH∫0t0ysds⊕∫t0tysds,∫0t0ysds⊕0-≤dH∫0t0ysds,∫0t0ysds+dH∫t0tysds,0-≤∫t0tdHys,0-ds≤Mt-t0<MεM=ε.If t<t0, by Theorem 21, we have(35)dH∫0tysds,∫0t0ysds=dH∫0tysds⊕0-,∫0tysds⊕∫tt0ysds≤dH∫0tysds,∫0tysds+dH0-,∫tt0ysds≤∫tt0dHys,0-ds≤-Mt-t0<MεM=ε.Hence, the map t↦∫0ty(s)ds is fuzzy continuous.Lemma 31.
Assuming thatf is a fuzzy function satisfying Assumption H, for a given initial function x0∈C0, one has a sequence of fuzzy function xn as defined in (24).Proof.
We show thatxn is fuzzy continuous for all n∈N by using mathematical induction.
Basis Step. Because x0∈C0 and x1 is defined as(36)x1t=S0,t⊙x0⊕∫0tSs,t⊙fs,x0sds∀t∈0,T,by Lemmas 26–30, x1 is fuzzy continuous.
Induction Step. For k>1, assuming that xk∈C0, since xk+1(t)=S(0,t)⊙x0⊕∫0tS(s,t)⊙f(s,xk(s))ds, by Lemmas 26–30, we have xk+1∈C0.
Therefore, by mathematical induction,xn∈C0 for all n∈N.Property 2.
We show thatxn is a Cauchy sequence in C0.Lemma 32.
Letf be a fuzzy function satisfying Assumption H and let x0∈C0 be a given initial function. Then, dC(xn,xn-1)≤APn-1 for all n∈N with P=lMT and A=dCx1,x0.Proof.
We show thatdC(xn,xn-1)≤APn-1 for all n∈N by mathematical induction.
Basis Step. For k=1, dCx1,x0=AP0.
Induction Step. For k>1, assuming that dC(xk,xk-1)≤APk-1, we have(37)dHxk+1t,xkt=dHS0,t⊙x0⊕∫0tSs,t⊙fs,xksds,S0,t⊙x0⊕∫0tSs,t⊙fs,xk-1sds=dH∫0tSs,t⊙fs,xksds,∫0tSs,t⊙fs,xk-1sds=supr∈0,1⁡max⁡∫0tSs,t⊙f_s,xksrds-∫0tSs,t⊙f_s,xk-1srds,∫0tSs,t⊙f-s,xksrds-∫0tSs,t⊙f-s,xk-1srds≤supr∈0,1⁡max⁡∫0tSs,tFs,x_ksr,x-ksr-Fs,x_k-1sr,x-k-1srds,∫0tSs,tGs,x_ksr,x-ksr-Gs,x_k-1sr,x-k-1srds≤Ml∫0tsupr∈0,1⁡max⁡x_ksr-x_k-1sr,x-ksr-x-k-1srds≤Ml∫0tdHxk-1s,xksds≤MlTdCxk,xk-1≤PAPk-1=APk.Thus, by mathematical induction, dC(xn,xn-1)≤APn-1 for all n∈N.Lemma 33.
Letf be a fuzzy function satisfying Assumption H and let x0∈C0 be a given initial function; one has dC(xn,x0)≤A1-Pn/1-P for all n∈N with P=lMT and A=dCx1,x0.Proof.
We show thatdC(xn,x0)≤A1-Pn/1-P for all n∈N by using mathematical induction.
Basis Step. For k=1, the above statement is true since dCx1,x0=A(1-P)/(1-P).
Induction Step. For m>1, assuming that dC(xj,x0)≤A(1-Pj)/1-P for all j∈1,2,…,m, by Lemma 32, for t∈[0,T], we have(38)dHxm+1t,x0t=dHxm+1t⊕xmt⊕⋯⊕x1t,x0t⊕xmt⊕⋯⊕x1t≤dHxm+1t,xmt+dHxmt,xm-1t+⋯+dHx1t,x0t≤APm+APm-1+⋯+A=A1-Pm+11-P,implying that dC(xm+1,x0)≤A1-Pm+1/1-P.
Therefore, by the principle of mathematical induction,dC(xn,x0)≤A1-Pn/1-P for all n∈N.Lemma 34.
Assume that Assumption H holds. Given an initial functionx0∈C0, xn is a bounded sequence.Proof.
Assumption H implies that there isl>0 such that P=lMT<1.
Therefore, there isB>0 such that A1-Pn/1-P≤B for all n∈N. By Lemma 33, we have(39)dCxn,0-dCx0,0≤dCxn,x0≤A1-Pn1-P≤B.Hence, dC(xn,0)≤B+dC(x0,0) for all n∈N.Lemma 35.
Assume that Assumption H holds. LetRn=A(1-Pn)/1-P and 0<c<1. Then, for a given initial function x0∈C0 and for each ε>0, there is N∈N such that 2RncN+1<ε for all n≥N with P=lMT and A=dC(x1,x0).Proof.
By Assumption H, there isl>0 such that P=lMT<1. Hence, there is B>0 such that Rn=A1-Pn/1-P≤B for all n∈N. Given any ε>0, choose N>logcε/2Bc.
Then,2RncN+1≤2BcNc<2Bcclogcε/2Bc=2Bcε/2Bc=ε.Next, let us define a mappingg with an initial value x0∈RF to be(40)gut≔S0,t⊙x0⊕∫0tSs,t⊙fs,usdsfor all u∈C[0,T],RF.Lemma 36.
Suppose that Assumption H holds. Then,g is a mapping from C0 to C0.Proof.
Letu∈C[0,T],RF. Similar to the proof of Lemma 31, we have g(u)∈C([0,T],RF). Let u1,u2∈C[0,T],RF such that u1=u2.
For anyt∈[0,T], we have(41)gu1t=S0,t⊙x0⊕∫0tSs,t⊙fs,u1sds=S0,t⊙x0⊕∫0tSs,t⊙fs,u2sds=gu2t.Therefore, g:C[0,T],RF→C[0,T],RF.Lemma 37.
Suppose that Assumption H holds. Then,g is a contraction mapping.Proof.
By Assumption H, there isl>0 such that P=lMT<1 and(42)max⁡Fs,u_sr,u-srr-Fs,v_sr,v-srr,Gs,u_sr,u-srr-Gs,v_sr,v-srr≤lmax⁡u_sr-v_sr,u-sr-v-sr.For all t∈[0,T], choose c=lMT. Then,(43)dHgut,gvt=dH∫0tSs,t⊙fs,usds,∫0tSs,t⊙fs,vsds=supr∈0,1⁡max⁡∫0tSs,t⊙f_s,usrds-∫0tSs,t⊙f_s,vsrds,∫0tSs,t⊙f-s,usrds-∫0tSs,t⊙f-s,vsrds≤supr∈0,1⁡max⁡∫0tSs,tFs,u_sr,u-sr-Fs,v_sr,v-srds,∫0tSs,tGs,u_sr,u-sr-Gs,v_sr,v-srds≤Ml∫0tsupr∈0,1⁡max⁡u_sr-v_sr,u-sr-v-srds≤cdCu,v.Therefore, g is a contraction mapping.Theorem 38.
Suppose that Assumption H holds. Then,xn is a Cauchy sequence in C0.Proof.
Given anyε>0. By Lemma 35, for all 0<c<1, there is N>0 such that 2RncN+1<ε for all n≥N, where Rn=A1-Pn/1-P with P=lMT and A=dC(x1,x0).
Letm,n∈N be such that m,n>N. WLOG, assume that n>m. By Lemma 33, we have(44)dCxm-N-1,xn-N-1=dCxm-N-1⊕x0,xn-N-1⊕x0≤dCxm-N-1,x0+dCxn-N-1,x0≤A1-Pm-N-11-P+A1-Pn-N-11-P<A1-Pn-N-11-P+A1-Pn-N-11-P=2A1-Pn-N-11-P<2A1-Pn1-P=2Rn.By definition of sequence xn and mapping g, we can write xn and xm as a composition of g,(45)xm=g∘g∘⋯∘g︸N+1xm-N-1,xn=g∘g∘⋯∘g︸N+1xn-N-1.Since g is a contraction mapping, there is some c, where 0<c<1(46)dCxm,xn=dCg∘g∘⋯∘gxm-N-1,g∘g∘⋯∘gxn-N-1≤cN+1dCxm-N-1,xn-N-1<2RncN+1<ε.Hence, xn is a Cauchy sequence in C[0,T],RF.By using Properties1 and 2, we prove the existence of a mild fuzzy solution to system (18) in the following theorem.Theorem 39.
If Assumption H holds, system (18) has a mild fuzzy solution; that is, there is x∈C[0,T],RF such that xt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds.Proof.
Given an initial functionx0∈C[0,T],RF and a sequence xn defined by(47)xnt=S0,t⊙x0⊕∫0tSs,t⊙fs,xn-1sds∀t∈0,T,by Theorem 38, xn is a Cauchy sequence in C[0,T],RF.
SinceC[0,T],RF is complete, xn converges in C[0,T],RF; that is, there is x∈C[0,T],RF such that limn→∞⁡xn=x.
Given anyε>0, since limn→∞⁡xn=x, there is N1∈N such that dCxn-1,x<ε/lMT for all n≥N1. Let wt≔S0,t⊙x0⊕∫0tSs,t⊙fs,xsds. We show that limn→∞⁡xn=w as follows. For each t∈[0,T], choose N=N1. Then,(48)dHxnt,wt=dH∫0tSs,t⊙fs,xn-1sds,∫0tSs,t⊙fs,xsds=supr∈0,1⁡max⁡∫0tSs,t⊙f_s,xn-1srds-∫0tSs,t⊙f_s,xsrds,∫0tSs,t⊙f-s,xn-1srds-∫0tSs,t⊙f-s,xsrds≤supr∈0,1⁡max⁡∫0tSs,tFs,x_n-1sr,x-n-1sr-Fs,x_sr,x-srds,∫0tSs,tGs,x_n-1sr,x-n-1sr-Gs,x_sr,x-srds≤Ml∫0tsupr∈0,1⁡max⁡x_n-1sr-x_sr,x-n-1sr-x-srds≤lMTdCxn-1,x<ε.Hence, xt=limn→∞⁡xnt=wt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds, implying that system (18) has a mild fuzzy solution.


## 3.1. Fuzzy Differential Equation

Consider an initial value problem of a fuzzy differential equation:(18)x′t=at⊙xt⊕ft,xt,t∈0,T,x0=x0,where x is a fuzzy state function of time variable t, f(t,x) is a fuzzy input function of variable t and x, x′ is the fuzzy derivative of x, x(0)=x0 is a fuzzy number, and a:[0,T]→R is a continuous function. Throughout this paper, we denote x by [x_,x-] and its r-cut by [x(t)]r=[x_(t)(r),x-(t)(r)] for all 0<r≤1.The fuzzy functionf(t,x) denotes [f_(t,x),f¯(t,x)] with(19)f_t,x=min⁡ft,u∣u∈xtr,f¯t,x=max⁡ft,u∣u∈xtr.The r-cut of f(t,x) for t∈[0,T] is given by(20)ft,xtr=f_t,xtr,f¯t,xtr∀0<r≤1.Consider the fuzzy derivative ofS(s,t)⊙x(s) for all s∈[0,t], where S(s,t)=e∫sta(τ)dτ. If x is fuzzy differentiable, that is, a solution to (18), using Theorem 19, we get(21)ddsSs,t⊙xs=Ss,t⊙x′s⊕dSs,tds⊙xs=Ss,t⊙as⊙xs⊕fs,xs⊕-Ss,tas⊙xs=Ss,tas⊙xs⊕Ss,t⊙fs,xs⊕-Ss,tas⊙xs=Ss,t⊙fs,xs.Using Theorem 20 and an initial value x(0)=x0, we obtain(22)xt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds.Hence, (22) is a fuzzy integral that corresponds to the fuzzy differential equation (18). Solution to (22) is a type of solutions to (18) that we define next.Definition 25.
Letx∈C([0,T],RF). x is calledfuzzy mild solution of the fuzzy differential equation (18) if x satisfies the fuzzy integral equation(23)xt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds,where S(s,t)=e∫sta(τ)dτ.In the next section, we prove the existence of a fuzzy mild solution of (18) under the following assumption.Assumption H.
It declares that iff(t,x)=[f_(t,x),f¯(t,x)] is a fuzzy function with f_(t,x)=min⁡ft,u∣u∈xtr=F(t,x_,x-), f¯(t,x)=max⁡ft,u∣u∈xtr=G(t,x_,x-), where F,G:[0,T]×C([0,T],C([0,1],RF))2→C([0,1],RF), then there is l>0 such that lMT<1 and F(t1,x1(t1)(r),y1(t1)(r))-F(t2,x2(t2)(r),y2(t2)(r))(r), G(t1,x1(t1)(r),y1(t1)(r))-G(t2,x2(t2)(r),y2(t2)(r))(r)≤l(t1-t2 + max⁡x1t1r-x2t2r,y1t1r-y2t2r), for all (t1,x1,y1),(t2,x2,y2)∈[0,T]×C([0,T],C([0,1],R))×C([0,T],C([0,1],R)), where M=sups,t∈[0,T]⁡S(s,t) and S(s,t)=e∫sta(τ)dτ.


## 3.2. Existence of a Solution

In this subsection, we prove the existence of a mild fuzzy solution to system (18) under Assumption H by using the method of successive approximation. Let us begin by defining a sequence of function xn for an initial value x0∈RF as(24)xnt=S0,t⊙x0⊕∫0tSs,t⊙fs,xn-1sds∀t∈0,T,where x0∈C0 is a given initial function. For any x0∈RF and t∈[0,T], we have xn:[0,T]→RF. Next, we show that the sequence xn has the following properties:(1)
xn∈C0 for all n∈N.(2)
xn is a Cauchy sequence in C0.Property 1.
We show thatxn∈C0 for all n∈N by referring to the following statements.Lemma 26.
Letf be a fuzzy function that satisfies Assumption H. Then, for each t, where t0∈[0,T], there exists l>0 such that(25)dHft,xt,ft0,xt0≤lt-t0+dHxt,xt0.Proof.
Lett,t0∈[0,T]. Since(26)dHft,xt,ft0,xt0=supr∈0,1⁡max⁡Ft,x_tr,x-trr-Ft0,x_t0r,x-t0rr,Gt,x_tr,x-trr-Gt0,x_t0r,x-t0rr,by Assumption H, there is l>0 such that(27)max⁡Ft,x_tr,x-trr-Ft0,x_t0r,x-t0rr,Gt,x_tr,x-trr-Gt0,x_t0r,x-t0rr≤lt-t0+max⁡x_tr-x_t0r,x-tr-x-t0r.Hence,(28)dHft,xt,ft0,xt0≤lt-t0+supr∈0,1⁡max⁡x_tr-x_t0r,x-tr-x-t0r=lt-t0+dHxt,xt0.Lemma 27.
Letf be a fuzzy function that satisfies Assumption H. Then, for each x∈C0, the map t↦f(t,x(t)) is fuzzy continuous.Proof.
Lett,t0∈[0,T]. By Lemma 26, there is l>0 such that(29)dHft,xt,ft0,xt0≤lt-t0+dHxt,xt0.Given any ε>0, by the fuzzy continuity of x, there exists δ1>0 such that, for all t∈[0,T], if t-t0<δ1, then dH(x(t),x(t0))<ε/2l. Choose δ=min⁡ε/2l,δ1. Then, for each t∈[0,T] such that t-t0<δ, we have(30)dHft,xt,ft0,xt0≤lt-t0+dHxt,xt0<lε2l+ε2l=ε.Therefore, the map t↦f(t,x(t)) is fuzzy continuous.Lemma 28.
Ify∈C0 and β∈C([0,T],R), then β⊙y∈C0.Proof.
Lett0∈[0,T]. Because β∈C([0,T],R) and y∈C0, there exist M1 and M2>0 such that βt≤M1 and dH(y(t),0-)≤M2 for all t∈[0,T]. Set B=max⁡{M1,M2}. Given any ε>0, by the continuity of y and β, there is δ1>0 for each t∈[0,T]. If t-t0<δ1, then(31)βt-βt0<ε2B,dHyt,yt0<ε2B.Choose δ=min⁡ε/2B,δ1. Then, for each t∈[0,T] such that t-t0<δ, we have(32)dHβt⊙yt,βt0⊙yt0=dHβt⊙yt+βt⊙yt0,βt0⊙yt0+βt⊙yt0≤dHβt⊙yt,βt⊙yt0+dHβt⊙yt0,βt0⊙yt0≤βtdHyt,yt0+βt-βt0dHyt0,0-<Bε2B+Bε2B=ε.Therefore, we can conclude that the map t↦β(t)⊙y(t) is fuzzy continuous.Lemma 29.
Ify1,y2∈C0, then y1⊕y2∈C0.Proof.
Lett0∈[0,T]. Given any ε>0, by the continuity of y1 and y2, there is δ1>0 for all t∈[0,T] such that t-t0<δ1, and so we obtain dH(y1(t),y1(t0)),dH(y2(t),y2(t0))<ε/2.
Chooseδ=min⁡ε/2,δ1. Then, for each t∈[0,T] such that t-t0<δ, we have(33)dHy1t⊕y2t,y1t0⊕y2t0≤dHy1t,y1t0+dHy2t,y2t0<ε2+ε2=ε.Hence, y1⊕y2∈C0.Lemma 30.
Ify∈C0, then the map t↦∫0ty(s)ds is fuzzy continuous.Proof.
Sincey∈C0, there is M>0 such that dH(y(s),0-)≤M for all s∈[0,T].
Lett0∈[0,T]. Given any ε>0, choose δ=ε/M. Then, for each t∈[0,T] such that t-t0<δ and t>t0, by Theorem 20, we have(34)dH∫0tysds,∫0t0ysds=dH∫0t0ysds⊕∫t0tysds,∫0t0ysds⊕0-≤dH∫0t0ysds,∫0t0ysds+dH∫t0tysds,0-≤∫t0tdHys,0-ds≤Mt-t0<MεM=ε.If t<t0, by Theorem 21, we have(35)dH∫0tysds,∫0t0ysds=dH∫0tysds⊕0-,∫0tysds⊕∫tt0ysds≤dH∫0tysds,∫0tysds+dH0-,∫tt0ysds≤∫tt0dHys,0-ds≤-Mt-t0<MεM=ε.Hence, the map t↦∫0ty(s)ds is fuzzy continuous.Lemma 31.
Assuming thatf is a fuzzy function satisfying Assumption H, for a given initial function x0∈C0, one has a sequence of fuzzy function xn as defined in (24).Proof.
We show thatxn is fuzzy continuous for all n∈N by using mathematical induction.
Basis Step. Because x0∈C0 and x1 is defined as(36)x1t=S0,t⊙x0⊕∫0tSs,t⊙fs,x0sds∀t∈0,T,by Lemmas 26–30, x1 is fuzzy continuous.
Induction Step. For k>1, assuming that xk∈C0, since xk+1(t)=S(0,t)⊙x0⊕∫0tS(s,t)⊙f(s,xk(s))ds, by Lemmas 26–30, we have xk+1∈C0.
Therefore, by mathematical induction,xn∈C0 for all n∈N.Property 2.
We show thatxn is a Cauchy sequence in C0.Lemma 32.
Letf be a fuzzy function satisfying Assumption H and let x0∈C0 be a given initial function. Then, dC(xn,xn-1)≤APn-1 for all n∈N with P=lMT and A=dCx1,x0.Proof.
We show thatdC(xn,xn-1)≤APn-1 for all n∈N by mathematical induction.
Basis Step. For k=1, dCx1,x0=AP0.
Induction Step. For k>1, assuming that dC(xk,xk-1)≤APk-1, we have(37)dHxk+1t,xkt=dHS0,t⊙x0⊕∫0tSs,t⊙fs,xksds,S0,t⊙x0⊕∫0tSs,t⊙fs,xk-1sds=dH∫0tSs,t⊙fs,xksds,∫0tSs,t⊙fs,xk-1sds=supr∈0,1⁡max⁡∫0tSs,t⊙f_s,xksrds-∫0tSs,t⊙f_s,xk-1srds,∫0tSs,t⊙f-s,xksrds-∫0tSs,t⊙f-s,xk-1srds≤supr∈0,1⁡max⁡∫0tSs,tFs,x_ksr,x-ksr-Fs,x_k-1sr,x-k-1srds,∫0tSs,tGs,x_ksr,x-ksr-Gs,x_k-1sr,x-k-1srds≤Ml∫0tsupr∈0,1⁡max⁡x_ksr-x_k-1sr,x-ksr-x-k-1srds≤Ml∫0tdHxk-1s,xksds≤MlTdCxk,xk-1≤PAPk-1=APk.Thus, by mathematical induction, dC(xn,xn-1)≤APn-1 for all n∈N.Lemma 33.
Letf be a fuzzy function satisfying Assumption H and let x0∈C0 be a given initial function; one has dC(xn,x0)≤A1-Pn/1-P for all n∈N with P=lMT and A=dCx1,x0.Proof.
We show thatdC(xn,x0)≤A1-Pn/1-P for all n∈N by using mathematical induction.
Basis Step. For k=1, the above statement is true since dCx1,x0=A(1-P)/(1-P).
Induction Step. For m>1, assuming that dC(xj,x0)≤A(1-Pj)/1-P for all j∈1,2,…,m, by Lemma 32, for t∈[0,T], we have(38)dHxm+1t,x0t=dHxm+1t⊕xmt⊕⋯⊕x1t,x0t⊕xmt⊕⋯⊕x1t≤dHxm+1t,xmt+dHxmt,xm-1t+⋯+dHx1t,x0t≤APm+APm-1+⋯+A=A1-Pm+11-P,implying that dC(xm+1,x0)≤A1-Pm+1/1-P.
Therefore, by the principle of mathematical induction,dC(xn,x0)≤A1-Pn/1-P for all n∈N.Lemma 34.
Assume that Assumption H holds. Given an initial functionx0∈C0, xn is a bounded sequence.Proof.
Assumption H implies that there isl>0 such that P=lMT<1.
Therefore, there isB>0 such that A1-Pn/1-P≤B for all n∈N. By Lemma 33, we have(39)dCxn,0-dCx0,0≤dCxn,x0≤A1-Pn1-P≤B.Hence, dC(xn,0)≤B+dC(x0,0) for all n∈N.Lemma 35.
Assume that Assumption H holds. LetRn=A(1-Pn)/1-P and 0<c<1. Then, for a given initial function x0∈C0 and for each ε>0, there is N∈N such that 2RncN+1<ε for all n≥N with P=lMT and A=dC(x1,x0).Proof.
By Assumption H, there isl>0 such that P=lMT<1. Hence, there is B>0 such that Rn=A1-Pn/1-P≤B for all n∈N. Given any ε>0, choose N>logcε/2Bc.
Then,2RncN+1≤2BcNc<2Bcclogcε/2Bc=2Bcε/2Bc=ε.Next, let us define a mappingg with an initial value x0∈RF to be(40)gut≔S0,t⊙x0⊕∫0tSs,t⊙fs,usdsfor all u∈C[0,T],RF.Lemma 36.
Suppose that Assumption H holds. Then,g is a mapping from C0 to C0.Proof.
Letu∈C[0,T],RF. Similar to the proof of Lemma 31, we have g(u)∈C([0,T],RF). Let u1,u2∈C[0,T],RF such that u1=u2.
For anyt∈[0,T], we have(41)gu1t=S0,t⊙x0⊕∫0tSs,t⊙fs,u1sds=S0,t⊙x0⊕∫0tSs,t⊙fs,u2sds=gu2t.Therefore, g:C[0,T],RF→C[0,T],RF.Lemma 37.
Suppose that Assumption H holds. Then,g is a contraction mapping.Proof.
By Assumption H, there isl>0 such that P=lMT<1 and(42)max⁡Fs,u_sr,u-srr-Fs,v_sr,v-srr,Gs,u_sr,u-srr-Gs,v_sr,v-srr≤lmax⁡u_sr-v_sr,u-sr-v-sr.For all t∈[0,T], choose c=lMT. Then,(43)dHgut,gvt=dH∫0tSs,t⊙fs,usds,∫0tSs,t⊙fs,vsds=supr∈0,1⁡max⁡∫0tSs,t⊙f_s,usrds-∫0tSs,t⊙f_s,vsrds,∫0tSs,t⊙f-s,usrds-∫0tSs,t⊙f-s,vsrds≤supr∈0,1⁡max⁡∫0tSs,tFs,u_sr,u-sr-Fs,v_sr,v-srds,∫0tSs,tGs,u_sr,u-sr-Gs,v_sr,v-srds≤Ml∫0tsupr∈0,1⁡max⁡u_sr-v_sr,u-sr-v-srds≤cdCu,v.Therefore, g is a contraction mapping.Theorem 38.
Suppose that Assumption H holds. Then,xn is a Cauchy sequence in C0.Proof.
Given anyε>0. By Lemma 35, for all 0<c<1, there is N>0 such that 2RncN+1<ε for all n≥N, where Rn=A1-Pn/1-P with P=lMT and A=dC(x1,x0).
Letm,n∈N be such that m,n>N. WLOG, assume that n>m. By Lemma 33, we have(44)dCxm-N-1,xn-N-1=dCxm-N-1⊕x0,xn-N-1⊕x0≤dCxm-N-1,x0+dCxn-N-1,x0≤A1-Pm-N-11-P+A1-Pn-N-11-P<A1-Pn-N-11-P+A1-Pn-N-11-P=2A1-Pn-N-11-P<2A1-Pn1-P=2Rn.By definition of sequence xn and mapping g, we can write xn and xm as a composition of g,(45)xm=g∘g∘⋯∘g︸N+1xm-N-1,xn=g∘g∘⋯∘g︸N+1xn-N-1.Since g is a contraction mapping, there is some c, where 0<c<1(46)dCxm,xn=dCg∘g∘⋯∘gxm-N-1,g∘g∘⋯∘gxn-N-1≤cN+1dCxm-N-1,xn-N-1<2RncN+1<ε.Hence, xn is a Cauchy sequence in C[0,T],RF.By using Properties1 and 2, we prove the existence of a mild fuzzy solution to system (18) in the following theorem.Theorem 39.
If Assumption H holds, system (18) has a mild fuzzy solution; that is, there is x∈C[0,T],RF such that xt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds.Proof.
Given an initial functionx0∈C[0,T],RF and a sequence xn defined by(47)xnt=S0,t⊙x0⊕∫0tSs,t⊙fs,xn-1sds∀t∈0,T,by Theorem 38, xn is a Cauchy sequence in C[0,T],RF.
SinceC[0,T],RF is complete, xn converges in C[0,T],RF; that is, there is x∈C[0,T],RF such that limn→∞⁡xn=x.
Given anyε>0, since limn→∞⁡xn=x, there is N1∈N such that dCxn-1,x<ε/lMT for all n≥N1. Let wt≔S0,t⊙x0⊕∫0tSs,t⊙fs,xsds. We show that limn→∞⁡xn=w as follows. For each t∈[0,T], choose N=N1. Then,(48)dHxnt,wt=dH∫0tSs,t⊙fs,xn-1sds,∫0tSs,t⊙fs,xsds=supr∈0,1⁡max⁡∫0tSs,t⊙f_s,xn-1srds-∫0tSs,t⊙f_s,xsrds,∫0tSs,t⊙f-s,xn-1srds-∫0tSs,t⊙f-s,xsrds≤supr∈0,1⁡max⁡∫0tSs,tFs,x_n-1sr,x-n-1sr-Fs,x_sr,x-srds,∫0tSs,tGs,x_n-1sr,x-n-1sr-Gs,x_sr,x-srds≤Ml∫0tsupr∈0,1⁡max⁡x_n-1sr-x_sr,x-n-1sr-x-srds≤lMTdCxn-1,x<ε.Hence, xt=limn→∞⁡xnt=wt=S0,t⊙x0⊕∫0tSs,t⊙fs,xsds, implying that system (18) has a mild fuzzy solution.


## 4. Fuzzy Delay System

In this section, we investigate a fuzzy system with delay:(49)x′t=at⊙xt⊕ft,xt,xt,0≤t≤T,xt=φt,-l≤t≤0,where x is a fuzzy state function of variable t and xt=x(t+θ), for -l≤θ≤0, is the state that is time-delayed. We may consider xt as a state in the past, before time t. Here, φ is a given fuzzy function of past state, before or at t=0. In this system, we assume that the fuzzy input function f(t,x,xt) depends on t, x, and xt, and the scalar function a:[0,T]→R is continuous. The fuzzy derivative of x with respect to t is denoted by x′. All functions are defined using the following notation.The fuzzy functionsx and φ are denoted by [x_,x-] and [φ_,φ-], respectively, with their r-cuts denoted by(50)xtr=x_tr,x-tr,φtr=φ_tr,φ-tr∀0<r≤1,respectively.The fuzzy functionf(t,x(t),xt) is denoted by [f_(t,x,xt),f¯(t,x,xt)] with(51)f_t,x,xt=min⁡ft,u,v∣u∈xtr,v∈xtr,f¯t,x,xt=max⁡ft,u,v∣u∈xtr,v∈xtr.The r-cut of f(t,x(t),xt) is denoted by [f(t,x(t),xt)]r=[f_(t,x(t),xt)(r),f¯(t,x(t),xt)(r)], for all 0<r≤1.LetS(s,t)=e∫sta(τ)dτ and assume that x is fuzzy differentiable that satisfies the conditions of system (49). Consider the fuzzy derivative of S(s,t)⊙x(s) for all s∈[0,t].By Theorem17, we have(52)ddsSs,t⊙xs=Ss,t⊙x′s⊕dSs,tds⊙xs=Ss,t⊙as⊙xs⊕fs,xs,xs⊖Ss,tas⊙xs=Ss,tas⊙xs⊕Ss,t⊙fs,xs,xs⊖Ss,tas⊙xs=Ss,t⊙fs,xs,xs.By Theorem 19 and the initial value x(0)=φ(0)=φ0, we get(53)xs=S0,t⊙φ0⊕∫0tSs,t⊙fs,xs,xsdsas a fuzzy integral equation satisfying system (49).Definition 40.
Letx∈C([-l,T],RF). x is called amild fuzzy solution to system (49), if x satisfies the fuzzy integral equation(54)xt=S0,t⊙φ0⊕∫0tSs,t⊙fs,xs,xsds,0≤t≤Tφt,-l≤t≤0.Next, we prove the existence of a mild fuzzy solution to system (49) under the following assumption.Assumption A.
LetM=sups,t∈[0,T]⁡S(s,t) and(55)B0,T,RF=x·:0,T⟶RF∣x∈C-l,T,RF,xt=xt+θ,-r≤θ≤0.Assume that f satisfies the following conditions:(A-1)
There is a constantk>0 such that(56)dHft,xt,yt,0-≤k1+dHxt,0-+dHyt,0-for allt∈[0,T],  x∈C([-l,T],RF), y(·)∈B([0,T],RF).(A-2)
There is a constantL>0 such that(57)dHft,x1t,y1t,fs,x2s,y2s≤Lt-s+dHx1t,x2t+dHy1t,y2tfor alls,t∈[0,T], x1,x2∈C([-l,T],RF), y1(·),y2(·)∈B([-l,T],RF).Definition 41.
If there existsτ0>0 such that x∈C([-l,τ0],RF) satisfies the fuzzy integral equation(58)xt=S0,t⊙φ0⊕∫0tSs,t⊙fs,xs,xsds,0≤t≤τ0φt,-l≤t≤0,one says that system (49) ismildly fuzzily solvable in [0,τ0] and x is called a mild fuzzy solution in [0,τ0].
For eachτ>0. Let us denote a weighted metric space C([-l,τ],RF) by Cτ. Its metric is defined by(59)dCu,v=supt∈0,τ⁡e-λtdHut,vt,for some given λ≥0. The metric space (C([-l,τ],RF),dC) is a complete metric space.
For eachγ>0, define Ω(γ,τ) to be(60)Ωγ,τ=y∈Cτ∣max0≤t≤τ⁡dHyt,φ0≤γ,yt=φt,-l≤t≤0.Then, Ω(γ,τ) is convex and closed.
Letτ>0. We define a mapping Q:Ω(γ,τ)→Cτ to be(61)Qyt=S0,t⊙φ0⊕∫0tSs,t⊙fs,ys,ysds,0≤t≤τφt,-l≤t≤0,for all y∈Ω(γ,τ). Then, Q is a bounded mapping. By assumption (A-1), there is a constant k>0 such that(62)dHfs,ys,ys,0-≤k1+dHys,0-+dHys,0-.Since y∈C([-l,τ],RF), there is N>0 such that dH(y(s),0-)+dH(ys,0-)≤N for all s∈[-l,τ]. Hence, for each t∈[0,τ], we have(63)dHQyt,0-=dHS0,t⊙φ0⊕∫0tSs,t⊙fs,ys,ysds,0-≤MdHφ0,0-+Mk∫0t1+dHys,0-+dHys,0-ds≤MdHφ0,0-+k1+Nτ<∞.Lemma 42.
The mappingQ:Ω(γ,τ)→Cτ is well-defined and there is τ0>0 such that Q(Ω(γ,τ0))⊆Ω(γ,τ0).Proof.
Letγ,τ>0, {yn} be a sequence in Ω(γ,τ), and let y∈Ω(γ,τ) be such that yn→y.
By condition (A-2), there isL>0 such that(64)dHfs,yns,yns,fs,ys,ys≤LdHyns,ys+dHyns,ys∀s∈0,τ.Given any ϵ>0, since yn→y, there is n0∈N such that dCyn,y<ε/2ML for all n≥n0. Therefore, for t∈[0,τ], we obtain(65)dHQynt,Qyt=dH∫0tSs,t⊙fs,yns,ynsds,∫0tSs,t⊙fs,ys,ysds≤∫0tSs,tdHfs,yns,yns,fs,ys,ysds≤ML∫0tdHyns,ys+dHyns,ysds≤2MLτdCyn,y<ε.This implies that the mapping Q:Ω(γ,τ)→Cτ is well-defined.
Next, we show that there existsτ0>0 such that Q(Ω(γ,τ0))⊆Ω(γ,τ0).
By conditions (A-1) and (A-2), there existL1 and L2>0 such that(66)dHf0,y0,y0,0-≤L11+dCφ,0-,dHfs,ys,ys,f0,y0,y0≤L2maxυ∈0,τ⁡dHyυ,φ0∀s∈0,τ.Hence,(67)dHQyt,φ0=dHS0,t⊙φ0⊕∫0tSs,t⊙fs,yns,ynsds,φ0≤dHS0,t⊙φ0,φ0+dH∫0tSs,t⊙f0,y0,y0,0-+dH∫0tSs,t⊙fs,ys,ysds,∫0tSs,t⊙f0,y0,y0ds≤supr∈0,1⁡max⁡S0,tφ_0r-φ_0r,S0,tφ-0r-φ-0r+ML1∫0t1+dCφ0,0-ds+M∫0tdHfs,ys,ys,f0,y0,y0ds≤S0,τ-1dCφ0,0-+ML1τ1+dCφ0,0-+ML2τmaxυ∈0,τ⁡dHyυ,φ0≡qτ.Since q(τ)→0 as τ→0+, there exists τ0>0 such that 0<q(τ0)<1.
Therefore, we can conclude that there isτ0>0 such that Q(Ω(γ,τ0))⊆Ω(γ,τ0).Lemma 43.
Assume that conditions (A-1) and (A-2) hold. Then, there existsτ0>0 such that system (49) is mildly fuzzily solvable in [0,τ0] and its mild fuzzy solution is unique.Proof.
Letτ>0. Define Ω(1,τ)=y∈Cτ∣max0≤t≤τ⁡dH(y(t),φ0)≤1,y(t)=φ(t),t∈[-l,0].
Then,Ω(1,τ) is convex and closed. Define a mapping Q:Ω(1,τ)→Cτ as that in (61).
By Lemma42, the mapping Q is well-defined on Ω(1,τ) and there is τ0 such that Q:Ω(1,τ0)→Ω(1,τ0). Let τ1≤τ0. We show that Q is a strong contraction mapping on Ω(1,τ2) for some τ2>0. Let y1,y2∈Ω(1,τ1). By condition (A-2), there is a(τ1)>0 such that(68)dHfs,y1s,y1s,fs,y2s,y2s≤aτ1dHy1s,y2s+dHy1s,y2s≤2aτ1dCy1,y2∀s∈0,τ1.Hence, dH(Qy1(t),Qy2(t))≤2Ma(τ1)τ1dCy1,y2=p(τ1)dCy1,y2 for all t∈[0,τ1] with p(τ1)=2Ma(τ1)τ1. Since p(τ1)=2Ma(τ1)τ1→0 as τ1→0+, there is τ2>0 such that p(τ2)<1. This implies that Q is a strong contraction mapping on Ω(1,τ2) for some τ2>0. By the contraction mapping principle, there exists a unique x∈Ω(1,τ2) such that Qx=x; that is,(69)xt=S0,t⊙φ0⊕∫0tSs,t⊙fs,xs,xsds,0≤t≤τ2φt,-l≤t≤0.Theorem 44.
Assume that conditions (A-1) and (A-2) hold. Then, system (49) is mildly fuzzily solvable in [0,T].Proof.
Let[-r,τmax⁡) be the biggest interval where system (49) is mildly fuzzily solvable.
We show thatτmax⁡>T by contradiction. Suppose that τmax⁡≤T. Then, limt→τmax⁡⁡dH(x(t),0-)=∞ because if limt→τmax⁡⁡dH(x(t),0-)<∞, then there exists a sequence tn and κ>0 such that t→τmax⁡ and dH(x(t),0-)≤κ for all n. So x can extend beyond [0,tn+δ] for some δ=δ(tn)>0. This implies that system (49) is mildly fuzzily solvable in [-r,τmax⁡+δ), which contradicts the definition of [-r,τmax⁡). However, the case that limt→τmax⁡⁡dH(x(t),0-)=∞ also contradicts the a priori boundary property of solution x. Hence, τmax⁡>T; that is, system (49) is mildly fuzzily solvable on [0,T].


## 5. Fuzzy Control Problem

In this section, we study a fuzzy differential equation system with time delay and regulation:(70)x′t=at⊙xt⊕ft,xt,xt,ut,0≤t≤T,xt=φt,-l≤t≤0.In the above equations, x is a fuzzy function of time variable t; xt (equaling x(t+θ) for some -l≤θ≤0) is a state of time delay (consider xt as a state in the past before time t); φ is a fuzzy history function before start time t=0; u is a fuzzy controller function of time variable t; and a:[0,T]→R is a given continuous function. In this study, we assume that the input function f(t,x,xt,u) is a fuzzy function of time variable t, state variable x, delay variable xt, and controller variable u∈Uad (Uad is an admissible control set). The fuzzy derivative of x is denoted by x′.From now on, we denotex, φ, and u by [x_,x-], [φ_,φ-], and [u_,u-], respectively.The fuzzy functionf(t,x,xt,u) is denoted by [f_(t,x,xt,u),f¯(t,x,xt,u)] with(71)f_t,x,xt,u=min⁡ft,x1,x2,x3∣x1∈xtr,x2∈xtr,x3∈ur,f¯t,x,xt,u=max⁡ft,x1,x2,x3∣x1∈xtr,x2∈xtr,x3∈ur.In this research, we investigate equation system (70) under the following assumptions.Assumption B.
(B-1) LetB([0,T],RF)=x(·):[0,T]→RF∣x∈C([-l,T],RF),xt=x(t+θ),-r≤θ≤0 and Uad=C[0,T],RF and let f be a fuzzy function; there exists a constant k>0 such that(72)dHft,xt,yt,zt,0-≤k1+dHxt,0-+dHyt,0-+dHzt,0-for all t∈[0,T], x∈C([-l,T],RF), y(·)∈B([0,T],RF), and z∈Uad.
(B-2) There exists a constantL>0 such that(73)dHfs,x1s,y1s,z1s,ft,x2t,y2t,z2t≤Ls-t+dHx1s,x2t+dHy1s,y2s+dHz1s,z2tfor all s,t∈[0,T], x1,x2∈C([-l,T],RF), y1(·),y2(·)∈B([0,T],RF), and z1,z2∈Uad.Definition 45.
Letx∈C([-l,T],RF) and u∈Uad. x is called amild fuzzy solution of system (70) with respect to control u in [-l,T], if x satisfies this system of fuzzy integral equation:(74)xt=S0,t⊙φ0⊕∫0tSs,t⊙fs,xs,xs,usds,0≤t≤Tφt,-l≤t≤0.Theorem 46.
Assume that Assumption B holds. Then, for eachu∈Uad, system (70) has a mild fuzzy solution with respect to control u.Proof.
Letu∈Uad. Define fu(t,x(t),xt) to be f(t,x(t),xt,u(t)). By Assumption B and the continuity of u, fu satisfies Assumption A. Therefore, by Theorem 44, system (70) has a mild fuzzy solution with respect to u.Note that for each solutionx with respect to a control u, we can denote x by xu and call the ordered pair (xu,u) apairwise control pair, sometimes written shortly as (x,u).Next, we investigate an optimization control problem, problem (B), orBolza problem.Problem P.
Problem P is to find the pairwise control pair(x0,u0)∈C[-l,T],RF×Uad such that(75)Jx0,u0≤Jxu,u∀u∈Uad,where J(xu,u)=∫0Tr(t,xu,xtu,u(t))dt+g(xu(T)) is a Bolza cost functional. The multivariable function r is called arunning function and the function g is called aterminal function. For convenience, J(xu,u) is written as J(u). We prove the existence of a solution to Problem P constrained by system (70) under the following assumptions.Assumption U.
Assume thatUad=C([0,T],RF):(U-1)
The running functionr:[0,T]×C([-l,T],RF)×B([0,T],RF)×Uad→(-∞,∞] is Borel measurable.(U-2)
The terminal functiong:C([-l,T],RF)→R is nonnegative and continuous.(U-3)
The running functionr(t,·,·,·) is sequentially lower semicontinuous on C([-l,T],RF)×B([0,T],RF)×Uad for almost every t∈[0,T].(U-4)
The running functionr(t,x,y(·),·) is convex on C([0,T],RF) for all x∈C([-l,T],RF), y(·)∈B([0,T],RF) and almost every t∈[0,T].(U-5)
There are constantsa,b,c>0 and λ∈L([0,T],R) such that(76)rt,x,y·,u≥λt+adHxt,0-+bdHyt,0-+cdHut,0-for allt∈[0,T], x∈C([-l,T],RF),  y(·)∈B([0,T],RF), and u∈Uad.Theorem 47.
Under Assumptions B and U, Problem P constrained by system (70) has at least one solution; that is, there exists a pairwise control pair (x0,u0)∈C[-l,T],RF×Uad such that J(x0,u0)≤J(xu,u) for all (xu,u)∈C[-l,T],RF×Uad.Proof.
Letm=inf⁡J(xu,u)∣u∈Uad. If m=+∞, the theorem is already true. Assume that m<+∞. By assumption (U-5), there are a,b,c>0 and λ∈L([0,T],R) such that(77)rt,xu,xtu,u≥λt+adCxu,0-+bdCxtu,0-+cdCu,0-for all t∈[0,T], xu,xtu∈C([-l,T],RF), and u∈Uad. Since g is a nonnegative function, we have(78)Jxu,u=∫0Trt,xu,xtu,utdt+gxuT≥∫0Tλtdt+a∫0TdHxut,0-dt+b∫0TdHxtu,0-dt+c∫0TdHut,0-dt+gxuT≥-ω>-∞,for someω>0,∀u∈Uad.So m≥ω>-∞. By the definition of minimum, there is a sequence of the minimum point, say un, of the cost functional J such that limn→∞⁡J(xun,un)=m and(79)Jxu,u≥∫0Tλtdt+a∫0TdHxunt,0-dt+b∫0TdHxtun,0-dt+c∫0TdHunt,0-dt+gxunT.Thus, there is N0>0 and m1>0 such that m+m1≥J(xun,un)≥c∫0TdH(un(t),0-)dt for all n≥N0. This implies that m+m1/c≥∫0TdH(un(t),0-)dt. Consequently, dC(un,0-)≤m+m1/Tc for all n≥N0. Hence, un is a bounded sequence in Uad⊂L2[0,T],RF with respect to the norm · defined by x=dC(x,0). Since L2[0,T],RF is a reflexive Banach space, there exists a subsequence unk of un such that unk→wu0 for some u0∈Uad.
Letx0∈C([0,T],RF) be a mild fuzzy solution with respect to a control u0 and let xnk be a sequence of mild fuzzy solution corresponding to the sequence of control unk; that is,(80)xnkt=S0,t⊙φ0⊕∫0tSs,t⊙fs,xnks,xnks,unksds,0≤t≤Tφt,-l≤t≤0.By assumption (B-2), for all 0≤t≤T, there is a constant a>0 such that(81)dHxnkt,x0t≤2Ma∫0tdHxnks,x0sds+∫0tdHunks,u0sds≤2Ma∫0tdHxnks,x0sds+TdCunk,u0.By Gronwall lemma, there is M1>0 such that(82)dCxnk,x0≤M1dCunk,u0.Since unk→wu0, xnk→wx0. By using assumptions (U-2) and (U-3), we obtain(83)m=lim_nk→∞⁡Jxnkunk,unk=lim_nk→∞⁡∫0Trt,xnkunk,xnktunk,unktdt+gxnkunkT≥∫0Tlim_nk→∞⁡rt,xnkunk,xnktunk,unktdt+glim_nk→∞⁡xnkunkT≥∫0Trt,x0,xt0,u0tdt+gx0T=Jx0,u0.Thus, J(x0,u0)=m; that is, J(x0,u0)≤J(xu,u) for all (xu,u)∈C[-l,T],RF×Uad.


## 6. Conclusion

This paper is concerned with proving of the existence and uniqueness of a mild solution to nonlinear fuzzy differential equation constrained by initial value. Then, we already proved the existence of a solution to the system that the initial value constraint was then replaced by delay function constraint. Furthermore, we prove the existence of a solution to optimal control problem of the latter type of equation. Last but not least we should be interested in studying applications and numerical method of these problems. Even though it seems likely that efforts in this direction can be successful, there is no guarantee for that. Therefore, we can only hope for the best and prepare for the worst.



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*Source: 659072-2015-12-15.xml*