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.

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. Let be a family of fuzzy subset of , called a fuzzy number space. It satisfies the following conditions, for each :(1) is normal; that is, there exists such that .(2) is a convex fuzzy set; that is, for all and .(3) is upper semicontinuous on ; that is, for each and for all sequences , if then .(4) is compact; that is, for all sequences , there is a subsequence such that .Notice that and .

Definition 2. Let and . The set of -cut of , denoted as , is defined by

According to Definition 1, conditions (1)–(4) imply that for all is a compact set. Hence, we can denote by a closed interval , where is a function satisfying the following conditions:(1) is a bounded, left continuous, and nondecreasing function on .(2) is a bounded, right continuous, and nonincreasing function on .(3) for all .

Next, we define addition and scalar multiplication for the set in the sense of Minkowski.

Definition 3. Let and be any nonempty subsets of and ; addition between and denoted by is defined byMultiplication of by a scalar denoted by is defined bywhere, for , summation of is denoted by ; that is,

Definition 4 (see [18]). Let and be any nonempty subsets of . A nonempty subset is called a Hukuhara difference between and if . A Hukuhara difference between and is denoted by .

Note the following: (1) may not exist even when definitely exists, so, for any and , and (2) .

Definition 5. Let . Zadeh’s extension of is the function (again, labeled as ) defined by

Theorem 6. Let be Zadeh’s extension of . Then, the set of -cut of is of the formfor all and .

Definition 5 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. Let and ; addition between and , denoted by , is defined byand multiplication of by a scalar , denoted by , is defined by

Theorem 8. Let and . Then, one haswhere is the addition identity on .

Theorem 9 (see [20]). Under addition and multiplication , one has the following:(1)No element of  , except  , has an inverse under .(2)For all such that both and or and, for all ,(3)For all and for all ,(4)For all and for all ,

Definition 10. Let . If there exists such that , then is called a Hukuhara difference (fuzzy) between and , denoted by .

Next, we define the distance between any two elements in . denotes .

Definition 11 (see [15]). Let . The distance (Hausdorff distance) between and is defined by , whereHence, according to the property of the distance , is a complete metric space.

Definition 12. A function is called fuzzy function and the -cut of for all is denoted by for all .

Definition 13. Let be a fuzzy function. is called fuzzy continuous on if for all and for all there exists such that for all if , then .

Definition 14. Let be a fuzzy function. is called fuzzy uniform continuous on if for all there exists such that for all if , then .

Definition 15. Let be a fuzzy function. One says that is bounded on if there is such that for all .

Definition 16 (see [18]). Let be a fuzzy function. One says that is fuzzy differentiable at if there is and such that for all if and exist, thenThe fuzzy number is called fuzzy derivative of at and is denoted by or ; that is,For the extremes of the interval , the fuzzy derivative of at is if exists, and the fuzzy derivative of at is if exists (the multiplier denotes a scalar fuzzy multiple).

Theorem 17 (see [15]). Let the following be true: ; is differentiable; and is fuzzy differentiable on ; then the following is true:(1).(2).(3).

Definition 18. For each , one says that is integrable if there exists a fuzzy function such that for all . The fuzzy function is called fuzzy antiderivative of and is denoted by .

Theorem 19 (see [15]). Let be fuzzy differentiable; then

Theorem 20 (see [15]). Let be fuzzy integrable and ; then

Theorem 21 (see [15]). If is fuzzy differentiable, then is fuzzy continuous.

Definition 22. A fuzzy sequence is a function from to . A fuzzy sequence (where for all ) is denoted by or more briefly by .

Definition 23. Let be a fuzzy sequence and . One says that converges to if and only if for all there exists such that for all , denoted by .

Definition 24. Let be a fuzzy sequence. One says that is bounded if there is such that for all .

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, denotes with a weighted metric defined by , where (which can be any given value). Since is complete, the space is also complete; see [15]. For convenience, we denote as .

3.1. Fuzzy Differential Equation

Consider an initial value problem of a fuzzy differential equation:where is a fuzzy state function of time variable , is a fuzzy input function of variable and , is the fuzzy derivative of , is a fuzzy number, and is a continuous function. Throughout this paper, we denote by and its -cut by for all .

The fuzzy function denotes withThe -cut of for is given by

Consider the fuzzy derivative of for all , where . If is fuzzy differentiable, that is, a solution to (18), using Theorem 19, we getUsing Theorem 20 and an initial value , we obtainHence, (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. Let . is called fuzzy mild solution of the fuzzy differential equation (18) if satisfies the fuzzy integral equationwhere .

In the next section, we prove the existence of a fuzzy mild solution of (18) under the following assumption.

Assumption H. It declares that if is a fuzzy function with , , where , then there is such that and , + , for all , where and .

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 for an initial value aswhere is a given initial function. For any and , we have . Next, we show that the sequence has the following properties:(1) for all .(2) is a Cauchy sequence in .

Property 1. We show that for all by referring to the following statements.

Lemma 26. Let be a fuzzy function that satisfies Assumption H. Then, for each , where , there exists such that

Proof. Let . Sinceby Assumption H, there is such thatHence,

Lemma 27. Let be a fuzzy function that satisfies Assumption H. Then, for each , the map is fuzzy continuous.

Proof. Let . By Lemma 26, there is such thatGiven any , by the fuzzy continuity of , there exists such that, for all , if , then . Choose . Then, for each such that , we haveTherefore, the map is fuzzy continuous.

Lemma 28. If and , then .

Proof. Let . Because and , there exist and such that and for all . Set . Given any , by the continuity of and , there is for each . If , thenChoose . Then, for each such that , we haveTherefore, we can conclude that the map is fuzzy continuous.

Lemma 29. If , then .

Proof. Let . Given any , by the continuity of and , there is for all such that , and so we obtain .
Choose . Then, for each such that , we haveHence, .

Lemma 30. If , then the map is fuzzy continuous.

Proof. Since , there is such that for all .
Let . Given any , choose . Then, for each such that and , by Theorem 20, we haveIf , by Theorem 21, we haveHence, the map is fuzzy continuous.

Lemma 31. Assuming that is a fuzzy function satisfying Assumption H, for a given initial function , one has a sequence of fuzzy function as defined in (24).

Proof. We show that is fuzzy continuous for all by using mathematical induction.
Basis Step. Because and is defined asby Lemmas 26–30, is fuzzy continuous.
Induction Step. For , assuming that , since , by Lemmas 26–30, we have .
Therefore, by mathematical induction, for all .

Property 2. We show that is a Cauchy sequence in .

Lemma 32. Let be a fuzzy function satisfying Assumption H and let be a given initial function. Then, for all with and .

Proof. We show that for all by mathematical induction.
Basis Step. For , .
Induction Step. For , assuming that , we haveThus, by mathematical induction, for all .

Lemma 33. Let be a fuzzy function satisfying Assumption H and let be a given initial function; one has for all with and .

Proof. We show that for all by using mathematical induction.
Basis Step. For , the above statement is true since .
Induction Step. For , assuming that for all , by Lemma 32, for , we haveimplying that .
Therefore, by the principle of mathematical induction, for all .

Lemma 34. Assume that Assumption H holds. Given an initial function , is a bounded sequence.

Proof. Assumption H implies that there is such that .
Therefore, there is such that for all . By Lemma 33, we haveHence, for all .

Lemma 35. Assume that Assumption H holds. Let and . Then, for a given initial function and for each , there is such that for all with and .

Proof. By Assumption H, there is such that . Hence, there is such that for all . Given any , choose .
Then, .

Next, let us define a mapping with an initial value to befor all .

Lemma 36. Suppose that Assumption H holds. Then, is a mapping from to .

Proof. Let . Similar to the proof of Lemma 31, we have . Let such that .
For any , we haveTherefore, .

Lemma 37. Suppose that Assumption H holds. Then, is a contraction mapping.

Proof. By Assumption H, there is such that andFor all , choose . Then,Therefore, is a contraction mapping.

Theorem 38. Suppose that Assumption H holds. Then, is a Cauchy sequence in .

Proof. Given any . By Lemma 35, for all , there is such that for all , where with and .
Let be such that . WLOG, assume that . By Lemma 33, we haveBy definition of sequence and mapping , we can write and as a composition of ,Since is a contraction mapping, there is some , where Hence, is a Cauchy sequence in .

By using Properties 1 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 such that .

Proof. Given an initial function and a sequence defined byby Theorem 38, is a Cauchy sequence in .
Since is complete, converges in ; that is, there is such that .
Given any , since , there is such that for all . Let . We show that as follows. For each , choose . Then,Hence, , implying that system (18) has a mild fuzzy solution.

4. Fuzzy Delay System

In this section, we investigate a fuzzy system with delay:where is a fuzzy state function of variable and , for , is the state that is time-delayed. We may consider as a state in the past, before time . Here, is a given fuzzy function of past state, before or at . In this system, we assume that the fuzzy input function depends on , , and , and the scalar function is continuous. The fuzzy derivative of with respect to is denoted by . All functions are defined using the following notation.

The fuzzy functions and are denoted by and , respectively, with their -cuts denoted byrespectively.

The fuzzy function is denoted by withThe -cut of is denoted by , for all .

Let and assume that is fuzzy differentiable that satisfies the conditions of system (49). Consider the fuzzy derivative of for all .

By Theorem 17, we haveBy Theorem 19 and the initial value , we getas a fuzzy integral equation satisfying system (49).

Definition 40. Let . is called a mild fuzzy solution to system (49), if satisfies the fuzzy integral equation

Next, we prove the existence of a mild fuzzy solution to system (49) under the following assumption.

Assumption A. Let andAssume that satisfies the following conditions:(A-1)There is a constant such that for all ,  , .(A-2)There is a constant such that for all , , .

Definition 41. If there exists such that satisfies the fuzzy integral equationone says that system (49) is mildly fuzzily solvable in and is called a mild fuzzy solution in .
For each . Let us denote a weighted metric space by . Its metric is defined byfor some given . The metric space is a complete metric space.
For each , define to beThen, is convex and closed.
Let . We define a mapping to befor all . Then, is a bounded mapping. By assumption (A-1), there is a constant such thatSince , there is such that for all . Hence, for each , we have

Lemma 42. The mapping is well-defined and there is such that .

Proof. Let , be a sequence in , and let be such that .
By condition (A-2), there is such thatGiven any , since , there is such that for all . Therefore, for , we obtainThis implies that the mapping is well-defined.
Next, we show that there exists such that .
By conditions (A-1) and (A-2), there exist and such thatHence,Since as , there exists such that .
Therefore, we can conclude that there is such that .

Lemma 43. Assume that conditions (A-1) and (A-2) hold. Then, there exists such that system (49) is mildly fuzzily solvable in and its mild fuzzy solution is unique.

Proof. Let . Define .
Then, is convex and closed. Define a mapping as that in (61).
By Lemma 42, the mapping is well-defined on and there is such that . Let . We show that is a strong contraction mapping on for some . Let . By condition (A-2), there is such thatHence, for all with . Since as , there is such that . This implies that is a strong contraction mapping on for some . By the contraction mapping principle, there exists a unique such that ; that is,

Theorem 44. Assume that conditions (A-1) and (A-2) hold. Then, system (49) is mildly fuzzily solvable in .

Proof. Let be the biggest interval where system (49) is mildly fuzzily solvable.
We show that by contradiction. Suppose that . Then, because if , then there exists a sequence and such that and for all . So can extend beyond for some . This implies that system (49) is mildly fuzzily solvable in , which contradicts the definition of . However, the case that also contradicts the a priori boundary property of solution . Hence, ; that is, system (49) is mildly fuzzily solvable on .

5. Fuzzy Control Problem

In this section, we study a fuzzy differential equation system with time delay and regulation:In the above equations, is a fuzzy function of time variable ; (equaling for some ) is a state of time delay (consider as a state in the past before time ); is a fuzzy history function before start time ; is a fuzzy controller function of time variable ; and is a given continuous function. In this study, we assume that the input function is a fuzzy function of time variable , state variable , delay variable , and controller variable ( is an admissible control set). The fuzzy derivative of is denoted by .

From now on, we denote , , and by , , and , respectively.

The fuzzy function is denoted by with

In this research, we investigate equation system (70) under the following assumptions.

Assumption B. (B-1) Let and and let be a fuzzy function; there exists a constant such thatfor all , , , and .
(B-2) There exists a constant such thatfor all , , , and .

Definition 45. Let and . is called a mild fuzzy solution of system (70) with respect to control in , if satisfies this system of fuzzy integral equation:

Theorem 46. Assume that Assumption B holds. Then, for each , system (70) has a mild fuzzy solution with respect to control .

Proof. Let . Define to be . By Assumption B and the continuity of , satisfies Assumption A. Therefore, by Theorem 44, system (70) has a mild fuzzy solution with respect to .

Note that for each solution with respect to a control , we can denote by and call the ordered pair a pairwise control pair, sometimes written shortly as .

Next, we investigate an optimization control problem, problem (B), or Bolza problem.

Problem P. Problem P is to find the pairwise control pair such thatwhere is a Bolza cost functional. The multivariable function is called a running function and the function is called a terminal function. For convenience, is written as . We prove the existence of a solution to Problem P constrained by system (70) under the following assumptions.

Assumption U. Assume that :(U-1)The running function is Borel measurable.(U-2)The terminal function is nonnegative and continuous.(U-3)The running function is sequentially lower semicontinuous on for almost every .(U-4)The running function is convex on for all , and almost every .(U-5)There are constants and such that for all , ,  , and .

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 such that for all .

Proof. Let . If , the theorem is already true. Assume that . By assumption (U-5), there are and such thatfor all , , and . Since is a nonnegative function, we haveSo . By the definition of minimum, there is a sequence of the minimum point, say , of the cost functional such that andThus, there is and such that for all . This implies that . Consequently, for all . Hence, is a bounded sequence in with respect to the norm defined by . Since is a reflexive Banach space, there exists a subsequence of such that for some .
Let be a mild fuzzy solution with respect to a control and let be a sequence of mild fuzzy solution corresponding to the sequence of control ; that is,By assumption (B-2), for all , there is a constant such thatBy Gronwall lemma, there is such thatSince , . By using assumptions (U-2) and (U-3), we obtainThus, ; that is, for all .

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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author gratefully acknowledges Mr. Pratana Kangsadal for improving English language of this paper. The author thanks King Mongkut’s Institute of Technology Ladkrabang, Thailand, for research funding.