# Fuzzy Optimal Control Problem of Several Variables

Authors: Altyeb Mohammed Mustafa; Zengtai Gong; Mawia Osman
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

The purpose of this paper is to establish the necessary conditions for a fuzzy optimal control problem of several variables. Also, we define fuzzy optimal control problems involving isoperimetric constraints and higher order differential equations. Then, we convert these problems to fuzzy optimal control problems of several variables in order to solve these problems using the same solution method. The main results of this paper are illustrated throughout three examples, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.


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## Body

## 1. Introduction

Optimal control theory is considered as a modern extension of the classical calculus of variations; however, it differs from calculus of variations in that it uses control variables to optimize the function. The development of the mathematical theory for optimal control began in the early 1950’s, partially in response to problems in various branches of engineering and economics. The study of classical optimal control theory from different viewpoints greatly attracted the attention of many mathematicians, and the detailed arguments can be found in many textbooks, for instance, [1], and references therein. Moreover, optimal control strategy, i.e., solving necessary conditions for optimality, can be applied in several fields, such as economy, biology, and process engineering (for more details, see [1–5]).On the other hand, uncertainty is inherent in most dynamical systems in its input, output, and manner, and fuzziness is a kind of uncertainty very common in real-world problems [6]. In 1965, Zadeh introduced the concepts of fuzzy sets and fuzzy numbers in [7], followed up in 1972 by Chang and Zadeh when they proposed the concept of the fuzzy derivative in [8]. A large number of researches have been studied in various aspects of the theory and applications of these notions; one of these research lines has been the fuzzy optimal control problem. In the past few decades, the fuzzy optimal control problem has received growing attention, and many results of researches have been reported in the literature ([9–19] and references therein).Recently, a lot of works done in the field of the fuzzy optimal control problem have only examined problems with one control and one dependent state variable; however, many times, we will wish to examine fuzzy optimal control problems which arise in a wide variety of scientific and engineering applications such as physics, chemical engineering, and economy, with more variables (more controls and more states). It seems that it is a good idea to consider fuzzy optimal control problems of several variables and discuss how to handle such problems. Further, treating a special type of fuzzy optimal control problems such as problems having a type of constraint known as an isoperimetric constraint and problems involving higher order differential equations has been presented. In [11], the modified fuzzy Euler-Lagrange condition was established for the fuzzy Isoperimetric Variational Problem (IVP), which is considered as a fuzzy constrained variational problem, but, in this paper, we overcome the fuzzy optimal control problem involving the isoperimetric constraint, which is considered as a fuzzy constrained optimal control problem.The main aim of this paper is to derive the necessary conditions of the fuzzy optimal control problem of several variables based on the concepts of differentiability and integrability of a fuzzy valued function parameterized by the left- and right-hand functions of itsα-level set and variational approaches, in order to provide the solutions of this problem. However, the solutions of the fuzzy optimal control problem of several variables, optimal controls, and corresponding optimal states are not always fuzzy functions. Thus, to guarantee that the solutions of the fuzzy optimal control problem of several variables are always fuzzy functions, we will introduce the concepts of strong (fuzzy) and weak solutions of this problem.The rest of this paper is organized as follows: In Section2, we recall some basic terminologies and definitions used in the present paper. In Section 3, we establish our main results concerning the necessary conditions of the fuzzy optimal control problem of several variables and treating two special cases of the fuzzy optimal control problem. Additionally, we propose the definitions of strong (fuzzy) and weak solutions of our problem. In Section 4, we give three examples that can serve to illustrate our main results. In Section 5, we present some concluding remarks.


## 2. Preliminaries

Throughout this paper,FR denotes the class of fuzzy subsets of the real axis. A fuzzy set v~ on R is a mapping v~:R⟶0,1. For each fuzzy set v~, we denote its α-level set by v~α and defined by v~α=vlα,vrα=x∈R:v~x≥α for any α∈0,1. The support of v~ we denote by suppv~, where suppv~=x∈R:v~x>0. The closure of suppv~ defines the 0-level set of v~; thus,
(1)v~α=x∈R:v~x≥α,if0<α≤1,clsuppv~,ifα=0,where clM denotes the closure of set M. Fuzzy set v~∈FR is called a fuzzy number if
(1)
v~ is a normal fuzzy set, i.e., there exists an x0∈R such that v~x0=1
(2)
v~ is a convex fuzzy set, i.e., v~rx+1−ry≥minv~x,v~y for any x,y∈R and r∈0,1
(3)
v~ is upper semicontinuous on R
(4)
v~0=clsuppv~=cl⋃α∈0,1v~α is compactIn the rest of this paper, we useE1 to denote the fuzzy number space.It is clear that theα−level set v~α=vlα,vrα is bounded closed interval in R for all α∈0,1, where vlα and vrα denote the left-hand and right-hand endpoints of v~α, respectively. Obviously, any v∈R can be regarded as a fuzzy number v~ defined by
(2)v~x=1,x=v,0,x≠v.In particular, fuzzy zero is defined as0~x=1 if x=0 and 0~x=0 otherwise.Leta~,b~∈E1 and k∈R. For any x∈R, we can define the addition a~+b~ and scalar multiplication ka~, respectively, as
(3)a~+b~x=sups+t=xmina~s,b~t,ka~x=a~xk,k≠0,0a~x=1,x=0,0,x≠0.Usingα−level set, we can also define the addition a~+b~ and scalar multiplication ka~, respectively, as
(4)a~+b~α=a~α+b~α=s+t:s∈a~α,t∈b~α,ka~α=ka~α=kx:x∈a~α.Leta~ be a fuzzy number, the opposite of a~ is denoted by −a~ and characterized by −a~x=a~−x [20]. In the case that a~α=alα,arα, we have −a~α=−arα,−alα for all α∈0,1.The binary operation “.” inR can be extended to the binary operation “⨀” of two fuzzy numbers by using the extension principle. Let a~ and b~ be fuzzy numbers, then
(5)α~⊙b~z=sups⋅t=zminα~s,b~t.Usingα-level set, the product α~⊙b~ is defined by
(6)α~⊙b~α=minαlαblα,αlαbrα,αrαblα,αrαbrα,maxαlαblα,αlαbrα,αrαblα,αrαbrα,in the case that a~α=alα,arα and b~α=blα,brα.Lemma 1 (see [21]).
If al:0,1⟶R and ar:0,1⟶Rsatisfy the following conditions:
(1)
al is a bounded increasing function
(2)
ar is a bounded decreasing function
(3)
al1≤ar1
(4)
limα→k−alα=alk and limα→k−arα=ark, for all 0<k≤1
(5)
limα→0+alα=al0 and limα→0+arα=ar0then a~:R⟶0,1 defined by a~x=supα∣alα≤x≤arα is a fuzzy number with a~α=alα,arα. Conversely, if a~:R⟶0,1 is a fuzzy number with a~α=alα,arα, then the functions alα and arα satisfy conditions (1)-(5).DefineD:E1×E1⟶R+∪0 by
(7)Da~,b~=supα∈0,1max∣alα−blα∣,∣arα−brα∣,where a~α=alα,arα and b~α=blα,brα. Da~,b~ is called the distance between fuzzy numbers a~ and b~.It should be noted thatD satisfies the following properties:
(1)
E1,D is a complete metric space
(2)
Da~+c~,b~+c~=Da~,b~
(3)
Dka~,kb~=∣k∣Da~,b~, where a~,b~,c~∈E1 and k∈RA special class of fuzzy numbers is the class of triangular fuzzy numbers. We say that the fuzzy numbera~ is triangular if al1=ar1, alα=al1−1−αal1−al0, and arα=al1+1−αar0−al1. The triangular fuzzy number a~ is generally denoted by a~=al0,al1,ar0.Definition 2 (partial ordering [9]).
Let a~,b~∈E1, we write a~≺¯b~, if alα≤blα and arα≤brα for all α∈0,1. We also write a~≺b~, if a~≺¯b~ and there exists α0∈0,1 such that alα0<blα0 or arα0<brα0. Furthermore, a~=b~, if a~≺¯b~ and a~≻¯b~. In other words, a~=b~, if a~α=b~α for all α∈0,1.
In the sequel, we say thata~,b~∈E1 are comparable if either a~≺¯b~ or a~≻¯b~ and noncomparable otherwise.Definition 3 (gH-difference [22]).
Suppose thata~,b~∈E1, where a~α=alα,arα and b~α=blα,brα for all α∈0,1, the generalized Hukuhara difference of two fuzzy numbers a~ and b~ (gH-difference for short) is defined by
(8)α~⊖gHb~=c~⟺1α~=b~+c~,or2b~=α~+−1c~.
Ifc~=α~⊖gHb~ exists as a fuzzy number, then its α-level set is
(9)clα=minalα−blα,arα−brα,crα=maxalα−blα,arα−brα,for all α∈0,1.Definition 4 (fuzzy valued function [9]).
The functionf~:t0,t1⟶E1 is called a fuzzy valued function if f~t is assigned a fuzzy number for any t∈t0,t1. We also denote f~tα=flt,α,frt,α, where flt,α=f~tlα=minf~tα and frt,α=f~trα=maxf~tα. Therefore, any fuzzy valued function f~ may be understood by flt,α and frt,α being, respectively, a bounded increasing function of α and a bounded decreasing function of α for α∈0,1. Also, it holds flt,α≤frt,α for any α∈0,1.Definition 5 (continuity of a fuzzy valued function [23]).
We say thatf~:t0,t1⟶E1 is continuous at t∈t0,t1, if both flt,α and frt,α are continuous functions at t∈t0,t1 for all α∈0,1.Definition 6 (gH-differentiability of a fuzzy valued function [24]).
Lett^∈t0,t1 and h be such that t^+h∈t0,t1, then the gH-derivative of a fuzzy valued function x~t:t0,t1⟶E1 at t^∈t0,t1 is defined as
(10)ẋ~t^=limh→0x~t^+h⊖gHx~t^h.
Ifẋ~t^∈E1, we say that x~ is generalized Hukuhara differentiable (gH-differentiable for short) at t^. Also, we say that x~ is 1-gH-differentiable at t^ if
(11)1ẋ~t^α=ẋlt^,α,ẋrt^,α,forα∈0,1,and x~ is 2-gH-differentiable at t^ if
(12)2ẋ~t^α=ẋrt^,α,ẋlt^,α,forα∈0,1.Definition 7 (nth order gH-differentiability of a fuzzy valued function [25]).
Letx~t:t0,t1⟶E1. We say that x~t is nth order gH-differentiable at t^ whenever the function x~t is gH-differentiable of the order i, i=1,2,⋯,n−1, at t^ and if there exist ẋ~nt^∈E1 such that
(13)ẋ~nt^=limh→0ẋ~n−1t^+h⊖gHẋ~n−1t^h.Definition 8 (switching point [26]).
We say that a pointt^0∈t0,t1 is a switching point for the differentiability of x~t if in any neighborhood N of t^0 there exist points t^1<t^0<t^2 such that
(i)
type(I): att^1 (11) holds while (12) does not hold and at t^2 (12) holds while (12) does not hold or
(ii)
type(II): att^1 (12) holds while (12) does not hold and at t^2 (11) holds while (12) does not holdDefinition 9 (see [21]).
Letf~:t0,t1⟶E1. We say that f~ is fuzzy-Riemann integrable to I∈E1 if for any ε>0, there exists δ>0 such that for any division P=i,j;ξ of t0,t1 with the norms Δp<δ, we have
(14)D∑P∗i−j⊙f~ξ;I<є,where Σ∗ denotes the fuzzy summation. We choose to write I≔∫t0t1f~tdt. Furthermore, for any α∈0,1,
(15)∫t0t1f~tαdt=∫t0t1flt,αdt,∫t0t1frt,αdt.Theorem 10 (see [24]).
Iff~t:t0,t1⟶E1 is gH-differentiable with no switching point in the interval t0,t1, then we have
(16)F~t=∫t0t1ḟ~xdx=f~t1⊖gHf~t0.Theorem 11 (see [24]).
Letf~t:t0,t1⟶E1 be a continuous fuzzy valued function. Then,
(17)F~t=∫t0tf~xdx,t∈t0,t1,is gH-differentiable and Ḟ~t=f~t.From now, we useCE1t0,t1 to denote the space of all fuzzy valued functions that have continuous gH-derivatives on t0,t1 and CE1nt0,t1 to denote the space of all fuzzy valued functions that have nth continuous gH-derivatives on t0,t1.


## 3. Fuzzy Optimal Control of Several Variables

This section is aimed at deriving the necessary conditions for the fuzzy optimal control problem of several variables. For this purpose, the fuzzy optimal control problem of several variables is introduced at first, then using fuzzy variational approaches, the problem is solved.Consider the following fuzzy optimal control problem of several variables:(18)minu~1,⋯,u~mJ~x~1,⋯,x~n,u~1,⋯,u~m=ϕ~x~1t1,⋯,x~nt1,t1+∫t0t1f~x~1t,⋯,x~nt,u~1t,⋯,u~mt,tdtsubjecttoẋ~jt=g~jx~1t,⋯,x~nt,u~1t,⋯,u~mt,tx~jt0=x~j0,x~jt1isfree,forj=1,2,⋯,n,where f~,g~:En×Em×R⟶E1 are assumed to be functions of class CE1t0,t1 with respect to all their arguments. The fuzzy states x~jt for j=1,2,⋯,n and the fuzzy controls u~kt for k=1,2,⋯,m are functions of t∈t0,t1⊆R. Furthermore, x~jt are assumed to be 1-gH-differentiable functions for all j=1,2,⋯,n. In problem (18), we make no requirements on n and m. In other words, n>m, n=m, or n<m are all acceptable.We say that an admissible fuzzy curvex~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗ is the solution of problem (18), if for all admissible curve x~1,⋯,x~n,u~1,⋯,u~m of problem (18)
(19)J~x~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗≺¯J~x~1,⋯,x~n,u~1,⋯,u~m.It is well know that, from Definition2, the above inequality holds if and only if
(20)Jlx1∗l,x1∗r,⋯,xn∗l,xn∗r,u1∗l,u1∗r,⋯,um∗l,um∗r,t,α≤Jlx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,t,α,Jrx1∗l,x1∗r,⋯,xn∗l,xn∗r,u1∗l,u1∗r,⋯,um∗l,um∗r,t,α≤Jrx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,t,α,for all α∈0,1, where the α−level set of fuzzy curves x~∗, x~, u~∗, and u~ are characterized, respectively, by
(21)x~j∗tα=xj∗lt,α,xj∗rt,α,x~jtα=xjlt,α,xjrt,α,u~k∗tα=uk∗lt,α,uk∗rt,α,u~ktα=uklt,α,ukrt,α,for j=1,2,⋯,n and k=1,2,⋯,m.Definition 12 (fuzzy Hamiltonian function).
We define fuzzy Hamiltonian function as(22)ℍx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,λ1l,⋯,λnl,λ1r,⋯,λnr,t,α=flx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,t,α+frx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,t,α+∑i=1nλilgilx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,t,α+λirgirx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,t,α.Theorem 13 (necessary conditions for problem (18)).
Assume that x~1∗t,⋯,x~n∗t is a vector of admissible fuzzy states and u~1∗t,⋯,u~m∗t is a vector of admissible fuzzy controls. Then, the necessary conditions for x~1∗t,⋯,x~n∗tand u~1∗t,⋯,u~m∗t to be optimal solutions for (18) are
(23)λ̇j∗lt,α=−∂ℍ∂xjlx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,λ1l,⋯,λnl,λ1r,⋯,λnr,t,α,(24)λ̇j∗rt,α=−∂ℍ∂xjrx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,λ1l,⋯,λnl,λ1r,⋯,λnr,t,α,(25)0=∂ℍ∂uklx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,λ1l,⋯,λnl,λ1r,⋯,λnr,t,α,(26)0=∂ℍ∂ukrx1l,x1r,⋯,xnl,xnr,u1l,u1r,⋯,uml,umr,λ1l,⋯,λnl,λ1r,⋯,λnr,t,α,(27)λjlt1,α=∂ϕl∂xjl+∂ϕr∂xjlt=t1,(28)λjrt1,α=∂ϕl∂xjr+∂ϕr∂xjrt=t1,for all α∈0,1, t∈t0,t1, j=1,2,⋯,n, and k=1,2,⋯,m.Proof.
Let us first consider the variation ofukl, ukr, xjl, and xjr defined, respectively, by
(29)ukl=uk∗l+δukl,ukr=uk∗r+δukr,xjl=xj∗l+δxjl,xjr=xj∗r+δxjr,for k=1,2,⋯,m and j=1,2,⋯,n. Using Theorems 10 and 11 and because the problem is a minimization problem, we can rewrite J~ as
(30)J~1x~1,⋯,x~n,u~1,⋯,u~m=∫t0t1f~x~1t,⋯,x~nt,u~1t,⋯,u~mt,t+ddtϕ~x~1t,⋯,x~nt,tdt.
The increment ofJ~1, denoted by ΔJ~1 is
(31)ΔJ~1=∫t0t1f~x~1∗+δx~1,⋯,x~n∗+δx~n,u~1∗+δu~1,⋯,u~m∗+δu~m,t+ddtϕ~x~1∗+δx~1,⋯,x~n∗+δx~n,tdt⊖gH∫t0t1f~x~1∗t,⋯,x~n∗t,u~1∗t,⋯,u~m∗t,t+ddtϕ~x~1∗t,⋯,x~n∗t,tdt=∫t0t1f~x~1∗+δx~1,⋯,x~n∗+δx~n,u~1∗+δu~1,⋯,u~m∗+δu~mα+ddtϕ~x~1∗+δx~1,⋯,x~n∗+δx~nαdt⊖gH∫t0t1f~x~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗α+ddtϕ~x~1∗,⋯,x~n∗αdt,where
(32)x~1∗+δx~1,⋯,x~n∗+δx~n,u~1∗+δu~1,⋯,u~m∗+δu~mα=x1∗l+δx1l,x1∗r+δx1r,⋯,xn∗l+δxnl,xn∗r+δxnr,u1∗l+δu1l,u1∗r+δu1r,⋯,um∗l+δuml,um∗r+δumr,t,α,x~1∗+δx~1,⋯,x~n∗+δx~nα=x1∗l+δx1l,x1∗r+δx1r,⋯,xn∗l+δxnl,xn∗r+δxnr,t,α,x~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗α=x1∗l,x1∗r,⋯,xn∗l,xn∗r,u1∗l,u1∗r,⋯,um∗l,um∗r,t,α,x~1∗,⋯,x~n∗α=x1∗l,x1∗r,⋯,xn∗l,xn∗r,t,α,ΔJ~1α=ΔJ1l,ΔJ1r.
Using the gH-difference, and without sake of generality, we consider(33)ΔJ1l=∫t0t1flx1∗l+δx1l,x1∗r+δx1r,⋯,xn∗l+δxnl,xn∗r+δxnr,u1∗l+δu1l,u1∗r+δu1r,⋯,um∗l+δuml,um∗r+δumr,t,α+ddtϕlx1∗l+δx1l,x1∗r+δx1r,⋯,xn∗l+δxnl,xn∗r+δxnr,t,αdt−∫t0t1flx1∗l,x1∗r,⋯,xn∗l,xn∗r,u1∗l,u1∗r,⋯,um∗l,um∗r,t,α+ddtϕlx1∗l,x1∗r,⋯,xn∗l,xn∗r,t,αdt,ΔJ1r=∫t0t1frx1∗l+δx1l,x1∗r+δx1r,⋯,xn∗l+δxnl,xn∗r+δxnr,u1∗l+δu1l,u1∗r+δu1r,⋯,um∗l+δuml,um∗r+δumr,t,α+ddtϕlx1∗l+δx1l,x1∗r+δx1r,⋯,xn∗l+δxnl,xn∗r+δxnr,t,αdt−∫t0t1frx1∗l,x1∗r,⋯,xn∗l,xn∗r,u1∗l,u1∗r,⋯,um∗l,um∗r,t,α+ddtϕrx1∗l,x1∗r,⋯,xn∗l,xn∗r,t,αdt.
SinceJ~1x~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗≤J~1x~1,⋯,x~n,u~1,⋯,u~m if and only if
(34)J1lx~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗α≤J1lx~1,⋯,x~n,u~1,⋯,u~mα,J1rx~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗α≤J1rx~1,⋯,x~n,u~1,⋯,u~mα,for all α∈0,1, then x~1∗,⋯,x~n∗,u~1∗,⋯,u~m∗α are optimal solutions for the crisp functions J1l and J1r. Suppose that δJ1l and δJ1r denote the first variation of J1l and J1r, respectively. From the classical theory of optimal control, we know that if u1∗l,u1∗r,⋯,um∗l,um∗r are optimal, then it is necessary that δJ1l and δJ1r are zero. In order to find the first variation of J1l and J1r, we need to evaluate the derivatives in the integrand of J1l and J1r along the optimal trajectory; then, we obtain
(35)ΔJ1l=∫t0t1∑j=1n∂fl∂xjlδxjl+∂fl∂xjrδxjr+∑k=1m∂fl∂uklδukl+∂fl∂ukrδukr+ddt∑j=1n∂ϕl∂xjlδxjl+∂ϕl∂xjrδxjrdt+Oδu1∗l2+Oδu1∗r2+⋯+Oδum∗l2+Oδum∗r2,ΔJ1r=∫t0t1∑j=1n∂fr∂xjlδxjl+∂fr∂xjrδxjr+∑k=1m∂fr∂uklδukl+∂fr∂ukrδukr+ddt∑j=1n∂ϕr∂xjlδxjl+∂ϕr∂xjrδxjrdt+Oδu1∗l2+Oδu1∗r2+⋯+Oδum∗l2+Oδum∗r2.
Subsequently, on optimal trajectories, the first variation ofJ1l and J1r is zero, i.e.,
(36)δJ1l=∫t0t1∑j=1n∂fl∂xjlδxjl+∂fl∂xjrδxjr+∑k=1m∂fl∂uklδukl+∂fl∂ukrδukr+ddt∑j=1n∂ϕl∂xjlδxjl+∂ϕl∂xjrδxjrdt=0,δJ1r=∫t0t1∑j=1n∂fr∂xjlδxjl+∂fr∂xjrδxjr+∑k=1m∂fr∂uklδukl+∂fr∂ukrδukr+ddt∑j=1n∂ϕr∂xjlδxjl+∂ϕr∂xjrδxjrdt=0,for all variations. Now, we are ready to introduce the fuzzy Lagrange multiplier functions λ~1t,⋯,λ~nt by considering the integral
(37)ψ~=∫t0t1λ~t⊙g~jx~1t,⋯,x~nt,u~1t,⋯,u~mt,t⊖gHẋ~jtdt,for j=1,2,⋯,n. Using the gH-difference and without sake of generality, we consider the α−level set of ψ~, respectively, as
(38)ψl=∫t0t1−λjlẋjl−gjldt,(39)ψr=∫t0t1−λjrẋjr−gjrdt,for j=1,2,⋯,n. In the remainder of the proof, we will ignore similar arguments. We start by computing the variation of (38):
(40)δψl=∫t0t1−δλjlẋjl−gjl−δẋjl−gjlλjldt,forj=1,2,⋯,n(41)=∫t0t1∑j=1n∑i=1nλjl∂gjl∂xilδxil+∂gjl∂xirδxir+∑k=1mλjl∂gjl∂uklδukl+∂gjl∂ukrδukr−∑j=1nλjlδẋjldt.
It is clear that from the definition of gH-difference, the first term of Equation (40) is zero. Integrating the last term on the RHS of (41) by parts and because x~jt0 is specified, i.e., δxjlt0=0 (and δxjrt0=0) for all j=1,2,⋯,n, then, we arrive at
(42)δψl=∫t0t1∑j=1n∑i=1nλjl∂gjl∂xilδxil+∂gjl∂xirδxir+∑k=1mλjl∂gjl∂uklδukl+∂gjl∂ukrδukr+∑j=1nλ̇jlδxjldt−∑j=1nλjlt1δxjlt1.
By considering the following summations,(43)∑j=1n∑i=1nλjl∂gjl∂xilδxil+∂gjl∂xirδxir=∑j=1n∑i=1nλil∂gil∂xjlδxjl+∂gil∂xjrδxjr,∑j=1n∑k=1mλjl∂gjl∂uklδukl+∂gjl∂ukrδukr=∑k=1m∑i=1nλil∂gil∂uklδukl+∂gil∂ukrδukr,we can rewrite δψl as
(44)δψl=∫t0t1∑j=1n∑i=1nλil∂gil∂xjlδxjl+∂gil∂xjrδxjr+∑k=1m∑i=1nλil∂gil∂uklδukl+∂gil∂ukrδukr+∑j=1nλ̇jlδxjldt−∑j=1nλjlt1δxjlt1.
Similarly, when we consider (39) with δxjrt0=0 for all j=1,2,⋯,n, and the same summation with small change (the right-hand functions of g~,λ~ instead of the left-hand functions), we arrive at
(45)δψr=∫t0t1∑j=1n∑i=1nλir∂gir∂xjlδxjl+∂gir∂xjrδxjr+∑k=1m∑i=1nλir∂gir∂uklδukl+∂gir∂ukrδukr+∑j=1nλ̇jrδxjrdt−∑j=1nλjrt1δxjrt1.
Sinceψl=0 and ψr=0 for all u1l,u1r,⋯,uml,umr, then δψl=0 and δψr=0. Further, we can replace the conditions δJ1l=0 and δJ1r=0 by δJ1l+δψl=0 and δJ1r+δψr=0, respectively [10]. Then, we have
(46)δJ1l+δψl=∫t0t1∑j=1n∂fl∂xjl+∑i=1nλil∂gil∂xjl+λ̇jlδxjl+∂fl∂xjr+∑i=1nλil∂gil∂xjrδxjr+∑k=1m∂fl∂ukl+∑i=1nλil∂gil∂uklδukl+∂fl∂ukr+∑i=1nλil∂gil∂ukrδukr+ddt∑j=1n∂ϕl∂xjlδxjl+∂ϕl∂xjrδxjrdt−∑j=1nλjlt1δxjlt1,(47)δJ1r+δψr=∫t0t1∑j=1n∂fr∂xjl+∑i=1nλir∂gir∂xjlδxjl+∂fr∂xjr+∑i=1nλir∂gir∂xjr+λ̇jrδxjr+∑k=1m∂fr∂ukl+∑i=1nλir∂gir∂uklδukl+∂fr∂ukr+∑i=1nλir∂gir∂ukrδukr+ddt∑j=1n∂ϕr∂xjlδxjl+∂ϕr∂xjrδxjrdt−∑j=1nλjrt1δxjrt1.
Summing Equations (46) and (47) and using the definition of the fuzzy Hamiltonian function, we have
(48)∫t0t1ddt∑j=1n∂ϕl∂xjlδxjl+∂ϕl∂xjrδxjr+∂ϕr∂xjlδxjl+∂ϕr∂xjrδxjr+∂ℍ∂xjl+λ̇jlδxjl+∂ℍ∂xjr+λ̇jrδxjr+∂ℍ∂uklδukl+∂ℍ∂ukrδukrdt−∑j=1nλjlt1δxjlt1+λjrt1δxjrt1=0.
Integrating the first term of (48) considering δxjlt0=0 and δxjrt0=0 for all j=1,2,⋯,n and removing the terms involving δxjl,δxjr,δukl, and δukr, therefore, the necessary conditions follow.Note 1.
It should be noted that in problem (18), if x~jt1 is specified, then λ~jt has no boundary conditions for all j=1,2,⋯,n.


### 3.1. Isoperimetric Constraints

In this part, we turn our attention to a special type of fuzzy optimal control problem, defined as(49)minu~J~u~=ϕ~x~t1,t1+∫t0t1f~x~t,u~t,tdtsubjecttoẋ~t=g~x~t,u~t,t,x~t0=x~0∫t0t1h~x~t,u~t,tdt=A~.This type of constraint is known as an isoperimetric constraint, wheref~,g~, and h~ are assumed to be functions of class CE1t0,t1 with respect to all their arguments. To establish the solution method for this type of problem, we convert this problem to a more familiar form, the fuzzy optimal control problem of several variables, by introducing a second state variable y~t, and let
(50)y~t=∫t0th~x~s,u~s,sds.Therefore, if we use Theorems10 and 11, then we have
(51)ẏ~t=h~x~t,u~t,t,y~t0=0~,y~t1=A~.Subsequently, problem (49) is transformed into
(52)minu~J~u~=ϕ~x~t1,t1+∫t0t1f~x~t,u~t,tdtsubjecttoẋ~t=g~x~t,u~t,t,x~t0=x~0ẏ~t=h~x~t,u~t,t,ẏ~t0=0~,y~t1=A~.This problem can now be solved using Theorem13, i.e., the developed method for solving fuzzy optimal control problems of several variables.


### 3.2. Higher Order Differential Equations

Here, we deal with problems involving higher order differential equations. Consider the following problem:(53)minu~1,⋯,u~mJ~u~1,⋯,u~m=∫t0t1f~x~t,x~1t,⋯,x~nt,u~1t,⋯,u~mt,tdtsubjecttox~n+1t=g~x~t,x~1t,⋯,x~nt,u~1t,⋯,u~mt,tx~t0=β~1,x~1t0=β~2,⋯,x~nt0=β~n+1,for n>1. Theorem 13 does not directly deal with this type of problem. But it is easy to convert this problem to the fuzzy optimal control problem of several variables by introducing n+1 state variables defined, respectively, by
(54)x~1t=x~t,x~2t=x~1t,⋯,x~n+1t=x~nt.Then, problem (53) is transformed into
(55)minu~1,⋯,u~mJ~u~1,⋯,u~m=∫t0t1f~x~1t,x~2t,⋯,x~n+1t,u~1t,⋯,u~mt,tdtsubjecttox~11t=x~2t,x~1t0=β~1x~21t=x~3t,x~2t0=β~2⋮x~n1t=x~n+1t,x~nt0=β~nx~n+11t=g~x~1t,x~2t,⋯,x~n+1t,u~1t,⋯,u~mt,t,x~n+1t0=β~n+1,which can be solved by using Theorem 13, i.e., the developed method for solving fuzzy optimal control problems of several variables.


### 3.3. The Strong (Fuzzy) and Weak Solutions

The developed method for solving the fuzzy optimal control problem of several variables provides the solutions of this problem, optimal fuzzy controls and corresponding fuzzy states, by solving the necessary conditions introduced in Theorem13. Meanwhile, to guarantee that the solutions of the fuzzy optimal control problem of several variables are always fuzzy functions, we propose the concepts of strong (fuzzy) and weak solutions of this problem. In the following definition, based on the conditions (1) and (2) of Lemma 1, we introduce the definition of strong (fuzzy) and weak solutions of the fuzzy optimal control problem of several variables (18).Definition 14 (strong (fuzzy) and weak solutions).
(1)
Strong (Fuzzy) Solution. We say that u~k∗tα and x~j∗tα are strong(fuzzy) solutions of problem (18) if uk∗lt,α,uk∗rt,α,xj∗lt,α, and xj∗rt,α, for k=1,2,⋯,m and j=1,2,⋯,n, obtained from (23), (24), (25), (26), (27), and (28) satisfy their related properties defined in the conditions (1) and (2) of Lemma 1, for all t∈t0,t1 and α∈0,1.
(2)
Weak Solution. We say that u~k∗tα and x~j∗tα are weak solutions of problem (18) if uk∗lt,α,uk∗rt,α, xj∗lt,α, and xj∗rt,α, for k=1,2,⋯,m and j=1,2,⋯,n, obtained from (23), (24), (25), (26), (27), and (28) do not satisfy their related properties in the conditions (1) and (2) of Lemma 1; thus, we define u~k∗tα and x~j∗tα as
(56)u~k∗tα=2uk∗rt,1−uk∗lt,α,uk∗rt,α,ifuk∗l,uk∗raredecreasingfunctionsofα,uk∗lt,α,2uk∗lt,1−uk∗rt,α,ifuk∗l,uk∗rareincreasingfunctionsofα,uk∗rt,α,uk∗lt,α,ifuk∗lisadecreasingfunctionanduk∗risanincreasingfunctionofα,x~j∗tα=2xj∗rt,1−xj∗lt,α,xj∗rt,α,ifxj∗l,xj∗raredecreasingfunctionsofα,xj∗lt,α,2xj∗lt,1−xj∗rt,α,ifxj∗l,xj∗rareincreasingfunctionsofα,xj∗rt,α,xj∗lt,α,ifxj∗lisadecreasingfunctionandxj∗risanincreasingfunctionofα,for all t∈t0,t1, α∈0,1, k=1,2,⋯,m, and j=1,2,⋯,n.Note 2.
In the next section, we will give three examples that can serve to illustrate our main results, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.


## 3.1. Isoperimetric Constraints

In this part, we turn our attention to a special type of fuzzy optimal control problem, defined as(49)minu~J~u~=ϕ~x~t1,t1+∫t0t1f~x~t,u~t,tdtsubjecttoẋ~t=g~x~t,u~t,t,x~t0=x~0∫t0t1h~x~t,u~t,tdt=A~.This type of constraint is known as an isoperimetric constraint, wheref~,g~, and h~ are assumed to be functions of class CE1t0,t1 with respect to all their arguments. To establish the solution method for this type of problem, we convert this problem to a more familiar form, the fuzzy optimal control problem of several variables, by introducing a second state variable y~t, and let
(50)y~t=∫t0th~x~s,u~s,sds.Therefore, if we use Theorems10 and 11, then we have
(51)ẏ~t=h~x~t,u~t,t,y~t0=0~,y~t1=A~.Subsequently, problem (49) is transformed into
(52)minu~J~u~=ϕ~x~t1,t1+∫t0t1f~x~t,u~t,tdtsubjecttoẋ~t=g~x~t,u~t,t,x~t0=x~0ẏ~t=h~x~t,u~t,t,ẏ~t0=0~,y~t1=A~.This problem can now be solved using Theorem13, i.e., the developed method for solving fuzzy optimal control problems of several variables.


## 3.2. Higher Order Differential Equations

Here, we deal with problems involving higher order differential equations. Consider the following problem:(53)minu~1,⋯,u~mJ~u~1,⋯,u~m=∫t0t1f~x~t,x~1t,⋯,x~nt,u~1t,⋯,u~mt,tdtsubjecttox~n+1t=g~x~t,x~1t,⋯,x~nt,u~1t,⋯,u~mt,tx~t0=β~1,x~1t0=β~2,⋯,x~nt0=β~n+1,for n>1. Theorem 13 does not directly deal with this type of problem. But it is easy to convert this problem to the fuzzy optimal control problem of several variables by introducing n+1 state variables defined, respectively, by
(54)x~1t=x~t,x~2t=x~1t,⋯,x~n+1t=x~nt.Then, problem (53) is transformed into
(55)minu~1,⋯,u~mJ~u~1,⋯,u~m=∫t0t1f~x~1t,x~2t,⋯,x~n+1t,u~1t,⋯,u~mt,tdtsubjecttox~11t=x~2t,x~1t0=β~1x~21t=x~3t,x~2t0=β~2⋮x~n1t=x~n+1t,x~nt0=β~nx~n+11t=g~x~1t,x~2t,⋯,x~n+1t,u~1t,⋯,u~mt,t,x~n+1t0=β~n+1,which can be solved by using Theorem 13, i.e., the developed method for solving fuzzy optimal control problems of several variables.


## 3.3. The Strong (Fuzzy) and Weak Solutions

The developed method for solving the fuzzy optimal control problem of several variables provides the solutions of this problem, optimal fuzzy controls and corresponding fuzzy states, by solving the necessary conditions introduced in Theorem13. Meanwhile, to guarantee that the solutions of the fuzzy optimal control problem of several variables are always fuzzy functions, we propose the concepts of strong (fuzzy) and weak solutions of this problem. In the following definition, based on the conditions (1) and (2) of Lemma 1, we introduce the definition of strong (fuzzy) and weak solutions of the fuzzy optimal control problem of several variables (18).Definition 14 (strong (fuzzy) and weak solutions).
(1)
Strong (Fuzzy) Solution. We say that u~k∗tα and x~j∗tα are strong(fuzzy) solutions of problem (18) if uk∗lt,α,uk∗rt,α,xj∗lt,α, and xj∗rt,α, for k=1,2,⋯,m and j=1,2,⋯,n, obtained from (23), (24), (25), (26), (27), and (28) satisfy their related properties defined in the conditions (1) and (2) of Lemma 1, for all t∈t0,t1 and α∈0,1.
(2)
Weak Solution. We say that u~k∗tα and x~j∗tα are weak solutions of problem (18) if uk∗lt,α,uk∗rt,α, xj∗lt,α, and xj∗rt,α, for k=1,2,⋯,m and j=1,2,⋯,n, obtained from (23), (24), (25), (26), (27), and (28) do not satisfy their related properties in the conditions (1) and (2) of Lemma 1; thus, we define u~k∗tα and x~j∗tα as
(56)u~k∗tα=2uk∗rt,1−uk∗lt,α,uk∗rt,α,ifuk∗l,uk∗raredecreasingfunctionsofα,uk∗lt,α,2uk∗lt,1−uk∗rt,α,ifuk∗l,uk∗rareincreasingfunctionsofα,uk∗rt,α,uk∗lt,α,ifuk∗lisadecreasingfunctionanduk∗risanincreasingfunctionofα,x~j∗tα=2xj∗rt,1−xj∗lt,α,xj∗rt,α,ifxj∗l,xj∗raredecreasingfunctionsofα,xj∗lt,α,2xj∗lt,1−xj∗rt,α,ifxj∗l,xj∗rareincreasingfunctionsofα,xj∗rt,α,xj∗lt,α,ifxj∗lisadecreasingfunctionandxj∗risanincreasingfunctionofα,for all t∈t0,t1, α∈0,1, k=1,2,⋯,m, and j=1,2,⋯,n.Note 2.
In the next section, we will give three examples that can serve to illustrate our main results, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.


## 4. Illustrative Examples

Example 1.
Find the fuzzy control that
(57)minimizeJ~u~t=∫01x~1t+x~2t⊖gHu~2tdtsubjecttoẋ~1t=x~2,x~10=0~,x~11isfreeẋ~2t=0,1,3u~t,x~20=0~,x~21isfree.Solution 1.
First, without loss of generality, we formulate the fuzzy Hamiltonian function as(58)H=fl+fr+λ1lg1l+λ2lg2l+λ1rg1r+λ2rg2r=x1l+x2l−ul2+x1r+x2r−ur2+λ1lx2l+λ2lαul+λ1rx2r+λ2r3−2αur.
In fact,ϕ~x~11,x~21,1=0~, then
(59)ϕlx1l1,x1r1,x2l1,x2r1,1,ϕrx1l1,x1r1,x2l1,x2r1,1=0,0.
The necessary conditions for optimality of Theorem13 with the initial conditions and the dynamical system of (57) give
(60)ẋ1l=x2l,x1l0=0,(61)ẋ1r=x2r,x1r0=0,(62)ẋ2l=αul,x2l0=0,(63)ẋ2r=3−2αur,x2r0=0,(64)λ̇1l=−∂H∂x1l⟹λ̇1l=−1,λ1l1=0,(65)λ̇1r=−∂H∂x1r⟹λ̇1r=−1,λ1r1=0,(66)λ̇2l=−∂H∂x2l⟹λ̇2l=−1+λ1l,λ2l1=0,(67)λ̇2r=−∂H∂x2r⟹λ̇2r=−1+λ1r,λ2r1=0,(68)∂H∂ul=0⟹−2ul+αλ2l=0⟹ul=α2λ2l,(69)∂H∂ur=0⟹−2ur+3−2αλ2r=0⟹ur=3−2α2λ2r.
We solve Equations (60), (61), (62), (63), (64), (65), (66), (67), (68), and (69) analytically; then, we obtain
(70)λ1∗lt,α=1−t,λ1∗rt,α=1−t,λ2∗lt,α=t2−4t+32,λ2∗rt,α=t2−4t+32,u∗lt,α=αt2−4t+34,u∗rt,α=3−2αt2−4t+34,x1∗lt,α=α2t4−8t3+18t248,x1∗rt,α=3−2α2t4−8t3+18t248,x2∗lt,α=α2t3−6t2+9t12,x2∗rt,α=3−2α2t3−6t2+9t12.
We can easily show thatu∗lt,α,x1∗lt,α, and x2∗lt,α are continuous increasing functions of α, and u∗rt,α,x1∗rt,α, and x2∗rt,α are continuous decreasing functions of α. Furthermore,
(71)u∗lt,1=u∗rt,1=t2−4t+32,x1∗lt,1=x1∗rt,1=t4−8t3+18t248,x2∗lt,1=x2∗rt,1=t3−6t2+9t12,for all t∈0,1. Therefore, the α-level set of optimal fuzzy control u~∗t and optimal fuzzy states x~1∗t and x~2∗t is characterized, respectively, by
(72)u~∗tα=αt2−4t+34,3−2αt2−4t+34,x~1∗tα=α2t4−8t3+18t248,3−2α2t4−8t3+18t248,x~2∗ta=α2t3−6t2+9t12,3−2α2t3−6t2+9t12,for all α,t∈0,1. Therefore, the above solutions are strong (fuzzy) solutions of problem (57).Example 2.
Find the fuzzy control that
(73)minimizeJ~u~t=∫01x~t⊖gH12u~2tdtsubjecttoẋ~t=u~t,x~0=0,1,2,x~11isfree∫01x~tdt=2,4,6.Solution 2.
First, we introduce a second state variabley~t by
(74)y~t=∫0tx~sds;then, problem (73) converts to
(75)J~u~t=∫01x~t⊝12u~2tdt,subjecttoẋ~t=u~t,x~0=0,1,2,x~1isfreeẏ~t=x~t,y~0=0~,y~1=2,4,6.Second, without loss of generality, we formulate the fuzzy Hamiltonian function as(76)H=xl−12ul2+xr−12ur2+λ1lul+λ2lxl+λ1rur+λ2rxr.It is clear that(77)ϕ~x~1,1α=0,0,0,1,2α=α,2−α,2,4,6α=2+2α,6−2α.Now, applying Theorem13, we find the following necessary conditions:
(78)ẋl=ul,xl0=α,(79)ẋr=ur,xr0=2−α,(80)ẏl=xl,yl0=0,(81)ẏr=xr,yr0=0,(82)λ̇1l=−∂H∂xl⟹λ̇1l=−1+λ2l,λ1l1=0,(83)λ̇1r=−∂H∂xr⟹λ̇1r=−1+λ2r,λ1r1=0,(84)λ̇2l=−∂H∂yl=0⟹λ2l=K1,forsomeconstantK1,(85)λ̇2r=−∂H∂yr=0⟹λ2r=K2,forsomeconstantK2,(86)∂H∂ul=0⟹−ul+λ1l=0⟹ul=λ2l,(87)∂H∂ur=0⟹−ur+λ1r=0⟹ur=λ2r.In the rest of the solution, we will ignore the similar cases; we only consider the left-hand functions ofα−level set.We begin by substituting (84) into differential Equation (82), and then, by solving the differential equation, we obtain
(88)λ1lt,α=1+K11−t.Substitutingλ1lt,α into (86) gives
(89)ult,α=1+K11−t.After that, we substituteult,α into Equation (78), then solve it with the appropriate condition, we obtain
(90)xlt,α=1+K1t−t22+α.By the same manner, we substitutexlt,α into Equation (80) and solve it with the appropriate condition; then, we have
(91)ylt,α=1+K1t22−t36+αt.To find the value of constant of integrationK1, we use the condition yl1,α=2+2α; thus,
(92)1+K112−16+α=2+2α⟹1+K1=6+3α.Therefore, for allα,t∈0,1, we arrive at
(93)u∗lt,α=6+3α1−t,x∗lt,α=6+3αt−t22+α,y∗lt,α=6+3αt22−t36+αt.Similarly, if we consider the right-hand functions ofα−level set and for all α,t∈0,1, we arrive at
(94)u∗rt,α=12−3α1−t,x∗rt,α=12−3αt−t22+2−α,y∗rt,α=12−3αt22−t36+2−αt.We can easily show thatu∗lt,α,x∗lt,α, and y∗lt,α are continuous increasing functions of α and u∗rt,α,x∗rt,α, and y∗rt,α are continuous decreasing functions of α. Moreover,
(95)u∗lt,1=u∗rt,1=91−t,x1∗lt,1=x1∗rt,1=9t−t22+1,x2∗lt,1=x2∗rt,1=9t22−t36+t,for all t∈0,1. Therefore, the α-level set of optimal fuzzy control u~∗t and optimal fuzzy states x~1∗t and x~2∗t is characterized, respectively, by
(96)u~∗tα=6+3α1−t,12−3α1−t,x~∗tα=6+3αt−t22+α,12−3αt−t22+2−α,y~∗ta=6+3αt22−t36+αt,12t22−t36+2−αt,for all α,t∈0,1. Therefore, the above solutions are strong (fuzzy) solutions of problem (73).Example 3.
Find the fuzzy control that(97)minimizeJ~u~t=∫01x~t+x~1t⊖gHu~2tdtsubjecttox~2t=0,1,3u~,x~0=0~,x~10=0~.Solution 3.
We letx~1t=x~t and x~2t=x~1t, then problem (97) converted to problem (57) in Example 1, i.e., problem (97) has the same solutions of problem (57).


## 5. Conclusion

In summary, we proved the necessary conditions for optimality of the fuzzy optimal control problem of several variables. Also, fuzzy optimal control problems involving isoperimetric constraint and higher order differential equation have been considered. By introducing new variables, we transformed these problems into fuzzy optimal control problems of several variables in order to use the developed method to solve these problems. The definitions of strong (fuzzy) and weak solutions of our problems have been introduced. By three examples, we discussed and summarized the applicability of our main results of this paper.



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*Source: 2182640-2019-12-29.xml*