Existence and uniqueness results for fuzzy linear differential-algebraic ...

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ScienceDirect Fuzzy Sets and Systems 245 (2014) 30–42 www.elsevier.com/locate/fss

Existence and uniqueness results for fuzzy linear differential-algebraic equations R. Alikhani a,b , F. Bahrami a , T. Gnana Bhaskar b,∗ a Department of Mathematics, University of Tabriz, Tabriz, Iran b Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

Received 15 November 2012; received in revised form 26 January 2014; accepted 11 March 2014 Available online 17 March 2014

Abstract We discuss the existence results for a fuzzy initial value problem of linear differential-algebraic equations and provide an explicit representation for the solution. A few illustrative examples are given. © 2014 Elsevier B.V. All rights reserved. Keywords: Differential-algebraic equations; Fuzzy differential equations; Initial value problems

1. Introduction Differential equations occur naturally in the modeling of dynamical behavior of physical processes. If the states of the physical systems are in some ways constrained, for instance, by conservation laws such as Kirchhoff’s laws in electrical networks, or by position constraints such as the movement of mass points on a surface, then the corresponding mathematical models contain algebraic equations to describe these constraints in addition to the differential equations that describe the dynamics of the system. Such systems, comprising of both differential and algebraic equations are called differential-algebraic equations (DAEs). Thus, the modeling of constrained mechanical systems, electrical circuits and chemical reaction kinetics, semi-discretization of systems of partial differential equations and singular perturbation problems see [6]), usually lead to systems of DAEs. For a study of fundamental properties such as the canonical forms, existence and uniqueness theory etc., of a system of linear DAEs we refer to [11] and [22]. The study of DAE’s is an active research area and there exist works that deal with different classes of DAEs such as partial DAEs [21], stochastic DAEs [24], functional DAEs [12], ill-posed DAEs [30]. For a numerical study of DAEs we refer to [6]. Recently, [26] dealt with the study of semi-explicit systems of nonlinear DAEs, where the coefficients of the system considered are estimated using stochastic collocation and Galerkin methods. * Corresponding author. Tel.: +1 (321) 674 7213; fax: +1 (321) 674 7412.

E-mail addresses: [email protected] (R. Alikhani), [email protected] (F. Bahrami), [email protected] (T. Gnana Bhaskar). http://dx.doi.org/10.1016/j.fss.2014.03.006 0165-0114/© 2014 Elsevier B.V. All rights reserved.


R. Alikhani et al. / Fuzzy Sets and Systems 245 (2014) 30–42

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On the other hand, in order to obtain realistic models of some dynamical processes, it is often necessary to take into account imprecision, randomness or uncertainty in the measurement of underlying parameters (see [9,10,16,23]). Such models require the study of qualitative and numerical aspects of fuzzy differential equations, see [1,2,5,7,18,19]. To the best of our knowledge, a systematic study of existence theory of fuzzy DAEs is yet to be done. Initiating such a study in this work, we study the existence results for the following initial value problem associated with a system of fuzzy linear differential-algebraic equations: E x(t) ˙ = Ax(t) + f (t) t ∈ I, x(0) = γ ,

(1)

¯ Rn ) and γ ∈ Rn , where x(t) = [x1 (t), . . . , xn (t)]T , E and A are n × n real matrices, E is a singular matrix, f ∈ C(I, F n ∈ N. Here the interval I = (0, T ) for some T > 0, or (0, ∞) and I¯ denotes the closure of I. In our work, we consider the general model in which the entries of the coefficient matrices are not necessarily positive. Our approach is based on the idea to separate the positive and negative entries by rearranging the unknowns in a suitable order. We replace the fuzzy system of order n × n by a crisp system of order (2n) × (2n) in which the positive and the negative entries are separated. We find the conditions that ensure that the solution obtained in the process is a valid fuzzy solution. The paper is organized as follows. In Section 2, we introduce some basic notions of fuzzy numbers and describe the existence theory of the crisp solution for linear differential-algebraic equations. In Section 3, which is the main section of the paper, we study the existence of solution of the fuzzy initial value problem (1). A few illustrative examples are presented in Section 4. 2. Preliminaries In this section we recall a few known results that are needed in our work. The space of fuzzy numbers (see [13]), denoted by RF , is the set of functions u : R → [0, 1] that have the following properties: (i) (ii) (iii) (iv)

u is normal, i.e. there exists t0 ∈ R such that u(t0 ) = 1 u is fuzzy convex, i.e. u(λt1 + (1 − λ)t2 ) min{u(t1 ), u(t2 )}, for any t1 , t2 ∈ R, λ ∈ [0, 1] u is upper semicontinuous [u]0 = cl{t ∈ R | u(t) > 0} is compact. (Here clA denotes the closure of A.)

For 0 < α 1, α-level set of u ∈ RF is defined by [u]α = t ∈ R u(t) α . For any α ∈ [0, 1], [u]α = [uα , uα ] is a bounded closed interval. For u, v ∈ RF , λ ∈ R, we define the addition u + v and scalar multiplication λ · u as: [u + v]α = [u]α + [v]α

and

[λ · u]α = λ[u]α , where [u]α + [v]α and λ[u]α mean the usual addition of two subsets of R and the usual product between a scalar and a subset of R respectively (see [13,29]). Let u, v ∈ RF . If there exists a unique fuzzy number w ∈ RF such that v + w = u, then w is called the H-difference of u, v and is denoted by u v (see [27]). The Hausdorff distance between u and v is given by D : RF × RF → R+ ∪ {0} D(u, v) = sup max uα − v α , uα − v α . α∈[0,1]

The space (RF , D) is a complete metric space.


32


The following lemma gives a parametric form of a fuzzy number. Lemma 2.1. (See [15].) Suppose that the bounded functions g : [0, 1] → R and g : [0, 1] → R satisfy the following conditions: (i) (ii) (iii) (iv) (v)

g is an increasing function. g is a decreasing function. g(1) g(1). For 0 < k 1, limα→k − g(α) = g(k) and limα→k − g(α) = g(k). limα→0+ g(α) = g(0) and limα→0+ g(α) = g(0).

Then [g(α), g(α)] is the parametric form of a fuzzy number. There exist several different notions of fuzzy derivative not all of which are equivalent. For example, we refer to [3,9,13,16,17]. More recently, another notion of generalized derivative has been introduced in [25]. For some recent contributions on the notion of generalized differentiability of fuzzy valued functions we refer to [4] and for contributions on differential equations over fuzzy spaces, see [28]. The notion of generalized differentiability introduced in [3] is widely used in the literature and in our work, we use it to study the existence of solutions to fuzzy DAEs. This notion of generalized differentiability is given below. Definition 2.2. (See [3].) Let g : (a, b) → RF and t0 ∈ (a, b). We say g is strongly generalized differentiable at t0 , if there exists an element g (t0 ) ∈ RF , such that (i) for all h > 0 sufficiently small, the differences g(t0 + h) g(t0 ), g(t0 ) g(t0 − h) and the following limits (in the metric D) exist: g(t0 + h) g(t0 ) g(t0 ) g(t0 − h) = lim = g (t0 ), h 0 h 0 h h lim

or (ii) for all h > 0 sufficiently small, the differences g(t0 ) g(t0 + h), g(t0 − h) g(t0 ) and the following limits (in the metric D) exist: g(t0 ) g(t0 + h) g(t0 − h) g(t0 ) = lim = g (t0 ), h 0 h 0 −h −h lim

or (iii) for all h > 0 sufficiently small, the differences g(x0 + h) g(t0 ), g(t0 − h) g(t0 ) and the following limits (in the metric D) exist: g(t0 + h) g(t0 ) g(t0 − h) g(t0 ) = lim = g (t0 ), h 0 h 0 h −h lim

or (iv) for all h > 0 sufficiently small, the differences exist g(t0 ) g(t0 + h), g(t0 ) g(t0 − h) and the following limits (in the metric D) exist: g(t0 ) g(t0 + h) g(t0 ) g(t0 − h) = lim = g (t0 ). h 0 h 0 −h h lim

Remark 2.3. We say that a function is (i)-differentiable if it is strongly generalized differentiable as the case (i) of the definition above, etc. We define the space of continuous fuzzy functions as ¯ RF ) = {g : I¯ → RF | g is continuous}, C(I,



33

which is a complete metric space endowed with the following metric ¯ RF ). H (g, h) = sup D g(t), h(t) , for g, h ∈ C(I, t∈I¯

We denote by C 1 (I, RF ) the set of continuous functions g : I → RF such that g is (i)-differentiable or (ii)-differentiable on I and g : I → RF is a continuous function. Lemma 2.4. (See [8].) Let g : (a, b) → RF be a generalized differentiable fuzzy function such that [g(t)]α = [g α (t), g α (t)] for each α ∈ [0, 1]. 1. If g is (i)-differentiable, then g α , g α are differentiable functions and α α g (t) = g (t), g α (t) . 2. If g is (ii)-differentiable, then g α , g α are differentiable functions and α α g (t) = g (t), g α (t) . Before we end this section, we introduce some preliminary ideas in the study of crisp DAEs. Let Mn (R) for n ∈ N denote the space of n × n matrices with real entries. For M ∈ Mn (R), M T denotes the AB A transpose of M. A matrix M ∈ Mn (R) may also be represented as a block matrix. For example, M = C ,M= B D and M = [A B] where A, B, C, D are the blocks of M with compatible sizes. Furthermore, we use notation |M|, for determinant of matrix M. We may also view an element of M ∈ Mn (R) as an ordered collection of column vectors, that is, M = [m(1) m(2) . . . m(n) ], where element m(i) is a n × 1 column vector. By a subcolumn of the lth column m(l) , (l) we mean m(k) = [mk,l mk+1,l . . . mn,l ]T . Similarly, viewing the matrix M as an ordered collection of row vectors, we write M = [m(1) m(2) . . . m(n) ]. Using the above notations, we define a particular submatrix of M. For fixed l, k ∈ {1, . . . , n}, let index set J = (j ) {jl , jl+1 , . . . , jn } be some nonempty subset of {1, . . . , n}. Let M(k)l be a submatrix of M, given by: (j ) (j ) (j ) (J ) M(k) = m(k)l m(k)l+1 . . . m(k)n , (j )

(j

)

(j )

where m(k)l , m(k)l+1 , . . . , m(k)n are column vectors of M. Consider the following linear system of differential-algebraic equations with constant coefficients E x˙ = Ax + f (t),

(2)

with the crisp initial condition: x(0) = x0 ,

(3)

¯ Rn ), and where E, A ∈ Mn (R), E is a singular matrix, f ∈ C(I, T T x(t) = x1 (t), . . . , xn (t) and x0 = x10 , . . . , xn0 . The following definitions and results can be found in [6,11,22]. Definition 2.5. Let E, A ∈ Mn (R). The matrix pair (E, A) is called regular if the characteristic polynomial p defined by p(λ) = |λE − A|, is not the zero polynomial. The following lemma gives a representation for a regular matrix pair (E, A) in terms of two matrices P and Q (cf. Theorem 2.7 [20]).


34


Lemma 2.6. Let E, A ∈ Mn (R) and (E, A) be regular. Then there exist nonsingular matrices P and Q such that

I 0 J 0 E=P Q A=P Q, 0 N 0 I where I is an identity matrix, J and N are matrices in Jordan canonical form; N is a nilpotent matrix with an index of nilpotency ν. If N = 0, then take ν = 1. Remark 2.7. For a given regular matrix pair (E, A), the pair

I 0 J 0 , , 0 N 0 I is called Weierstraß canonical form of (E, A) and the index of the matrix pair (E, A) is taken to be the index of nilpotency of N . That is ind(E, A) = ν. The matrices P , Q are called transformation matrices and J, N are referred as elements of Weierstraß canonical form of (E, A). Let P , Q and J, N be the transformation matrices and the elements of Weierstraß canonical form of (E, A) respectively. Denote

f˜1 y1 0 y1 (t) = Qx(t), = Qx0 and = P −1 f (t). y2 0 y2 (t) f˜2 Using the explicit representation of E and A given in Lemma 2.6, we obtain the following Weierstraß canonical form of the IVP (2)–(3). This canonical form consists of an IVP for a system of linear ODEs, that is uniquely solvable and a system of DAEs, given below: y˙1 = Jy1 + f˜1 ,

Subproblem-I

N y˙2 = y2 + f˜2 ,

Subproblem-II

y1 (0) = y1 0 ,

(4)

y2 (0) = y2 0 .

(5)

Clearly, subproblem (I) has a unique solution if f˜ is continuous. On the other hand, for subproblem II we have the following result that is easy to prove. ¯ .), the space of continuous crisp vector functions on I¯ that have ν continuous Lemma 2.8. Consider (5) with f˜2 ∈ C ν (I, derivatives, where ν be the index of nilpotency of N . Then (5) has the unique solution y2 (t) = −

ν−1

Ni

i=0

provided y2 0 = −

di ˜ f2 (t), dt i

ν−1 i=0

d ˜ N i dt i f2 (0). i

For the system of DAEs (2), the initial condition (3) is consistent if and only if y2 0 = −

ν−1

i=0

Ni

di ˜ f2 (0). dt i

That is, we need to ensure that the set of consistent initial values x0 is nonempty for the given DAEs. 3. Fuzzy initial value problem for the linear DAEs Consider the following fuzzy system: E x(t) ˙ = Ax(t) + f (t)

t ∈ I,

x(0) = γ ,

(6)

where x(t) = [x1 (t), . . . , xn E = (eij ) and A = (aij ) in Mn (R), E is a singular matrix, f ∈ n γ ∈ RF . Here I is an interval (0, T ) for some T > 0, or (0, ∞) and I¯ denotes the closure of I. (t)]T ,

¯ Rn ) C(I,

and



35

¯ RF ) is a solution of (6) if x and x˙ satisfy (6). We say that x ∈ C 1 (I, To find the solution x for (6) that is (i)-differentiable, we use the technique applied in [14] and Lemma 2.4 to transform the initial value problem for FDAE into a system of DAEs as follows: E X˙ α = AX α + F (t), X α (0) = X0α

(7)

α ), 1 i, j 2n and where E = (eij ) and A = (aij ), X α = (xiα ), X0α = (x0i

α

α

γ (t) x (t) f (t) α α , X0 = X = α , F (t) = f (t) x (t) γ α (t)

(8)

where x α (t) = [x α1 (t), x α2 (t), . . . , x αn (t)]T , γ α (t) = [γ α1 (t), γ α2 (t), . . . , γ αn (t)]T and x α (t), γ α (t) are defined similarly. And eij and aij are determined as follows: If eij 0

⇒

eij = eij ,

ei+n,j +n = eij ,

if eij < 0

⇒

ei,j +n = eij ,

ei+n,j = eij ,

and and

any eij not determined by above is equal to zero. aij is determined similarly. Then, the structure of E, A may be described as follows:

E 1 E2 A1 A2 E= , A= E2 E1 A2 A1

(9)

where E1 , A1 contain the positive entries of E, A respectively and E2 , A2 the negative entries of E, A. Clearly, E = E1 + E2 , A = A1 + A2 . To find the solution x for (6) that is (ii)-differentiable, using a similar approach as above the initial value problem for FDAE is transformed into a system of DAEs as follows: HX˙ α = AX α + F (t) X α (0) = X0α α ), 1 i, j 2n and where H = (hij ) and A = (aij ), X α = (xiα ), X0α = (x0i α

α

γ (t) x (t) f (t) α α , X0 = X = α , F (t) = f (t) x (t) γ α (t)

(10)

(11)

where x α (t) = [x α1 (t), x α2 (t), . . . , x αn (t)]T , γ α (t) = [γ α1 (t), γ α2 (t), . . . , γ αn (t)]T and x α (t), γ α (t) are defined similarly. The structure of H, A may be described as follows

E 2 E1 A1 A2 H= , A= (12) E1 E2 A2 A1 where E1 , A1 contain the positive entries of E, A respectively and E2 , A2 the negative entries of E, A and E = E1 + E2 , A = A1 + A2 . We need the following result from [14]. Theorem 3.1. The matrix pair (E, A) and (H, A) are regular if and only if there exists a λ such that |λE − A| = 0, λ(E1 − E2 ) − (A1 − A2 ) = 0,

(13)

i.e. the matrix pairs (E, A) and (E1 − E2 , A1 − A2 ) are both regular with the same λ. Our goal is to obtain an exact representation for the solutions for DAEs (7) and (10) when (E, A) and (H, A) are regular. If J is a matrix in Jordan canonical form, in general, an explicit expression for the (fuzzy) solution of the IVP (7) and (10) are complicated. Therefore, for simplicity we discuss the results for a diagonal matrix and obtain a simpler explicit form for the (fuzzy) solutions of (7) and (10).


36


So, for the rest of the paper, let J = diag(θ1 , θ2 , . . . , θm ). That is, J = (dij ) with θi , for i = j , for i, j = 1, . . . , m. dij = 0, for i = j We now present our main result below. Theorem 3.2. Let the pair (E, A) of square matrices be regular with P , Q as the transformation matrices, J , N as ¯ Rn ). Then the initial value the elements of Weierstraß canonical form of (E, A). Let ν = ind(E, A) and f ∈ C ν (I, problem (7) with a consistent initial condition is uniquely solvable. Moreover, let J = diag(θ1 , θ2 , . . . , θm ) where θi = θj for i = j and m = 2n − ν. Then, there exists an index set {jm+1 , . . . , j2n } ⊂ {1, . . . , 2n} such that the unique solution X α (t) of (7) is given by the following (explicit) expression: m 2n

α i θi t α Ck,l e + Rl (t), (14) x0k xl = k=1 k={jm+1 ,...,j2n }

i=1

i depend on the transformation matrix Q and R (t) depends on transformation matrices, for l = 1, . . . , 2n. Here, Ck,l l the elements of Weierstraß canonical form and f .

Proof. Let α ∈ [0, 1] be fixed. Since (E, A) is regular, the existence of X α (t) as solution of (7) is a direct consequence of (4) and Lemma 2.8. Let P and Q be the transformation matrices of (E, A). Letting Y α = QX α and utilizing Lemma 2.6, DAE (7) may be written in the following canonical form:

J 0 I 0 ˙α Y α (t) + F˜ (t), Y (t) = (15) 0 I 0 N with the initial condition Y0α = QX0α where α T Y α = y1α , . . . , y2n ; F˜ (t) = P −1 F (t),

α α T Y0α = y01 , . . . , y02n .

By the regularity of (E, A), there exists a λ ∈ R such that |λE − A| = 0. Since J is a diagonal matrix and N is a strictly upper triangular nilpotent matrix of index of nilpotency ν, there exist exactly m nonzero distinct eigenvalues of (A − λ0 E)−1 E and (2n − m) zero eigenvalues. We suppose H = (hi,j )2n×2n be the eigenvectors matrix of (A − λ0 E)−1 E . Now, we can rewrite (15) as the following two initial value problems: Y˙ 1α (t) = J Y1α (t) + F1 (t),

(16)

Y1α (0) = ßα1 ,

(17)

N Y˙ 2α (t) = Y2α (t) + F2 (t),

(18)

Y2α (0) = ßα2 ,

(19)

and

where T α T α α Y1α (t) = y1α (t), . . . , ym (t) , Y2α (t) = ym+1 (t), . . . , y2n (t) , T T F1 (t) = f˜1 (t), . . . , f˜m (t) , F2 (t) = f˜m+1 (t), . . . , f˜2n (t) , α α α T α T , . . . , y0m , and ßα2 = y0m+1 , . . . , y02n . ßα1 = y01 The components yiα (t) of the solution of (16)–(17) are given by t α yiα (t) = eθi t y0i + 0

f˜i (s)eθi s ds,

for i = 1, . . . , m.

(20)



37

Using Lemma 2.8, the solution of the problem (18)–(19) is obtained as: Y2α (t) = −

ν−1

Ni

i=0

di F2 (t) := G(t). dt i

(21)

Since the initial conditions are assumed to be consistent, we have here ßα2 = G(0).

(22)

In particular, denoting G(t) = [gm+1 (t), . . . , g2n

(t)]T ,

this condition is the following:

α α α α q2n,1 x01 + q2n,2 x02 + · · · + q2n,j2n x0j + · · · + q2n,2n x02n = g2n (0). 2n

Since Q is nonsingular and we have q2nj2n = 0 for some j2n ∈ {1, . . . , 2n}. Then, α x0j 2n

2n

q2n,i α g2n (0) = − x . q2n,j2n q2n,j2n 0i

α = Using this in y0l

2n

(23)

i=1 i=j2n

we get the following (2n − 1) equations: 2n

ql,k ql,j2n α ql,j2n g2n (0) 1 α x , y0l = + q 0k q2n,j2n q2n,j2n q 2nk 2n,j 2n k=1 α i=1 ql,i x0i ,

(24)

k=j2n

for l = 1, . . . , 2n − 1. Again, invoking the nonsingularity of Q, there exists j2n−1 ∈ {1, 2, . . . , 2n} \ {j2n } such that q2n−1,j2n−1 q2n−1,j2n (j2n−1 ) = Q (2n−1) = 0. q q 2n,j2n−1

2n,j2n

From l = (2n − 1) in (24) we get, α x0j 2n−1

=

|G(2n−1) (0)

(j )

2n Q(2n−1) |

(j2n−1 ) |Q(2n−1) |

−

2n

i=1 i={j2n−1 ,j2n }

(i)

|q(2n−1)

(j

2n) Q(2n−1) |

(j2n−1 ) |Q(2n−1) |

α . x0i

Using this in (24), we get the following system: for l = 1, . . . , 2n − 2. (j ) q (qˆ (ji ) )T (qˆ(l)2n−1 )T l,i (l) (j ) |G(i) (0) Q(ji+1 ) | (j2n−1 ) (i) (i) i 2n n

q(2n−1) Q(2n−1) Q(i+1) α α y0l = y0i + . (j2n−1 ) (ji+1 ) (j ) |Q(2n−1) | |Q(i+1) ||Q(i)i | i=1 i=2n−1 i={j2n−1 ,j2n }

Proceeding thus, using the nonsingularity of Q and notation for sub matrices given in Section 2, we find for k = (jm+1 ) |= 0. Thus, finally, for l = 1, . . . , m we 2n − 2, . . . , m + 1, jk ∈ {1, 2, . . . , 2n} \ {jk+1 , . . . , j2n } such that |Q(m+1) have: (j ) q (qˆ (ji ) )T (qˆ(l)m+1 )T l,i (l) (j ) |G(i) (0) Q(ji+1 ) | (j ) (i) q m+1 (i) 2n 2n Q i

Q(m+1) (m+1) (i+1) α α y0l = + y . (25) 0i (jm+1 ) (ji+1 ) (ji ) |Q | |Q ||Q | i=1 i=m+1 (i) (m+1) (i+1) i={jm+1 ,...,j2n }

α , i = m + 1, . . . , 2n. We see that the (2n − m) omitted components of X0α are x0j i Now, we are in a position to present the exact representation of the solution X α . Since X α = Q−1 Y α = H Y α , from (20) and (21) we conclude: ⎤ ⎡ x α (t) ⎤ ⎡ m h y α eθi t + t f˜ (s)eθi s ds + 2n i=1 1i 0i i=m+1 h1i gi (t) 1 0 i ⎥ .. ⎣ .. ⎦ = ⎢ ⎣ ⎦. . . t m 2n α α θi t + θi s ˜ x2n (t) i=1 h2ni y0i e i=m+1 h2ni gi (t) 0 fi (s)e ds +


38


α (i = 1, . . . , m) from (25) for the system above, we get, for l = 1, 2, . . . , 2n: Substituting y0i (j ) q (qˆ (jk ) )T (qˆ(i)m+1 )T i,k (i) (j ) |G(k) (0) Q(jk+1 ) | (j ) (k) m+1 (k) k 2n 2n m m

q(m+1) Q(m+1) Q(k+1) α θi t α θi t xl = hli hli e e x0k + (jm+1 ) (jk+1 ) (j ) |Q(m+1) | |Q(k+1) ||Q(k)k | k=1 i=1 i=1 k=m+1 k={jm+1 ,...,j2n }

+

m

t

f˜i (s)eθi s ds +

i=1 0

2n

hli gi (t).

i=m+1

We can write the equation above in the following compact form m 2n

i θi t α Ck,l e + Rl (t), xlα = x0k k=1 k={jm+1 ,...,j2n }

for l = 1, . . . , 2n.

i=1

2

We present below a result similar to Theorem 3.2 for IVP (10) involving the notion of (ii)-differ Theorem 3.3. Let the pair (H, A) of square matrices be regular with P , Q as transformation matrices and J , N as ¯ Rn ). Then the initial value the elements of Weierstraß canonical form of (H, A). Let ν = ind(H, A) and f ∈ C ν (I, problem (10) with a consistent initial condition is uniquely solvable. Moreover, let J = diag(θ1 , θ2 , . . . , θm ) where θi = θj for i = j and m = 2n − ν. Then there exists an index set {jm+1 , . . . , j2n } ⊂ {1, . . . , 2n} such that the unique solution X α (t) of (10) is given by the following (explicit) representation: m 2n

i θi t α xlα = Ck,l e + Rl (t), (26) x0k k=1 k={jm+1 ,...,j2n }

i=1

i depend on the transformation matrix Q and R (t) depends on transformation matrices, for l = 1, . . . , 2n. Here Ck,l l the elements of Weierstraß canonical form and f .

Remark 3.4. Since the fuzzy problem (6) is not equivalent to the crisp problems (7) or (10), we require additional conditions to ensure that there exist two fuzzy solutions for the fuzzy initial value problem (6). Theorem 3.5. Let the pair (E, A) be as in the theorem above. Let X α (t) = (xlα (t)), where m 2n

α i θi t α xl = Ck,l e + Rl (t) l ∈ {1, . . . , n}, x0k k=1 k={jm+1 ,...,j2n }

i=1

be unique solution of the problem (7). Furthermore, let X0α be differentiable with respect to α. For some T1 > 0 assume ¯ that the following conditions hold on I1 = [0, T1 ] ⊂ I: 2n m

i θt ∂ α i Ck,l − Ck,l+n ei (27) x 0, ∂α 0k k=1 k={jm+1 ,...,j2n } n

k=1 k={jm+1 ,...,j2n } n

k=1 k={jm+1 ,...,j2n }

i=1

m

i i 1 Ck,l eθi t xok − Ck,l+n + Rl (t) − Rl+n (t) 0,

(28)

i=1

m

i Ck,l

i=1

i − Ck,l+n

θi e

θi t

∂ α x 0 ∂α 0k

(29)


R. Alikhani et al. / Fuzzy Sets and Systems 245 (2014) 30–42 n

m

i Ck,l

k=1 k={jm+1 ,...,j2n }

i − Ck,l+n

θi e

θi t

1 ˙ l (t) − R ˙ l+n (t) 0. + R x0k

39

(30)

i=1

Then the fuzzy initial value problem (6) with consistent initial condition has a unique local solution that is (i)-differentiable. Proof. Let x(t) = [x1 (t), . . . , xn (t)]T with α-level sets [xl (t)]α = [x αl (t), x¯lα (t)] where x αl and x¯lα , l ∈ {1, . . . , n} are as in the theorem. We assert that under the assumptions of the theorem, xl ∈ C(I¯1 , RF ) and x˙l ∈ C(I1 , RF ) and x(t) satisfies the fuzzy initial value problem (6) on I1 . Since γk , k = 1, . . . , n are fuzzy numbers, the conditions (iv) and (v) of Lemma 2.1 for x αl and x¯lα are fulfilled. For a fixed l ∈ {1, . . . , n}, from (27) we conclude ∂ α ∂ α α (t) x l (t) − x¯lα (t) = xl (t) − xn+l ∂α ∂α m 2n

θt ∂ α i i i = Ck,l − Ck,l+n e x 0. ∂α 0k k=1 k={jm+1 ,...,j2n }

Also, from (28) we get n

x¯lα (t) − x αl (t) =

k=1 k={jm+1 ,...,j2n }

i=1

m

i i 1 Ck,l eθi t x0k − Ck,l+n + Rl (t) − Rl+n (t) 0.

i=1

Hence the conditions (i)–(iii) for Lemma 2.1 are verified and [x αl (t), x¯lα (t)] are valid α-level sets of the fuzzy number α valued function xl (t). Also from (30) and (29), [x˙ αl (t), x¯˙ l (t)] are valid level sets of the fuzzy number valued function x˙l (t) for t ∈ I1 . Then, from [17] and using the stacking theorem we can construct the solution of the problem (6). Therefore xl ∈ C 1 (I1 , RF ) ∩ C(I¯1 , RF ). The proof of assertion is complete and we conclude that x is a solution of the system (6). That the solution is unique follows from Theorem 3.2. 2 The following result is an analogue of Theorem 3.5, for (10) involving (ii)-differentiability. We omit its proof. Theorem 3.6. Let the pair (H, A) be as in the above Theorem 3.3. Let X α (t) = (xlα (t)), where m 2n

i θi t α Ck,l e + Rl (t) l ∈ {1, . . . , n}, x0k xlα = k=1 k={jm+1 ,...,j2n }

i=1

be unique solution of the problem (10). Furthermore, let X0α is differentiable relative to α. For some T1 > 0 assume that the following conditions hold on I1 = [0, T1 ] ⊂ I¯ m 2n

θt ∂ α i i Ck,l − Ck,l+n e i (31) x 0, ∂α 0k k=1 k={jm+1 ,...,j2n } n

k=1 k={jm+1 ,...,j2n } n

k=1 k={jm+1 ,...,j2n } n

k=1 k={jm+1 ,...,j2n }

i=1

m

i i 1 Ck,l eθi t xok − Ck,l+n + Rl (t) − Rl+n (t) 0,

(32)

i=1

m

i i Ck,l θi eθi t − Ck,l+n

i=1

m

i Ck,l

i=1

i − Ck,l+n

θi e

θi t

∂ α x 0 ∂α 0k

(33)

1 ˙ l (t) − R ˙ l+n (t) 0. + R x0k

(34)


40


Then the fuzzy initial value problem (6) with consistent initial condition has a unique local solution that is (ii)-differentiable. Remark 3.7. Thus, a method of finding a unique solution to the fuzzy IVP (6) is obtained in Theorems 3.2 and 3.5. It is interesting to note that although the method depends on the transformation matrices, P and Q, does not depend on P and Q. 4. Examples In this section, we present two examples of fuzzy IVPs and illustrate the ideas discussed in Section 3. Example 4.1. Consider the following linear fuzzy initial value problem −1 0 x˙1 1 0 x1 x1 (0) γ1 = . = 1 0 0 1 x˙2 x2 γ2 x2 (0) Using the method presented in Section 3, we rewrite the above problem as the following system of DAEs: ⎛ α⎞ ⎛ α ⎞ ⎛ α⎞ ⎛ α⎞ x˙ 1 γ1 x01 x1 ⎜ x˙ α ⎟ α α α⎟ ⎜ ⎟ ⎟ ⎜ ⎜ x2 ⎟ ⎜ ⎟ ⎜ x02 ⎟ = ⎜ γ 2 ⎟ , E ⎜ 2α ⎟ = A ⎜ α α ⎝x ⎠ ⎝γα ⎠ ⎝x ⎠ ⎝ x˙ 1 ⎠ 1 03 1 α ˙x α2 x α2 x04 γ α2 where

⎞ 0 0 −1 0 ⎜ 1 0 0 0⎟ E =⎝ ⎠ −1 0 0 0 0 0 1 0

⎛

⎛

and

1 ⎜0 A=⎝ 0 0

0 1 0 0

0 0 1 0

⎞ 0 0⎟ ⎠. 0 1

The pair (E, A) is regular especially with λ = 0. The matrix A−1 E = E has two distinct nonzero eigenvalues λ1 = 1, λ2 = −1 with eigenvector matrix ⎛ ⎞ 1 1 0 0 ⎜ 1 −1 1 0 ⎟ H =⎝ ⎠. −1 1 0 0 −1 −1 1 1 Corresponding to this example, we have the transformation matrix Q = H −1 and J = diag(θ1 , θ2 ) where θ1 = 1 and θ2 = −1. Furthermore, the consistency conditions are γ α2 = −γ α1 and γ α2 = −γ α1 . Then, by Theorem 3.2 and Theorem 3.5, it follows that the problem has the following unique solution on interval [0, ∞). ⎧ 1 t 1 −t 1 t 1 −t α α α ⎪ ⎪ e + e +γ1 − e + e , ⎨ x 1 (t) = γ 1 2 2 2 2 1 1 1 t 1 −t ⎪ ⎪ ⎩ x α1 (t) = γ α1 − et + e−t + γ α1 e + e , 2 2 2 2 ⎧ 1 t 1 −t 1 t 1 −t α α α ⎪ ⎪ e − e +γ1 − e − e , ⎨ x 2 (t) = γ 1 2 2 2 2 1 1 1 t 1 −t ⎪ ⎪ ⎩ x α2 (t) = γ α1 − et − e−t + γ α1 e − e . 2 2 2 2 Next, we consider the following example where E and A are matrices with positive entries. Here, adopting a slightly different approach, using Lemma 2.4, the fuzzy initial value problem is split into two crisp initial value problems. Let [x(t)]α = [x α (t), x¯ α (t)] and [γ ]α = [γ α , γ¯ α ]. If x is differentiable, then the fuzzy problem (6) splits into two crisp initial value problems as follows:



E x˙ α (t) = Ax α (t) + f (t)

41

t ∈ I,

x (0) = γ , α

α

(35)

where x α (t) = [x α1 (t), . . . , x αn (t)]T and γ α = [γ α1 , . . . , γ αn ]T and E x˙¯ (t) = Ax¯ α (t) + f (t) α

t ∈ I,

x¯ (0) = γ¯ . α

α

(36)

Here x¯ α (t) = [x¯1α (t), . . . , x¯nα (t)]T and γ¯ α = [γ¯1α , . . . , γ¯nα ]T . Under suitable conditions on the problem a unique fuzzy solution for the fuzzy initial value problem (6) is guaranteed. Example 4.2. Consider the following linear fuzzy initial value problem x1 x˙1 γ1 1 0 0 x1 (0) 1 1 1 0 2 0 x2 (0) = γ2 . x˙2 = 0 1 0 x2 0 0 1 0 1 0 x˙3 x3 γ3 x3 (0) Since the coefficient matrices in this problem have only nonnegative entries, this can be treated as two separate crisp problems as mentioned above. Again, the pair (E, A) is regular with λ = 0. The matrix A−1 E has two distinct nonzero eigenvalues λ1 = 2, λ2 = 1 with eigenvectors matrix H and H −1 are given by: ⎛ ⎞ 0 0 12 3 1 1 H= 2 0 0 , Q = H −1 = ⎝ 1 −2 1 ⎠ 1 0 −1 0 1 −1 2

The element J of Weierstraß canonical form is given by: J = diag(θ1 , θ2 ) where θ1 = consistency conditions are γ α2 = 2γ α3 and γ α2 = 2γ α3 . Then the solution of problem is given as: ⎧ 3 α θ1 t 3 α θ2 t α α ⎪ ⎪ ⎪ x 1 (t) = 2 γ 2 e + γ 1 − 2 γ 2 e , ⎪ ⎨ x α2 (t) = γ α2 eθ1 t , ⎪ ⎪ ⎪ ⎪ α 1 ⎩ x 3 (t) = γ α2 eθ1 t , 2 and

1 2

and θ2 = 1. Furthermore, the

⎧ 3 α θ1 t 3 α θ2 t α α ⎪ ⎪ x 1 (t) = γ 2 e + γ 1 − γ 2 e , ⎪ ⎪ 2 2 ⎨ α α θ1 t x 2 (t) = γ 2 e , ⎪ ⎪ ⎪ ⎪ ⎩ x α (t) = 1 γ α eθ1 t . 3 2 2

If we consider γ1 = γ2 , all the conditions of Lemma 2.1 are satisfied on interval [0, 2 ln 32 ] and the problem has the fuzzy unique solution on interval [0, 2 ln 32 ]. If γ1 = 2γ2 the problem has the unique solution on [0, ∞) where as if γ1 = 12 γ2 , then the problem has no solution. Acknowledgements The authors are thankful to the referees for their careful reading of the paper and valuable suggestions. The first author (R. Alikhani) wishes to thank the Department of Mathematical Sciences, Florida Tech, for their kind hospitality during her visit.


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