Approximate Solution of LR Fuzzy Sylvester Matrix Equations

The fuzzy Sylvester matrix equation A?̃? + ?̃?B = ?̃? in which A, B are m × m and n × n crisp matrices, respectively, and ?̃? is an m× n LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.


Introduction
System of simultaneous matrix equations plays a major role in various areas such as mathematics, physics, statistics, engineering, and social sciences.In many problems in various areas of science, which can be solved by solving a linear matrix equation, some of the system parameters are vague or imprecise, and fuzzy mathematics is better than crisp mathematics for mathematical modeling of these problems, and hence solving a linear matrix equation where some or all elements of the system are fuzzy is important.The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [1], Dubois and Prade [2], and Nahmias [3].A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [4], Goetschel and Voxman [5], and Wu and Ma [6,7].
Since Friedman et al. [8] proposed a general model for solving an  ×  fuzzy linear systems whose coefficients matrix are crisp and the right-hand side is an arbitrary fuzzy number vector by a embedding approach, some works [9][10][11][12][13][14][15][16][17][18][19][20] have been done about how to deal with some advanced fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), full fuzzy linear systems (FFLS), dual full fuzzy linear systems (DFFLS), and general dual fuzzy linear systems (GDFLS).However, for a fuzzy linear matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past decades.In 2009, Allahviranloo et al. [21] studied the fuzzy linear matrix equation, (FLME) of the form  X = C.By using the parametric form of the fuzzy number, they derived necessary and sufficient conditions for the existence of the set of fuzzy solutions and designed a numerical procedure for calculating the solutions of the fuzzy matrix equations.In 2011, Guo and Gong [22,23] investigated a class of simple fuzzy matrix equations  X = B by the undetermined coefficients method and studied least squares solutions of the inconsistent fuzzy matrix equation  X = B by using generalized inverses of matrices.In 2011, Guo [24] studied the approximate solution of fuzzy Sylvester matrix equations with triangular fuzzy numbers.Lately, Guo and Shang [25] considered the fuzzy symmetric solutions of fuzzy matrix equation  X = B.
The LR fuzzy number and its arithmetic operations were first introduced by Dubois and Prade.We know that triangular fuzzy numbers are just specious cases of LR fuzzy numbers.In particular, Allahviranloo et al. [26] have showed us that the weak fuzzy solution of fuzzy linear systems x = b does not exist sometimes when x, b are denoted by triangular fuzzy numbers.Recently, he also considered the LR fuzzy linear systems [27] by the linear programming with equality constraints.In this paper we consider the fuzzy approximate solution of LR fuzzy Sylvester matrix equation  X + X = C.In fact, the fuzzy Sylvester matrix equation  X + X = C has numerous applications in control theory, signal processing, filtering, model reduction, decoupling techniques for ordinary and partial differential equations, and block-diagonalization of matrices and so on.But there was little research work on it.The contributions of this paper is to generalize Dubois' definition and arithmetic operation of LR fuzzy numbers and then use this result to solve fuzzy Sylvester matrix systems numerically.The structure of this paper is organized as follows.
In Section 2, we recall the LR fuzzy number, generalize the definition of LR fuzzy numbers, and present the concept of the LR fuzzy Sylvester matrix equation.The model to the fuzzy Sylvester matrix equation is proposed in detail and the fuzzy approximate solution of the original fuzzy matrix systems is derived from solving the crisp systems of linear equations in Section 3. Some examples are given in Section 4 and the conclusion is drawn in Section 5.
Let  1 be the set of all fuzzy numbers on .
The definition of a right shape function (⋅) is usually similar to that of (⋅).
Noticing that ,  > 0 in Definition 2, which limits its applications, we extend the definition of LR fuzzy numbers as follows.

Some Results on Matrix Theory.
Let  be an  ×  real matrix and (⋅) ⊤ denote the transpose of a matrix (⋅).We recall that a generalized inverse  of  is an  ×  matrix which satisfies one or more of Penrose equations For a subset {, , } of set {1, 2, 3, 4}, the set of  ×  matrices satisfying the equations contained in {, , } is denoted by {, , }.A matrix in {, , } is called an {, , }-inverse of  and is denoted by {, , }.In particular, the matrix  is called a {1}-inverse or a -inverse of  if it satisfies (1).As usual, the -inverse of  is denoted by  − .If  satisfies (2) then it is called a {2}-inverse and If  satisfies (1) and ( 2) then it is called a reflexive inverse or a {1, 2}inverse of .The Moore-Penrose inverse of  is the matrix  which satisfies (1)-( 4).Any matrix  admits a unique Moore-Penrose inverse, denoted by  † .Lemma 4 (see [28]).For a system of linear equations When it is consistent, its solution can be expressed by  =  in which  ∈ {1}; when it has an infinite number of solutions, its minimal norm solution can be expressed by  =  in which  ∈ {1, 4}.When it is inconsistent, its least squares solutions can be expressed by  =  in which  ∈ {1, 3}.In particular,  =  †  is the minimum norm least square solution to the above linear system.

The Fuzzy Sylvester Matrix Equation Definition 6.
A matrix Ã = (ã  ) is called an LR fuzzy matrix, if each element ã of Ã is an LR fuzzy number.
Definition 7. The matrix system where  and  are real matrices and  is an LR fuzzy matrix, that is,   ,   ∈ , c ∈ , is called an LR fuzzy Sylvester matrix equations (LRFSMEs).
Using matrix notation, we have Up to the rest of this paper, we suppose that C is a positive LR fuzzy numbers matrix and use the formulas given in Definition 3.An LR fuzzy numbers matrix is called an LR solution of the fuzzy Sylvester matrix equation (11) if X satisfies (12).

Method for Solving LRFSMEs
In this section we investigate the fuzzy Sylvester matrix equations (12).Firstly, we set up a computing model for solving LRFSME.Then we define the LR fuzzy solution of LRFSME and obtain its solution representation by means of generalized inverses of matrices.Finally, we give a sufficient condition for strong fuzzy approximate solution to the fuzzy Sylvester matrix equation.
is called the extension on column of the fuzzy matrix Ã.
Proof.Setting  =   in (15), we have Similarly, the result is obvious when we replace  with   in (15).We combine ( 17) and ( 18) and obtain the following conclusion: Theorem 11.The matrix X ∈  × is the solution of the fuzzy linear matrix equation (12) if and only if that x = Vec( X) is the solution of the following [LR] linear fuzzy system: where  = (  ⊗  +   ⊗   ) and ỹ = Vec( C).
Proof.Applying the extension operation to two sides of ( 12) and according to the Definition 8 and Theorem 10, we have where  = (  ⊗  +   ⊗   ) is an  ×  matrix and ỹ = Vec( C) is an  LR fuzzy numbers vector.Thus the X is the solution of ( 16) is equivalent to that x = Vec( X) is the solution of (20).
For simplicity, we denote  =  in (20); thus Secondly, we extend the fuzzy LR linear system (16) into two systems of linear equations according to arithmetic operations of LR fuzzy numbers.
Meanwhile, the left spread   = (  1 , . . .,    ) and the right spread   = (  1 , . . .,    )  of the solution are obtained by solving the following crisp linear systems: in which (  ), 1 ≤ ,  ≤ 2 are determined as follows: and any   which is not determined by the above items is zero, 1 ≤ ,  ≤ 2.Moreover,  has the following structure: and satisfies  =  − .

Computing the Model.
In order to solve the LR fuzzy Sylvester matrix equation ( 12), we need to consider the LR fuzzy system of linear equations (20).In order to solve (20), we need to consider the systems of linear equations ( 23) and (24).For instance, when matrix  in (23) and matrix  in (24) are both invertible, their solutions are expressed by The following Lemma shows when the matrix  is nonsingular and how to calculate  −1 .
Lemma 13 (see [8]).The matrix  is nonsingular if and only if the matrices  =  −  and  +  are both nonsingular.If  −1 exists it must have the same structure as , that is, It seems that we have obtained the solution of the original fuzzy linear matrix system (20) as follows: where   is a  order unit matrix and  is a  order null matrix.But the solution vector may still not be an appropriate LR fuzzy numbers vector except for   ≥ 0,   ≥ 0. So one gives the definition of LR fuzzy solutions to (11)  ⊤ are an exact solution of (24); respectively, such that   ≥ 0,   ≥ 0, we call x = (,   ,   ) LR an LR fuzzy solution of (11).
When linear equation ( 23) or ( 24) is inconsistent, we can consider its approximate solution.An approximation solution which is often used is the least squares solution of ( 23) or (24), defined by minimizing the Frobenius norm of ( − ) or ( − ). For By the generalized inverse theory [28], we know where  {1,3} is the least squares inverse of the matrix .Now we define the LR fuzzy approximate solution of the fuzzy matrix equations (11) from the fuzzy linear systems (20).23) and (24); respectively, such that   ≥ 0,   ≥ 0, then we say x = (,   ,   ) LR is an LR fuzzy approximate solution of (11).
Let  belong to  × and ỹ be a  arbitrary LR fuzzy numbers vector.Then the solutions of linear systems ( 23) and ( 24) can be expressed uniformly by respectively, no matter ( 23) and ( 24) are consistent or not.
It seems that the solution of the LR fuzzy linear system (20) can be expressed as But it is not the case except for   ≥ 0 and   ≥ 0. In this case, we have a fuzzy set solution not fuzzy number solution.To find the fuzzy number solution, we can approximate the fuzzy set solution by a fuzzy number by one of the approximation methods finally.Now we give a sufficient condition for LR fuzzy solution to the fuzzy Sylvester matrix equation.

A Sufficient Condition for LR Fuzzy Solution.
To illustrate the expression (43) of an appropriate LR fuzzy solution vector, we now discuss the generalized inverses of matrix  in a special structure.
Then the matrix is the Moore-Penrose inverse of the matrix , where ( + ) † , ( − ) † are Moore-Penrose inverses of matrices  +  and  − , respectively.
The key points to make the solution vector be an LR fuzzy solution are   ≥ 0 and   ≥ 0. Since   ≥ 0,   ≥ 0 and the nonnegativity of   and   is equivalent to the condition  † ≥ 0. By the above analysis, one has the following conclusion.
Theorem 17.Let  belong to  × and  † nonnegative.Then the solution of the LR fuzzy linear system (12) is expressed by and it admits an LR fuzzy approximate solution.
Theorem 19 (see [29]).The inverse of a nonnegative matrix  is nonnegative if and only if  is a generalized permutation matrix.
Theorem 20 (see [30]).The matrix , of rank  with no zero row or zero column, admits a nonnegative {1, 3}-inverse if and only if there exist some permutation matrices ,  such that where  is a direct sum of  positive, rank-one matrices.
where each   has rank 1 and the rows of   are orthogonal to the rows of   , whenever  ̸ = , the zero matrix may be absent.