Positive definite solution of the matrix equation X = Q − A ^∗ X ⁻¹ A + B ^∗ X ⁻¹ B via Bhaskar-Lakshmikantham fixed point theorem

Purpose

The purpose of this paper is to study the existence and uniqueness of a positive definite solution to the nonlinear matrix equation X = Q − A^∗X⁻¹A + B^∗X⁻¹B, which is a special stochastic rational Riccati equation arising in stochastic control theory.

Methods

Our technique is based on the Bhaskar and Lakshmikantham coupled fixed point theorem.

Results

A new result on the existence of a unique positive definite solution is derived. An iterative method is constructed to compute the unique positive definite solution. Finally, some numerical examples are used to show that the iterative method is feasible.

Conclusion

Coupled fixed point theory on ordered metric spaces can be a useful tool to solve some classes of nonlinear matrix equations.

We consider the matrix equation:

where Q is an n × n Hermitian positive definite matrix, and A and B are arbitrary n × n matrices. Equation (1) is a special stochastic rational Riccati equation arising in stochastic control theory, and it can be described below. Some stochastic control problems lead to computing the positive definite solution of the following stochastic rational Riccati equation [1]:

where Z ⁺ stands for the Moore-Penrose inverse of a matrix Z and C; P, S, R and L are given matrices of size n×nn×mn×nm×m, and n×m, respectively, such that

is a Hermitian matrix, and the operator

is positive, i.e. X ≥ 0 implies π(X) ≥ 0. Consider the following case: C is the identity matrix, P is an n × n nonsingular matrix, S is an n × n positive definite matrix, L is the zero matrix, and π₁₂(X) = π₂(X) = 0,π₁(X) = (R + P^∗XP)⁻¹, where R + P^∗XP is positive definite for all positive semidefinite matrices X. Meanwhile, the stochastic rational Riccati Equation (2) has the form

Set

then

By Equations 3 to 5, we have

which implies that

Set

then Equation 3 can be equivalently written as Equation 1. Therefore, Equation 1 is a special stochastic rational Riccati equation (Equation 2). Moreover, some special cases of Equation 1 are also problems of practical importance, such as the matrix equation X + M^∗X⁻¹M = Q that arises in the control theory, ladder networks, dynamic programming, stochastic filtering, statistics, and so on [2-4]. The matrix equation X−M^∗X⁻¹M = Q arises in the analysis of stationary Gaussian reciprocal processes over a finite interval [5,6].

Since 1993, the matrix equations X + M^∗X⁻¹M = Q and X−M^∗X⁻¹M = Q have been extensively studied, and the research results mainly concentrated on the following:

sufficient conditions and necessary conditions for the existence of a (unique) positive definite solution [2,6-8];

numerical methods for computing the (unique) positive definite solution [4-6,9-13];

properties of the positive definite solution [2,4]; and

perturbation bound for the positive definite solution [3,14].

In addition, other nonlinear matrix equations such as AX² + BX + C = 0 [15], X^s ± A^∗X^−tA = Q[16,17], [18,19], X ± A^∗X^−qA = Q[3,20-27], [28], X + A^∗F(X)A = Q[29,30] have been investigated by many authors. However, results on the general nonlinear matrix equation (Equation 1) are few as far as we know.

In this paper, we first use the the Bhaskar and Lakshmikantham fixed point theorem to study the positive definite solution of the nonlinear matrix equation (Equation 1). A new sufficient condition for the existence of a unique positive definite solution to Equation 1 is derived. An iterative method is constructed to compute the unique Hermitian positive definite solution, and the error estimation formal is also given. In the end, we use some numerical examples to illustrate that the iterative method is feasible to compute the unique positive definite solution of Equation 1.

Throughout this paper, we denote by and the set of N × N complex and N × N Hermitian matrices, respectively. For , A ≥ 0 (A > 0) means that A is positive semi-definite (positive definite). Moreover, A ≥ B(A > B) means that A − B ≥ 0 (A − B > 0), and X∈[A,B] means A ≤ X ≤ B. A^∗ and r(A) denote the complex conjugate transpose and the spectral radius of A, respectively. We denote by ∥·∥ the spectral norm, i.e., , where λ ⁺ (A^∗A) is the largest eigenvalue of A^∗A. The N × N identity matrix will be written as I. We denote by ∥·∥_tr the trace norm. Recall that this norm is given by

where σ_j(A),j = 1,…,N are the singular values of A.

The following lemmas will be useful later.

Lemma 2.1 (See [31])

Let A ≥ 0 and B ≥ 0 be N × N matrices, then 0 ≤ tr(AB) ≤ ǁAǁ tr(B).

Lemma 2.2 (See [32])

If 0 < θ ≤ 1, and P and Q are positive definite matrices of the same order with P,Q ≥ bI > 0, then for every unitarily invariant norm |||P^θ − Q^θ||| ≤ θb^θ−1|||P − Q||| and |||P^−θ−Q^−θ||| ≤ θb^{−(θ + 1)}|||P−Q|||.

Lemma 2.3 (See [32])

Let satisfying − I < A < I, then ∥A∥ < 1.

Let (X,≼) be a partially ordered set and F:X × X → X be a given mapping. We say that F has the mixed monotone property if for any x,y∈X,

We say that (x,y) is a coupled fixed point of F if x = F(x,y) and y = F(y,x).

The proof of our main result is based on the following two fixed point theorems.

Theorem 2.1 ( [33])

Let (X,≼) be a partially ordered set endowed with a metric d such that (X,d) is complete. Let F:X × X → X be a continuous mapping having the mixed monotone property on X. Assume that there exists a δ ∈[0,1), such that

for all (x,y),(u,v) ∈ X × X with x ≽ uand y ≼ v. We suppose that there exist x₀, y₀ ∈ X, such that x₀ ≼ F(x₀,y₀) and y₀ ≽ F(y₀,x₀). Then,

(a) F has a coupled fixed point ; and

(b) the sequences {x_n} and {y_n} defined by x_{n + 1} = F(x_n,y_n) and y_{n + 1} = F(y_n,x_n) converge respectively to and .

In addition, suppose that every pair of elements has a lower bound and an upper bound, then

(c) F has a unique coupled fixed point ;

(d) ; and

(e) we have the following estimate:

For other results concerning fixed point theorems on ordered sets, we refer to [34-37].

Theorem 2.2 (Schauder Fixed point theorem)

Let S be a nonempty, compact, convex subset of a normed vector space. Every continuous function f:S → S mapping S into itself has a fixed point.

There exist a > 0, b > 0 (real numbers), such that the following assumptions were considered:

1. a⁻¹A^∗A + aI ≤ Q ≤ bI

2. bA^∗A−aB^∗B ≤ ab(Q − aI)

3. bB^∗B − aA^∗A ≤ ab(bI − Q)

4. , .

We denote by Ωthe set of matrices defined by

Our main result is discussed below:

Theorem 3.1

Under the assumptions 1 to 4, we have

(I) Equation 1 has a unique solution

(II)

(III) the sequences {X_n} and {Y_n} defined by

converge to , that is,

and the error estimation is given by

where 0 < δ < 1.

Proof

For all , let

□

We claim that F(Ω × Ω) ⊂ Ω. Indeed, let X,Y ∈ Ω, that is, X ≥ aI and Y ≥ aI. This implies that

On the other hand, from assumption 1, we have

Thus, we have

which implies that F(X,Y) ∈ Ω. Then, our claim holds.

Now, the mapping F :  Ω  × Ω→Ω is well defined. Let X,Y,U,V ∈ Ω, such that X ≥ Uand Y ≤ V. We have

Since U⁻¹ − X⁻¹ ≥ 0 and Y⁻¹ − V⁻¹ ≥ 0, using Lemma 2.1, we get

On the other hand, since X,Y,U,V ≥ aI, using Lemma 2.2, we have

and

Thus, we get

This implies that

where

From condition 4 and Lemma 2.3, we can easily show that 0 ≤ δ < 1. Now, taking X₀ = aI and Y₀ = bI, from conditions 2 and 3, we can easily show that X₀ ≤ F(X₀,Y₀) and Y₀ ≥ F(Y₀,X₀). On the other hand, for every , there is a greatest lower bound and a least upper bound. Note also that F is a continuous mapping. Now, (I) and (III) follow immediately from Theorem 2.1. Let be the unique solution to Equation 1 in Ω.

To prove (II), we shall use the Schauder fixed point theorem. We define the mapping G:[F(aI,bI),F(bI,aI)] → Ω by

We claim that G([F(aI,bI),F(bI,aI)]) ⊆[F(aI,bI),F(bI,aI)]. Let X∈[F(aI,bI),F(bI,aI)], that is,

Using the mixed monotone property of F, we get

On the other hand, from conditions 2 and 3, we have

Again, using the mixed monotone property of F, we get

From Equations 7 and 8, it follows that

Thus, our claim that G([F(aI,bI),F(bI,aI)]) ⊆ [F(aI,bI),F(bI,aI)] holds.

Now, G maps the compact convex set [F(aI,bI),F(bI,aI)] into itself. Since G is continuous, it follows from Schauder fixed point theorem (see Theorem 2.2 ) that G has at least one fixed point in this set. However, fixed points of G are solutions of Equation 1, and we proved already that Equation 1 has a unique solution in Ω. Thus, this solution must be in the set [F(aI,bI),F(bI,aI)], that is,

Thus, we proved (II). This makes end to the proof. □

The following results are immediate consequences of our Theorem 3.1.

Theorem 3.2

Consider Equation 1 with Q = I. Suppose that

(1) ; and

(2) .

Then, items I to III of Theorem 3.1 hold.

Theorem 3.3

Consider Equation 1 with A and B which are unitary matrices. Suppose that

(1) ; and

(2) (a⁻¹ + a)I ≤ Q ≤ (b + b⁻¹−a⁻¹)I.

Then, items I to III of Theorem 3.1 hold.

Theorem 3.4

Consider Equation 1 with A = 0. Suppose that

(1) aI ≤ Q ≤ bI;

(2) B^∗B ≤ a(bI − Q); and

(3) .

Then, items I to III of Theorem 3.1 hold.

Theorem 3.5

Consider Equation 1 with B = 0. Suppose that

(1) a⁻¹A^∗A + aI ≤ Q ≤ bI;

(2) A^∗A ≤ a(Q − aI); and

(3) .

Then, items I to III of Theorem 3.1 hold.

Numerical experiments

All programs are written in MATLAB version 7.1.

Example 1

In this example, we consider Equation 1 with

All the hypotheses of Theorem 3.1 are satisfied with a = 5 and b = 14. We consider the sequences {X_n} and {Y_n} defined in item III of Theorem 3.1 with X₀ = aI and Y₀ = bI. For each iteration k, we consider the errors

and

After 23 iterations, we get

with

Example 2

In this example, we consider Equation 1 with

All the hypotheses of Theorem 3.2 are satisfied with a = 0.5 and b = 5. After 20 iterations, we get

with

Example 3

We consider Equation 1 with

In this case, A and B are unitary matrices. All the hypotheses of Theorem 3.3 are satisfied with a = 1.514 and b = 101.5. After 7 iterations, we get

with

Example 4

We consider Equation 1 with

All the hypotheses of Theorem 3.4 are satisfied with a = 3.5 and b = 300. After 3 iterations, we get

with

Example 5

We consider Equation 1 with

All the hypotheses of Theorem 3.5 are satisfied with a = 2 and b = 100. After 10 iterations, we get

with

Fixed point theory on ordered metric spaces can be a useful tool to solve various classes of nonlinear matrix equations. In this work, to solve the nonlinear matrix equation X = Q − A^∗X⁻¹A + B^∗X⁻¹B, we suggested an iterative method based on a coupled fixed point theorem of Bhaskar and Lakshmikantham for mixed monotone mappings. The numerical experiments demonstrated that the proposed method is satisfactory.

The authors declare that they have no competing interests.

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

The second author acknowledges the supports from the National Natural Science Foundation of China (grant no.: 11101100) and the Natural Science Foundation of Guangxi Province (grant no.: 2012GXNSFBA053006).

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