Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 54, 1-18; http://www.math.u-szeged.hu/ejqtde/

On Ψ−stability of nonlinear Lyapunov matrix differential equations

Aurel Diamandescu University of Craiova

Abstract

We prove necessary and sufficient conditions for Ψ−(uniform) stability of the trivial solution of a nonlinear Lyapunov matrix differential equation. Key Words: Ψ−stability, Ψ−uniform stability, Lyapunov matrix differential equation. 2000 Mathematics Subject Classification: 34D20, 34D05.

1 Introduction

Recently, in [7] were studied the problems of Ψ−boundedness and Ψ−stability of the solutions of the corresponding Kronecker product system (3) associated with (1) in case F(t,X) = F(t) (i.e. a linear nonhomogeneous differential system of the form x′ = G(t)x + f(t)). But the obtained results in [7] are particular cases of our general results stated in [3], [4]. Indeed, if in Theorems 2.1 and 2.2 ([3]), the fundamental matrix Y is replaced with the fundamental matrix Z⊗Y of the linear system (5), Theorems 1 and 2 ([7]) concerning Ψ−boundedness follow. Similarly, if in Theorems 3.1 and 3.3 ([4]), the fundamental matrix Y is replaced with the fundamental matrix Z⊗Y of the linear system (5), Theorems 3 and 4 ([7]) concerning Ψ−stability follow. In addition, in all results from [7] there are a few mistakes in connection with the matrix Ψ.

In our paper [5] it is proved a necessary and sufficient condition so that the linear nonhomogeneous Lyapunov matrix differential equation Z′ = A(t)Z + ZB(t) + F(t) have at least one Ψ−bounded solution on ℝ_{+} for every Lebesgue Ψ−integrable matrix-valued function F on ℝ_{+}.

The purpose of present paper is to prove (necessary and) sufficient conditions for Ψ−(uniform) stability of trivial solution of nonlinear Lyapunov matrix differential equation

Z′ = A(t)Z + ZB(t) + F(t,Z),  (1)

which can be seen as a perturbed equation of

Z′ = A(t)Z + ZB(t).  (2)

We investigate conditions on the fundamental matrices of the systems X′ = A(t)X and Y′ = YB(t) and on the function F under which the trivial solutions of the equations (1) and (2) are Ψ−(uniformly) stable on ℝ_{+}. Here, Ψ is a matrix function whose introduction permits us obtaining a mixed asymptotic behavior for the components of solutions.

The main tool used in this paper is the technique of Kronecker product of matrices which has been successfully applied in various fields of matrix theory.

2 Preliminaries

In this section we present some basic definitions, notations, hypotheses and results which are useful later on.

Let ℝ^{n} be the Euclidean n - space. For x = (x_{1}, x_{2}, x_{3}, ...,x_{n})^{T} ∈ ℝ^{n}, let ∥x∥ = max{|x_{1}|, |x_{2}|, |x_{3}|, ..., |x_{n}|} be the norm of x ( ^{T} denotes transpose).

Let 𝕄_{m×n} be the linear space of all m×n real valued matrices. For a matrix A = (a_{ij}) ∈ 𝕄_{m×n}, we define the norm |A| by |A| = sup_{∥x∥≤1} ∥Ax∥ .

It is well-known that |A| = max_{1≤i≤n}{∑_{j=1}^{n} |a_{ij}|}.

Def inition 1. ([1]) Let A = (a_{ij}) ∈ 𝕄_{m×n} and B = (b_{ij}) ∈ 𝕄_{p×q}. The Kronecker product of A and B, written A⊗B, is defined to be the partitioned matrix

A ⊗ B = ( \begin{matrix}a_{11}B & a_{12}B & ⋯ & a_{1n}B \\ a_{21}B & a_{22}B & ⋯ & a_{2n}B \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m1}B & a_{m2}B & ⋯ & a_{mn}B \end{matrix} )

Obviously, A⊗B ∈ 𝕄_{mp×nq}.

Lemma 1. The Kronecker product has the following properties and rules, provided that the dimension of the matrices are such that the various expressions exist:

1). A⊗(B⊗C) = (A⊗B)⊗C; 2). (A⊗B)^{T} = A^{T}⊗B^{T}; 3). (A⊗B)·(C⊗D) = (A·C)⊗(B·D); 4). (A⊗B)^{−1} = A^{−1}⊗B^{−1}; 5). A⊗(B + C) = A⊗B + A⊗C; 6). (A + B)⊗C = A⊗C + B⊗C;

7). I_{p}⊗A = ( \begin{matrix}A & O & ⋯ & O \\ O & A & ⋯ & O \\ ⋮ & ⋮ & ⋮ & ⋮ \\ O & O & ⋯ & A \end{matrix} ); 8). (A(t)⊗B(t))′ = A′(t)⊗B(t) + A(t)⊗B′(t); (here, ′ denotes the derivative \frac{d}{dt}). Proof. See in [1].

Def inition 2. The application Vec : 𝕄_{m×n} ⟶ ℝ^{mn}, defined by

Vec(A) = (a_{11},a_{21}, ⋯, a_{m1},a_{12},a_{22}, ⋯, a_{m2}, ⋯, a_{1n},a_{2n}, ⋯, a_{mn})^{T},

where A = (a_{ij}) ∈ 𝕄_{m×n}, is called the vectorization operator.

Lemma 2. ([5]) The vectorization operator Vec : 𝕄_{n×n} ⟶ ℝ^{n^{2}}, is a linear and one-to-one operator. In addition, Vec and Vec^{−1} are continuous operators.

Lemma 3. A function F : ℝ_{+} ⟶ 𝕄_{n×n} is a continuous (differentiable) matrix function on ℝ_{+} if and only if the function f = Vec(F) : ℝ_{+} ⟶ ℝ^{n^{2}} is a continuous (differentiable) vector function on ℝ_{+}.

Proof. It is a simple exercise.

We recall that the vectorization operator Vec has the following properties as concerns the calculations (see in [7]):

Lemma 4. If A, B, M ∈ 𝕄_{n×n}, then 1). Vec(AMB) = (B^{T}⊗A)·Vec(M); 2). Vec(MB) = (B^{T}⊗I_{n})·Vec(M); 3). Vec(AM) = (I_{n}⊗A)·Vec(M); 4). Vec(AM) = (M^{T}⊗A)·Vec(I_{n}). Proof. It is a simple exercise.

In the systems (1) and (2) we assume that A and B are continuous n×n matrices on ℝ_{+} = [0,∞) and F : ℝ_{+} × 𝕄_{n×n} ⟶ 𝕄_{n×n} is a continuous n×n matrix such that F(t,O_{n}) = O_{n} (null matrix of order n×n).

By a solution of the equation (1) we mean a continuous differentiable n×n matrix function satisfying the equation (1) for all t ≥ 0.

Let Ψ_{i} : ℝ_{+} ⟶ (0,∞), i = 1,2,...,n, be continuous functions and

Ψ = diag [Ψ_{1}, Ψ_{2}, ⋯ Ψ_{n}].

Def inition 3. ([4]). The trivial solution of the vector differential equation x′ = f(t,x) (where x ∈ ℝ^{n} and f is a continuous n vector function) is said to be Ψ−stable on ℝ_{+} if for every ε > 0 and every t_{0} ∈ ℝ_{+}, there exists δ = δ(ε,t_{0}) > 0 such that any solution x(t) of the equation which satisfies the inequality ∥ Ψ(t_{0})x(t_{0})∥ < δ, exists and satisfies the inequality ∥ Ψ(t)x(t)∥ < ε for all t ≥ t_{0}.

The trivial solution of the vector differential equation x′ = f(t,x) is said to be Ψ−uniformly stable on ℝ_{+} if it is Ψ−stable on ℝ_{+} and the above δ is independent of t_{0}.

Extend this definition for a matrix differential equation.

Def inition 4. The trivial solution of the matrix differential equation X′ = F(t,X) (where X ∈ 𝕄_{n×n} and F is a continuous n×n matrix function) is said to be Ψ−stable on ℝ_{+} if for every ε > 0 and every t_{0} ∈ ℝ_{+}, there exists δ = δ(ε,t_{0}) > 0 such that any solution X(t) of the equation which satisfies the inequality | Ψ(t_{0})X(t_{0})| < δ, exists and satisfies the inequality | Ψ(t)X(t)| < ε for all t ≥ t_{0}.

The trivial solution of the matrix differential equation X′ = F(t,X) is said to be Ψ−uniformly stable on ℝ_{+} if it is Ψ−stable on ℝ_{+} and the above δ is independent of t_{0}.

The following lemmas play a vital role in the proofs of main results.

The first result is done in [7]. Because the proof is incomplete, we present it with a complete proof.

Lemma 5. The matrix function Z(t) is a solution of (1) on the interval J ⊂ ℝ_{+} if and only if the vector valued function z(t) = VecZ(t) is a solution of the differential system

z′ = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n}) z + f(t,z)  (3)

where f(t,z) = VecF(t,Z), on the same interval J.

Proof. It is similar with the proof of Lemma 7, [5].

Def inition 5. The above system (3) is called ”corresponding Kronecker product system associated with (1)”.

Lemma 6. For every matrix function M : ℝ_{+} ⟶ 𝕄_{n×n}, we have

\frac{1}{n} | Ψ(t)M(t) | ≤ ∥ (I_{n} ⊗ Ψ(t)) VecM(t) ∥_{ℝ^{^{n}^{2}}} ≤ | Ψ(t)M(t) | , t ≥ 0. (4)

Proof. From the proof of Lemma 2, it results that

\frac{1}{n} | A | ≤ ∥ VecA ∥_{ℝ^{^{n}^{2}}} ≤ | A | ,

for every A ∈ 𝕄_{n×n}.

Setting A = Ψ(t)M(t) for t ≥ 0 and using Lemma 4, the inequality (4) follows immediately.

Lemma 7. The trivial solution of the equation (1) is Ψ−(uniformly) stable on ℝ_{+} if and only if the trivial solution of the corresponding Kronecker product system (3) is I_{n} ⊗ Ψ−(uniformly) stable on ℝ_{+}.

Proof. First, suppose that the trivial solution of the equation (1) is Ψ−stable on ℝ_{+}. According to Definition 4, for a given ε > 0 and t_{0} ∈ ℝ_{+}, we choose δ_{0}(ε,t_{0}) = \frac{1}{n}δ(ε,t_{0}). Let z(t) be a solution of (3) such that ∥ (I_{n} ⊗ Ψ(t_{0}))z(t_{0})∥_{ℝ^{^{n}^{2}}} < δ_{0}(ε,t_{0}). From Lemma 5 and Lemma 6, Z(t) = Vec^{−1}z(t) is a solution on ℝ_{+} of the equation (1) such that | Ψ(t_{0})Z(t_{0})| < δ(ε,t_{0}). It follows that | Ψ(t)Z(t)| < ε for all t ≥ t_{0}.

From Lemma 6, we have that ∥ (I_{n} ⊗ Ψ(t))z(t)∥_{ℝ^{^{n}^{2}}} < ε for all t ≥ t_{0}. Thus, the trivial solution of the system (3) is I_{n} ⊗ Ψ−stable on ℝ_{+}.

Suppose, conversely, that the trivial solution of the system (3) is I_{n} ⊗ Ψ−stable on ℝ_{+}. According to Definition 3, for a given ε > 0 and t_{0} ∈ ℝ_{+}, we choose δ_{0}(ε,t_{0}) = δ(\frac{ε}{n},t_{0}). Let Z(t) be a solution of (1) such that | Ψ(t_{0})Z(t_{0})| < δ_{0}. From Lemma 5 and Lemma 6, z(t) = VecZ(t) is a solution on ℝ_{+} of the system (3) such that ∥ (I_{n} ⊗ Ψ(t_{0}))z(t_{0})∥_{ℝ^{^{n}^{2}}} < δ_{0}. It follows that ∥ (I_{n} ⊗ Ψ(t))z(t)∥_{ℝ^{^{n}^{2}}} < \frac{ε}{n} for all t ≥ t_{0}. From Lemma 6, we have that | Ψ(t)Z(t)| < ε for all t ≥ t_{0}. Thus, the trivial solution of the equation (1) is Ψ−stable on ℝ_{+}.

Since the proof of the Ψ−uniform stability is similar to the above proof, we omit it here.

The proof is now complete.

The next result is Lemma 1 of [7]. Because the proof is incomplete, we present it with a complete proof.

Lemma 8. Let X(t) and Y(t) be a fundamental matrices for the equations

X′ = A(t)X  (5)

and

Y′ = YB(t)  (6)

respectively.

Then, the matrix Z(t) = Y^{T}(t)⊗X(t) is a fundamental matrix for the system

z′ = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n}) z  (7)

Proof. See Lemma 6, [5].

3 Ψ− stability of the linear Lyapunov matrix differential equations

The purpose of this section is to study conditions for Ψ−(uniform) stability of trivial solution of linear Lyapunov matrix differential equation (2). These conditions can be expressed in terms of a fundamental matrices for the equations (5) and (6).

Theorem 1. Let X(t) and Y(t) be a fundamental matrices for the equations (5) and (6) respectively. Then,

a). The trivial solution of (2) is Ψ−stable on ℝ_{+} if and only if there exists a positive constant K such that

| Y^{T}(t) ⊗ (Ψ(t)X(t)) | ≤ K, for all t ∈ ℝ_{+}.  (8)

b). The trivial solution of (2) is Ψ−uniformly stable on ℝ_{+} if and only if there exists a positive constant K such that

| Y^{T}(t)(Y^{T}(s))^{−1} ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ≤ K, for all t ≥ s ≥ 0. (9)

Proof. a). Suppose that the trivial solution of (2) is Ψ−stable on ℝ_{+}. From Lemma 7, it follows that the trivial solution of (7) is I_{n} ⊗ Ψ−stable on ℝ_{+}. From Theorem 3.1 ([4]), it follows that the fundamental matrix Z(t) of (7) satisfies the condition

| (I_{n} ⊗ Ψ(t))Z(t) | ≤ K, for all t ∈ ℝ_{+},

where K is a positive constant.

From Lemma 8, we replace Z(t) = Y^{T}(t)⊗X(t) as a fundamental matrix on ℝ_{+} for the system (7). After computation, it follows that (8) holds.

Now, suppose that (8) holds for some K > 0 and for all t ∈ ℝ_{+}. From Lemma 8 and Theorem 3.1 ([4]), it follows that the trivial solution of (7) is I_{n} ⊗ Ψ−stable on ℝ_{+}. From Lemma 7, it follows that the trivial solution of (2) is Ψ−stable on ℝ_{+}.

b). Since the proof for the Ψ−uniform stability is similar to the above proof, we omit it here.

The proof is now complete.

Remark. 1. In the same manner as in classical stability, we can speak about Ψ−

(uniform) stability of the linear differential equation (2).

2. It is easy to see that if | Ψ(t) | and | Ψ^{−1}(t) | are bounded on ℝ_{+}, then the Ψ−(uniform) stability of the linear equation (2) is equivalent with the classical (uniform) stability of the linear equation (2).

Remark. Theorem 1 generalizes Theorem 3.1 ([4]).

Indeed, in particular case B(t) = O_{n}, we have Y = I_{n} and then Z(t) = I_{n}⊗X(t). On the other hand, it is easy to see that the solutions of (2) are Z(t) = X(t)C, C being a n×n constant matrix. Now, the conditions (8) and (9) become the conditions from Theorem 3.1 ([4]) concerning Ψ−(uniform) stability of the linear system x′ = A(t)x, because | I_{n}⊗M | = |M|

Corollary. If the equation (5) is Ψ−(uniformly) stable on ℝ_{+} and the equation (6) is (uniformly) stable on ℝ_{+}, then the equation (2) is Ψ−(uniformly) stable on ℝ_{+}.

Proof. It results from the above Theorem 1 and from the inequality | A⊗B | ≤ |A||B|, for all A, B ∈ 𝕄_{n×n}.

Sufficient conditions for the Ψ−(uniform) stability of trivial solution of the equation (2) are given by the next theorems.

Theorem 2. Let X(t) and Y(t) be a fundamental matrices for the equations (5) and (6) respectively. Suppose that there exist a continuous function φ : ℝ_{+} ⟶ (0,∞) and the constants p ≥ 1 and M > 0 which fulfil one of the following conditions:

(i). ∫ _{0}^{t} φ(s) | (Y^{T}(t)(Y^{T}(s))^{−1})⊗(Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) |^{p} ds ≤ M, for all t ≥ 0,

(ii). ∫ _{0}^{t} φ(s) | ((Y^{T}(s))^{−1}Y^{T}(t))⊗(X^{−1}(s)Ψ^{−1}(s)Ψ(t)X(t)) |^{p} ds ≤ M,for all t ≥ 0.

Then, the trivial solution of the equation (2) is Ψ−stable on ℝ_{+}.

Proof. From the hypotheses, Lemma 8 and Theorem 3.3 ([4]), it follows that the system (7) is I_{n} ⊗ Ψ−stable on ℝ_{+}. From this and Lemma 7, it follows that the equation (2) is Ψ−stable on ℝ_{+}.

Remark. 1. The function φ can serve to weaken the required hypotheses on the fundamental metrices X and Y.

2. In the conditions of Theorem 2, the equation (2) can not be Ψ−uniformly stable on ℝ_{+}. This is shown in the next example, adapted from J. L. Massera and J. J. Schäffer [6].

Example. Let a(t) be a real, continuously differentiable function, equal to 1 except in the intervals J_{n} = [n − 2^{−4n},n + 2^{−4n}], n = 1, 2,...; in J_{n}, a(t) lies between 1 and 4^{n} and a(n) = 4^{n}.

Consider the equation (2) with

A(t) = ( \begin{matrix}−(3 + \frac{a′(t)}{a(t)}) & 0 \\ 0 & −5 \end{matrix} ) , B(t) = ( \begin{matrix}4 & 3 \\ −10 & −7 \end{matrix} ) .

The matrix B has the eigenvalues λ_{1} = −1, λ_{2} = −2 and the Jordan canonical

form L = diag [−1, −2]. We have B^{T} = ULU^{−1}, where U = ( \begin{matrix}2 & 5 \\ 1 & 3 \end{matrix} ) .

The fundamental matrices for the equations (5) and (6) are

X(t) = ( \begin{matrix}\frac{e^{−3t}}{a(t)} & 0 \\ 0 & e^{−5t} \end{matrix} )

and

Y^{T}(t) = Ue^{Lt}U^{−1} = ( \begin{matrix}6e^{−t} − 5e^{−2t} & 10e^{−2t} − 10e^{−t} \\ 3e^{−t} − 3e^{−2t} & 6e^{−2t} − 5e^{−t} \end{matrix} )

respectively.

Consider

Ψ(t) = ( \begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix} ) .

For t ≥ s ≥ 0, we have

Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s) = ( \begin{matrix}\frac{a(s)}{a(t)}e^{−2(t − s)} & 0 \\ 0 & e^{−6(t − s)} \end{matrix} )

and

Y^{T}(t)(Y^{T}(s))^{−1} = ( \begin{matrix}6e^{−(t − s)} − 5e^{−2(t − s)} & 10e^{−2(t − s)} − 10e^{−(t − s)} \\ 3e^{−(t − s)} − 3e^{−2(t − s)} & 6e^{−2(t − s)} − 5e^{−(t − s)} \end{matrix} ) .

It follows that

| (Y^{T}(t)(Y^{T}(s))^{−1}) ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ≤

≤ 31 max \{\frac{a(s)}{a(t)}e^{−3(t − s)}, e^{−7(t − s)}\} , t ≥ s ≥ 0,

and then

∫ _{0}^{t} | (Y^{T}(t)(Y^{T}(s))^{−1}) ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ds ≤ 31, for all t ≥ 0.

Thus, the condition (i) of Theorem 2 is satisfied with φ = 1, p = 1, M = 31. It follows that the equation (2) is Ψ−stable on ℝ_{+}. On the other hand, for s = n, t = n + 2^{−4n} we have

| (Y^{T}(t)(Y^{T}(s))^{−1}) ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ≥
≥ |(6e^{−2^{−4n}} − 5e^{−2^{−4n + 1}}) 4^{n}e^{−2^{−4n + 1}} | ⟶ ∞.

From Theorem 1, it follows that the equation (2) is not Ψ−uniformly stable on ℝ_{+}.

Remark. Theorem 2 generalizes Theorem 3.3 ([4]).

Theorem 3. Suppose that the linear equation (2) is Ψ−uniformly stable on ℝ_{+} and A_{1}, B_{1} are continuous n×n matrices on ℝ_{+} such that

∫ _{0}^{∞} | I_{n} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{n} | dt < + ∞.

Then, the linear Lyapunov matrix differential equation

Z′ = (A(t) + A_{1}(t))Z + Z(B(t) + B_{1}(t))  (10)

is also Ψ−uniformly stable on ℝ_{+}.

Proof. The corresponding Kronecker product system associated with (10) can be seen in the form

z′ = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n}) z + (I_{n} ⊗ A_{1}(t) + B_{1}^{T}(t) ⊗ I_{n}) z,  (11)

i.e. as a perturbed system of the corresponding Kronecker product system associated with (2),

z′ = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n}) z.  (12)

From the hypothesis and Lemma 7, this system is I_{n} ⊗ Ψ−uniformly stable on ℝ_{+}.

On the other hand, from hypothesis, the matrix I_{n}⊗A_{1}(t) + B_{1}^{T}(t)⊗I_{n} satisfies the conditions of Theorem 3.4 ([4]).

Now, from this Theorem it follows that the system (11) is I_{n} ⊗ Ψ−uniformly stable on ℝ_{+}.

From Lemma 7 again, it follows that the system (10) is Ψ−uniformly stable on ℝ_{+}.

The proof is now complete.

Remark. If the linear equation (2) is only Ψ−stable on ℝ_{+}, then the linear equation (10) can not be Ψ−stable on ℝ_{+}. This is shown by the next Example, adapted from an example due to O. Perron [8] and Example 3.5 ([4]).

Example. Consider the equation (10) with

A(t) = ( \begin{matrix}−1 & 0 \\ 0 & sin ln(t + 1) + cos ln(t + 1) − \frac{3}{2} \end{matrix} ) , A_{1}(t) = ( \begin{matrix}0 & 0 \\ e^{− \frac{1}{2} (t + 1)} & 0 \end{matrix} ) ,

B(t) = \frac{1}{2}I_{2} and B_{1}(t) = O_{2}.

Consider Ψ(t) = ( \begin{matrix}e^{^{\frac{1}{2}} (t + 1)} & 0 \\ 0 & 1 \end{matrix} ) .

The equation (2) becomes

Z′ = ( \begin{matrix}−\frac{1}{2} & 0 \\ 0 & sin ln(t + 1) + cos ln(t + 1) − 1 \end{matrix} ) Z.

From the Example 3.5, ([4]), it follows that the equation (2) is Ψ−stable on ℝ_{+}, but it is not Ψ−uniformly stable on ℝ_{+}. In addition, the condition

∫ _{0}^{∞} | I_{2} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{n} | dt < + ∞

holds.

Finally, the perturbed equation (10) is not Ψ−stable on ℝ_{+} (see the Example 3.5, ([4])).

Remark. Theorem 3 is no longer true if we require that

lim_{t⟶∞} | I_{n} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{n} | = 0,

instead of the condition

∫ _{0}^{∞} | I_{n} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{n} | dt < + ∞.

This is shown by the next Example, adapted from Example 3.6 ([4]).

Example. Consider the equation (10) with

A(t) = ( \begin{matrix}0 & 1 \\ −1 & −\frac{2}{t + 1} \end{matrix} ), A_{1}(t) = ( \begin{matrix}−\frac{1}{t + 1} & 0 \\ 0 & \frac{1}{t + 1} \end{matrix} ) ,

B(t) = −\frac{1}{t + 1}I_{2} and B_{1}(t) = \frac{2}{t + 1}I_{2}.

Consider Ψ(t) = ( \begin{matrix}t + 1 & 0 \\ 0 & t + 1 \end{matrix} ) .

For the corresponding equation (2), we have that

X(t) = ( \frac{(t + 1) cos (t + 1) − sin (t + 1)}{(t + 1)^{2}} −\frac{(t + 1) sin (t + 1) + cos (t + 1)}{(t + 1)^{2}}^{\frac{sin (t + 1)}{t + 1}  \frac{cos (t + 1)}{t + 1}} )

and Y(t) = \frac{1}{t + 1}I_{2} are the fundamental matrices for the systems (5) and (6) respectively.

From the Theorem 1, it follows that the equation (2) is Ψ−uniformly stable on ℝ_{+}.

In addition, we have that

∫ _{0}^{∞} | I_{2} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{2} | dt = + ∞

and

lim_{t⟶∞} | I_{2} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{2} | = 0.

On the other hand, the solutions of the equation (10) are

Z(t) = ( \begin{matrix}sin t & cos t \\ cos t & − sin t \end{matrix} ) · C,

where C is a 2×2 constant matrix.

It is easy to see that the equation (10) is not Ψ−stable on ℝ_{+}.

Remark. Theorem 3 generalizes Theorem 3.4 ([4]).

Theorem 4. Suppose that: 1). There exist a continuous function φ : ℝ_{+} ⟶ (0,∞) and a positive constant

M such that the fundamental matrices X and Y for the equations (5) and (6) respectively satisfy the condition

∫ _{0}^{t} φ(s) | (Y^{T}(t)(Y^{T}(s))^{−1}) ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ds ≤ M, for all t ≥ 0;

2). A_{1}(t) and B_{1}(t) are continuous n×n matrices on ℝ_{+} such that

sup_{t≥0} \frac{1}{φ(t)} | I_{n} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{n} | < \frac{1}{M}.

Then, the linear Lyapunov matrix differential equation (10) is Ψ−stable on ℝ_{+}. Proof. From the hypothesis 1) and Theorem 2, it follows that the equation (2)

is Ψ−stable on ℝ_{+}. Thereafter, from Lemma 7, it follows that the equation (12) is I_{n} ⊗ Ψ−stable on ℝ_{+}. From Theorem 3.7 ([4]), it follows that the system (11) is I_{n} ⊗ Ψ−stable on ℝ_{+}. From Lemma 7 again, the equation (10) is Ψ−stable on ℝ_{+}.

The proof is now complete.

Remark. Theorem 4 is no longer true if we require that the equation (2) be Ψ−(uniformly) stable on ℝ_{+}, instead of the condition 1) of Theorem. This is shown by the next simple example, adapted from Example 3.8 ([4]).

Example. Consider the equation (2) with A(t) = B(t) = O_{2}. Then, a fundamental matrices for the equations (5) and (6) are X(t) = I_{2}, Y(t) = I_{2} respectively.

Consider Ψ(t) = ( \begin{matrix}1 & 0 \\ 0 & \frac{1}{t + 1} \end{matrix} ) .

Therefore,

(Y^{T}(t)(Y^{T}(s))^{−1}) ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & \frac{s + 1}{t + 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{s + 1}{t + 1} \end{matrix} )

is bounded for 0 ≤ s ≤ t < + ∞.

From Theorem 1, it follows that the equation (2) is Ψ−uniformly stable on ℝ_{+}. Now, we consider the equation (10) with

A_{1}(t) = ( \begin{matrix}0 & 0 \\ 0 & \frac{a}{\sqrt{t + 1}} \end{matrix} ) and B_{1}(t) = O_{2},

a being a positive constant.

It is easy to see (by reduction to absurdity) there is no function φ which satisfies the conditions of Theorem 4.

The solutions of the equation (10) are

Z(t) = ( \begin{matrix}1 & 0 \\ 0 & e^{2a^{\sqrt{t + 1}}} \end{matrix} ) · C,

where C is a 2×2 constant matrix.

It is easy to see that the trivial solution of (10) is not Ψ−stable on ℝ_{+}. Finally, we have

sup_{t≥0} | I_{2} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{2} | = a

and

lim_{t⟶∞} | I_{2} ⊗ (Ψ(t)A_{1}(t)Ψ^{−1}(t)) + B_{1}^{T}(t) ⊗ I_{2} | = 0.

Remark. Theorem 4 generalizes Theorem 3.7 ([4]).

4 Ψ− stability of the nonlinear Lyapunov matrix differential equations

The purpose of this section is to study the Ψ−(uniform) stability of trivial solution of the equation (1), where A, B and F are matrix functions. It will be assumed that A and B are continuous for t ∈ ℝ_{+} and that F is continuous for t ∈ ℝ_{+} and Z ∈ 𝕄_{n×n}. This will ensure the local existence of a solution passing through any given point (t_{0},Z_{0}) of the domain of definition of F, but it does not guarantee that the solution is unique or that it can be continued for all large values of t.

Thus, we state the following hypothesis:

(H) For all t_{0} ∈ ℝ_{+} and Z_{0} ∈ 𝕄_{n×n}, then there exists a unique solution Z(t) of the equation (1) such that Z(t_{0}) = Z_{0}.

Theorem 5. Suppose that:

a). the hypothesis (H) is satisfied;

b). the trivial solution of (2) is Ψ−uniformly stable on ℝ_{+};

c). the matrix function F satisfies the inequality

| Ψ(t)F(t,Z) | ≤ γ(t) | Ψ(t)Z |,

for all t ∈ ℝ_{+} and for all Z ∈ 𝕄_{n×n}, where γ is a continuous nonnegative function

on ℝ_{+} such that L = ∫ _{0}^{∞} γ(t)dt < +∞.

Then, the trivial solution of the equation (1) is Ψ−uniformly stable on ℝ_{+}. Proof. From Lemma 5, it follows that if Z(t) is the unique solution of (1) with

Z(t_{0}) = Z_{0}, then, z(t) = VecZ(t) is the unique solution of the system (3) with z(t_{0}) = z_{0} = VecZ_{0}. Therefore, z(t) is also a solution of the inhomogeneous linear system

z′ = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n}) z + f(t,z(t)), t ∈ [t_{0},t_{1}),  (13)

[t_{0},t_{1}) being the existence interval of solution z(t).

If U(t) is a fundamental matrix for the system (7), i.e. the homogeneous system associated with (13), by the variation of constant formula ([2], Chapter II, section 2(8)),

z(t) = U(t)U^{−1}(t_{0})z_{0} + ∫ _{t_{0}}^{t} U(t)U^{−1}(s)f(s,z(s))ds, t ∈ [t_{0},t_{1}).

From Lemma 8, we replace U(t) = Y^{T}(t)⊗X(t), X(t) and Y(t) being fundamental matrices for the equations (5) and (6) respectively. After computation, it follows that

z(t) = ((Y^{T}(t)(Y^{T}(t_{0}))^{−1}) ⊗ (X(t)X^{−1}(t_{0})))z_{0} +

+ ∫ _{t_{0}}^{t} ((Y^{T}(t)(Y^{T}(s))^{−1}) ⊗ (X(t)X^{−1}(s)))f(s,z(s))ds, t ∈ [t_{0},t_{1}). (14)

From hypothesis b) and Theorem 1, it follows that there exists a positive constant K such that

| Y^{T}(t)(Y^{T}(s))^{−1} ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ≤ K,

for all t ≥ s ≥ 0.

From (14), it follows that

(I_{n}⊗Ψ(t))z(t) = ((Y^{T}(t)(Y^{T}(t_{0}))^{−1})⊗(Ψ(t)X(t)X^{−1}(t_{0})Ψ^{−1}(t_{0})))(I_{n}⊗Ψ(t_{0}))z_{0}+

+∫ _{t_{0}}^{t}((Y^{T}(t)(Y^{T}(s))^{−1})⊗(Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)))(I_{n}⊗Ψ(s))f(s,z(s))ds, t ∈ [t_{0},t_{1})

and then,

∥ (I_{n} ⊗ Ψ(t))z(t) ∥_{ℝ^{^{n}^{2}}} ≤ K∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} +
+ K∫ _{t_{0}}^{t} ∥ (I_{n} ⊗ Ψ(s))f(s,z(s)) ∥_{ℝ^{^{n}^{2}}}ds, t ∈ [t_{0},t_{1}).

From hypothesis c) and Lemma 6, it follows that

∥ (I_{n} ⊗ Ψ(t))f(t,z) ∥_{ℝ^{^{n}^{2}}} = ∥ (I_{n} ⊗ Ψ(t))VecF(t,Z) ∥_{ℝ^{^{n}^{2}}} ≤
≤ | Ψ(t)F(t,Z) | ≤ γ(t) | Ψ(t)Z | ≤
≤ nγ(t) ∥ (I_{n} ⊗ Ψ(t))VecZ ∥_{ℝ^{^{n}^{2}}} = nγ(t) ∥ (I_{n} ⊗ Ψ(t))z ∥_{ℝ^{^{n}^{2}}},

for t ∈ ℝ_{+} and z ∈ ℝ^{n^{2}}.

Then,

∥ (I_{n} ⊗ Ψ(t))z(t) ∥_{ℝ^{^{n}^{2}}} ≤ K ∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} +
+ nK ∫ _{t_{0}}^{t} γ(s) ∥ (I_{n} ⊗ Ψ(s))z(s) ∥_{ℝ^{^{n}^{2}}} ds, t ∈ [t_{0},t_{1})

and therefore by Gronwall’s inequality,

∥ (I_{n} ⊗ Ψ(t))z(t) ∥_{ℝ^{^{n}^{2}}} ≤ K ∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} ·e^{nK∫ _{t_{0}}^{t} γ(s)ds} ≤

≤ Ke^{nKL} ∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} , for all t ∈ [t_{0},t_{1}).

This inequality shows that t_{1} = +∞ and hence, the solution z is defined on [t_{0}, +∞).

Therefore, it follows that the trivial solution of the system (13) is I_{n}⊗Ψ−uniformly stable on ℝ_{+}.

From Lemma 7, it follows that the trivial solution of (1) is Ψ−uniformly stable on ℝ_{+}.

The proof is now complete.

Example. Consider the equation (1) with

A(t) = ( \begin{matrix}1 & 1 \\ −1 & 1 \end{matrix} ) , B(t) = ( \begin{matrix}−6 & −3 \\ 10 & 5 \end{matrix} ) and F(t,Z) = ( \begin{matrix}e^{−t}z_{1} & 0 \\ 0 & \frac{sin z_{4}}{1 + t^{2}} \end{matrix} ) .

The matrices A, B^{T} have the eigenvalues λ_{1} = 1 + i, λ_{2} = 1 − i and μ_{1} = −1, μ_{2} = 0 respectively. The fundamental matrices for the systems (5) and (6) are

X(t) = ( \begin{matrix}e^{t} cos t & e^{t} sin t \\ −e^{t} sin t & e^{t} cos t \end{matrix} ) and Y(t) = ( \begin{matrix}2e^{−t} & e^{−t} \\ 5 & 3 \end{matrix} )

respectively.

Consider

Ψ(t) = ( \begin{matrix}e^{−t} & 0 \\ 0 & e^{−t} \end{matrix} ) .

We have

Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s) = ( \begin{matrix}cos (t − s) & sin (t − s) \\ − sin (t − s) & cos (t − s) \end{matrix} )

and

Y^{T}(t)(Y^{T}(s))^{−1} = ( \begin{matrix}6e^{−(t − s)} − 5 & 10 − 10e^{−(t − s)} \\ 3e^{−(t − s)} − 3 & 6 − 5e^{−(t − s)} \end{matrix} ) .

From Corollary of Theorem 1, it follows that the equation (2) is Ψ−uniformly stable on ℝ_{+}.

Further, the function F satisfies a Lipschitz condition and

| Ψ(t)F(t,Z) | = max \{e^{−2t} | z_{1} |, \frac{e^{−t} | sin z_{4} |}{1 + t^{2}}\} ≤ \frac{1}{1 + t^{2}} | Ψ(t)Z |

for all t ∈ ℝ_{+} and Z ∈ 𝕄_{2×2}.

From these, it is easy to see that the function F satisfies all the hypotheses of Theorem 5.

Thus, the trivial solution of the equation (1) is Ψ−uniformly stable on ℝ_{+}.

Theorem 6. Suppose that: a). the hypothesis (H) is satisfied; b). the trivial solution of (5) is Ψ−uniformly stable on ℝ_{+}; c). the matrix function F satisfies the inequality

| Ψ(t)F(t,Z) | ≤ γ(t) | Ψ(t)Z |,

for all t ∈ ℝ_{+} and for all Z ∈ 𝕄_{n×n}, where γ is a continuous nonnegative function

on ℝ_{+} such that ∫ _{0}^{∞} γ(t)dt < +∞;

d). the matrix function B satisfies the condition ∫ _{0}^{∞} |B(t)|dt < +∞.

Then, the trivial solution of the equation (1) is Ψ−uniformly stable on ℝ_{+}. Proof. It is similar to the proof of the above Theorem 5.

Remark. Theorems 5 and 6 generalize Theorem 6 ([2], Chapter III, Section 3).

Theorem 7. Suppose that: a). the hypothesis (H) is satisfied; b). there exist a continuous function φ : ℝ_{+} ⟶ (0,∞) and a positive constant

M such that the fundamental matrices X and Y for the equations (5) and (6) respectively satisfy the condition

∫ _{0}^{t} φ(s) | (Y^{T}(t)(Y^{T}(s))^{−1}) ⊗ (Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ds ≤ M, for all t ≥ 0;

c). the matrix function F satisfies the inequality

| Ψ(t)F(t,Z) | ≤ γ(t) | Ψ(t)Z |,

for all t ∈ ℝ_{+} and for all Z ∈ 𝕄_{n×n}, where γ : ℝ_{+} ⟶ [0,+∞) is a continuous function such that

q = sup_{t≥0} \frac{γ(t)}{φ(t)} < \frac{1}{nM}.

Then, the trivial solution of the system (1) is Ψ−stable on ℝ_{+}. Proof. From Lemma 5, it follows that if Z(t) is the unique solution of (1) with

Z(t_{0}) = Z_{0}, then, z(t) = VecZ(t) is the unique solution of the system (3) with z(t_{0}) = z_{0} = VecZ_{0}. Therefore, z(t) is also a solution of the inhomogeneous linear equation

z′ = (I_{n} ⊗ A(t) + B^{T}(t) ⊗ I_{n}) z + f(t,z(t)), t ∈ [t_{0},t_{1}),

[t_{0},t_{1}) being the existence interval of solution z(t).

If U(t) is a fundamental matrix for the system (7), by the variation of constant formula ([2], Chapter II, section 2(8)),

z(t) = U(t)U^{−1}(t_{0})z_{0} + ∫ _{t_{0}}^{t} U(t)U^{−1}(s)f(s,z(s))ds, t ∈ [t_{0},t_{1}).

From Lemma 8, we replace U(t) = Y^{T}(t)⊗X(t), as a fundamental matrix on ℝ_{+} for the system (7). After computation, it follows that

z(t) = (Y^{T}(t) ⊗ X(t)) · ((Y^{T}(t_{0}))^{−1} ⊗ X^{−1}(t_{0})) · z_{0} +
+ ∫ _{t_{0}}^{t} (Y^{T}(t) ⊗ X(t)) · ((Y^{T}(s))^{−1} ⊗ X^{−1}(s))f(s,z(s))ds, t ∈ [t_{0},t_{1}).  (15)

From hypothesis b) and Theorems 2 and 1, it follows that there exists a positive constant K such that

| Y^{T}(t) ⊗ (Ψ(t)X(t)) | ≤ K,

for all t ≥ 0.

From (15), it follows that

(I_{n} ⊗ Ψ(t))z(t)=(Y^{T}(t)⊗Ψ(t)X(t))((Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}))(I_{n} ⊗ Ψ(t_{0}))z_{0}+

+∫ _{t_{0}}^{t}((Y^{T}(t)(Y^{T}(s))^{−1})⊗(Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)))(I_{n}⊗Ψ(s))f(s,z(s))ds, t ∈ [t_{0},t_{1})

and then,

∥ (I_{n} ⊗ Ψ(t))z(t) ∥_{ℝ^{^{n}^{2}}} ≤ K| (Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}) |∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} +

+ ∫ _{t_{0}}^{t} φ(s)| (Y^{T}(t)(Y^{T}(s))^{−1})⊗(Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ·
·\frac{1}{φ(s)} ∥ (I_{n} ⊗ Ψ(s))f(s,z(s)) ∥_{ℝ^{^{n}^{2}}}ds, t ∈ [t_{0},t_{1}).

From hypothesis c) and Lemma 6, it follows that

∥ (I_{n} ⊗ Ψ(t))f(t,z) ∥_{ℝ^{^{n}^{2}}} ≤ nγ(t) ∥ (I_{n} ⊗ Ψ(t))z ∥_{ℝ^{^{n}^{2}}},

for t ∈ ℝ_{+} and z ∈ ℝ^{n^{2}}. Then, for t ∈ [t_{0},t_{1}) we have

∥ (I_{n} ⊗ Ψ(t))z(t) ∥_{ℝ^{^{n}^{2}}} ≤ K| (Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}) |∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} +

+ ∫ _{t_{0}}^{t} φ(s)| (Y^{T}(t)(Y^{T}(s))^{−1})⊗(Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ·

·\frac{nγ(s)}{φ(s)} ∥ (I_{n} ⊗ Ψ(s))z(s) ∥_{ℝ^{^{n}^{2}}}ds ≤
≤ K| (Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}) |∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} +
+ ∫ _{t_{0}}^{t} φ(s)| (Y^{T}(t)(Y^{T}(s))^{−1})⊗(Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s)) | ·

·\frac{nγ(s)}{φ(s)} sup_{t_{0}≤s≤t} ∥ (I_{n} ⊗ Ψ(s))z(s) ∥_{ℝ^{^{n}^{2}}}ds ≤
≤ K| (Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}) |∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} +
+ nqM sup_{t_{0}≤s≤t} ∥ (I_{n} ⊗ Ψ(s))z(s) ∥_{ℝ^{^{n}^{2}}}

and hence

sup_{t_{0}≤s≤t} ∥ (I_{n} ⊗ Ψ(s))z(s) ∥_{ℝ^{^{n}^{2}}} ≤
≤ (1 − nqM)^{−1}K| (Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}) |∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} .

Thus, for t ∈ [t_{0},t_{1}) we have

∥ (I_{n} ⊗ Ψ(t))z(t) ∥_{ℝ^{^{n}^{2}}} ≤
≤ (1 − nqM)^{−1}K| (Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}) |∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}} .

This inequality shows that t_{1} = +∞ and hence, the solution z is defined on [t_{0},+∞).

From the inequality

∥ (I_{n} ⊗ Ψ(t))z(t) ∥_{ℝ^{^{n}^{2}}} ≤

≤ (1 − nqM)^{−1}K| (Y^{T}(t_{0}))^{−1}⊗X^{−1}(t_{0})Ψ^{−1}(t_{0}) |∥ (I_{n} ⊗ Ψ(t_{0}))z_{0} ∥_{ℝ^{^{n}^{2}}}, t ≥ t_{0},

it follows that the trivial solution of the system (3) is I_{n} ⊗ Ψ−stable on ℝ_{+}.

From Lemma 7, it follows that the trivial solution of the equation (1) is Ψ−stable on ℝ_{+}.

The proof is now complete.

Theorem 8. Suppose that: a). the hypothesis (H) is satisfied; b). there exist a continuous function φ : ℝ_{+} ⟶ (0,∞) and a positive constant

M such that the fundamental matrix X of the system (5) satisfies the inequality

∫ _{0}^{t} φ(s) | Ψ(t)X(t)X^{−1}(s)Ψ^{−1}(s) | ds ≤ M,

for all t≥0; c). the matrix function F satisfies the inequality

| Ψ(t)F(t,Z) | ≤ γ(t) | Ψ(t)Z |,

for all t ∈ ℝ_{+} and for all Z ∈ 𝕄_{n×n}, where γ : ℝ_{+} ⟶ [0,+∞) is a continuous function such that

q = sup_{t≥0} \frac{| B(t) | + γ(t)}{φ(t)} < \frac{1}{nM}.

Then, the trivial solution of the equation (1) is Ψ−stable on ℝ_{+}. Proof. It is similar to the proof of the above Theorem 7.

Acknowledgments. The author would like to thank the anonymous referees for careful reading, valuable comments, suggestions and advice for improvement of previous version of this paper.

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(Received June 14, 2009)

Aurel Diamandescu University of Craiova, Department of Applied Mathematics, 13, ”Al. I. Cuza” st., 200585 Craiova, Romania. E - mail address: adiamandescu@central.ucv.ro