On Ψ−stability of nonlinear Lyapunov matrix differential equations

We prove necessary and sufficient conditions for Ψ−(uniform) stability of the trivial solution of a nonlinear Lyapunov matrix differential equation.


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), which can be seen as a perturbed equation of Z ′ = A(t)Z + ZB(t).
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.

Preliminaries
In this section we present some basic definitions, notations, hypotheses and results which are useful later on.
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 Ax .
It is well-known that |A| = max 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 .
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.
Proof. It is a simple exercise.
We recall that the vectorization operator Vec has the following properties as concerns the calculations (see in [7]): 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.
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 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 : Proof. From the proof of Lemma 2, it results that for every A ∈ M n×n . Setting A = Ψ(t)M(t) for t ≥ 0 and using Lemma 4, the inequality (4) follows immediately. 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 ) = 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 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 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 and respectively. Then, the matrix Z(t) = Y T (t)⊗X(t) is a fundamental matrix for the system 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 b). The trivial solution of (2) is Ψ−uniformly stable on R + if and only if there exists a positive constant K such that 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 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) (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).
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) (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: 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].
Consider the equation (2) with The matrix B has the eigenvalues λ 1 = −1, λ 2 = −2 and the Jordan canonical The fundamental matrices for the equations (5) and (6) are For t ≥ s ≥ 0, we have It follows that 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 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 Then, the linear Lyapunov matrix differential equation is also Ψ−uniformly stable on R + . Proof. The corresponding Kronecker product system associated with (10) can be seen in the form 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.
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 T 1 (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]).

Remark. Theorem 3 is no longer true if we require that
This is shown by the next Example, adapted from Example 3.6 ( [4]).
From the Theorem 1, it follows that the equation (2) is Ψ−uniformly stable on R + .
In addition, we have that On the other hand, the solutions of the equation (10) are 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 2). A 1 (t) and B 1 (t) are continuous n×n matrices on R + such that 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]).
From Theorem 1, it follows that the equation (2) is Ψ−uniformly stable on R + . Now, we consider the equation (10) with 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 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

Remark. Theorem 4 generalizes Theorem 3.7 ([4]). 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 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 < +∞.
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.
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 + . 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.