On the Ψ − uniform asymptotic stability of nonlinear Lyapunov matrix differential equations

. This paper deals with obtaining (necessary and) su ﬃ cient conditions for Ψ − uniform asymptotic stability of solutions of nonlinear Lyapunov matrix di ﬀ erential equations.


Introduction
The Lyapunov matrix differential equations occur in many branches of applied matematics.
The purpose of this paper is to provide sufficient conditions for Ψ−uniform asymptotic stability of trivial solution of nonlinear Lyapunov matrix differential equations of the form Z = A(t)Z + F(t, Z) (1) Z = A(t)Z + ZB(t) + t 0 G(t, s, Z(s))ds as a perturbed equations of linear matrix differential equations We investigate conditions on the fundamental matrices of the equations (4), (5) and (6) and on the coefficients of equations under which the trivial solutions of the equations (1) - (6) are Ψ−uniformly asymptotically stable on R + .
Here, Ψ is a matrix function whose introduction permits us obtaining a mixed asymptotic behavior for the components of solutions.
The results obtained in this work generalize results from works [3], [4], [5], [7]. The main tools used in this paper are the variation of constants formula, Gronwall's inequality and the technique of Kronecker product of matrices, which have been successfully applied in various fields of matrix theory.

Preliminaries
In this section we present some basic notations, definitions, hypotheses and results which are useful later on.
Let M n×n be the linear space of all real n × n matrices. For A = (a i j ) ∈ M n×n , we define the norm | A | by formula | A | = sup x ≤1 Ax . It is By a solution of the equation (1) (or (2) -(6)) we mean a continuous differentiable n × n matrix function Z(t) satisfying the equation (1) (or (2) -(6) respectively) for all t ∈ R + = [0, ∞).
In equations (4) -(6), we assume that A and B are continuous n×n matrix functions on R + . It is well-known that these conditions ensure the existence and uniqueness of the solutions of these equations passing through any given point (t 0 , Z 0 ) ∈ R + × M n×n and is defined on R + .
In addition, for equations (1) -(3), we assume that F : R + × M n×n −→ M n×n is continuous matrix function such that F(t, O n ) = O n and G : D −→ M n×n , D = {(t, s, Z) | 0 ≤ s ≤ t < +∞, Z ∈ M n×n }, is continuous matrix function such that G(t, s, O n ) = O n . It is well-known that these conditions ensure the local existence of a solution of these equations passing through any given point (t 0 , Z 0 ) ∈ R + × M n×n , but it does not guarantee that the solution is unique or that it can be continued for large values of t ∈ R + .
(H 3 ) For all t 0 > 0, Z 0 ∈ M n×n and ρ > 0, if | Ψ(t 0 )Z 0 |< ρ, there exists a unique solution Z(t) on R + of the equation (3) which satisfies the equality Z(t 0 ) = Z 0 and the inequality | Ψ(t)Z(t) |≤ ρ for all t ∈ [0, t 0 ]. Now, we recall the definitions of various types of Ψ− stability that we need in what follows. In this Definition, the equation Z = F(t, Z) is a general matrix differential equation.
The trivial solution of the equation Z = F(t, Z) is said to be Ψ−stable over R + if for each ε > 0 and each t 0 ∈ R + , there is a a corresponding δ = δ(ε, t 0 ) > 0 such that any solution Z(t) of equation which satisfies the inequality | Ψ(t 0 )Z(t 0 ) |< δ, exists and satisfies the inequality | Ψ(t)Z(t) |< ε for all t ≥ t 0 .
(ii) The trivial solution of the equation Z = F(t, Z) is said to be Ψ−uniformly stable over R + if it is Ψ−stable over R + and the above δ is independent of t 0 .
Remark 2.1 1. Our Definition extends the definition of ( uniform asymptotic, exponential) stability from (vector) differential equations to matrix differential equations. 2. For Ψ = I d , one obtain the notion of classical stability (see [3], [4]). 3. It is easy to see that if Ψ and Ψ −1 are bounded on R + , then the Ψ− stability is equivalent with the classical stability.
We now give some definitions and properties in connection with Kronecker product of matrices and vectorization operator.
The Kronecker product has the following properties and rules, provided that the dimension of the matrices are such that the various expressions exist:  11 , a 21 , · · · , a m1 , a 12 , a 22 , · · · , a m2 , · · · , a 1n , a 2n , · · · , a mn ) T , where A = (a i j ) ∈ M m×n , is called the vectorization operator.

Lemma 2.2 The vectorization operator
is a linear and one-to-one operator. In addition, Vec and Vec −1 are continuous operators.
Proof. See Lemma 2, [6]. We recall that the vectorization operator Vec has the following properties as concerns the calculations. The Lemmas which follows are one of the most useful technical results in the proofs of main results of present paper.

Lemma 2.4 ([6])
The matrix function Z(t) is a solution on R + of (2) if and only if the vector function z(t) = Vec(Z(t)) is a solution of the differential system where f (t, z) = Vec (F(t, Z)) , on the same interval R + .
The above system (7) is called "corresponding Kronecker product system associated with (2)".

Lemma 2.7 ([6]). Let X(t) and Y(t) be a fundamental matrices for the equations
respectively. Then, the matrix Z(t) = Y T (t) ⊗ X(t) is a fundamental matrix for the linear differential system (i.e. for homogeneous differential system associated with (7)).
At the end of this section, we recall a result which is useful in the proof of one of our main result.
where K and λ are positive constants; 3). the vector function f (t, s, x) satisfies the condition Then, the trivial solution of equation s, x(s))ds is Ψ−uniformly asymptotically stable on R + .

Ψ−uniform asymptotic stability of the linear (Lyapunov) matrix differential equations
The purpose of this section is to provide (necessary and) sufficient conditions for Ψ−uniform asymptotic stability of linear (Lyapunov) matrix differential equations (4) − (6).
Theorem 3.1 Let X(t) be a fundamental matrix for equation (4). Then, the trivial solution of equation (4) is Ψ−uniformly asymptotically stable on R + if and only if there exist the constants K > 0 and α > 0 such that Proof. The solution of equation (4) which takes the value C 0 ∈ M n×n at t 0 ≥ 0 is Suppose first that the trivial solution of equation (4) is Ψ−uniformly asymptotically stable on R + . From Definition 2.1, this solution is Ψ−uniformly stable on R + and there is a δ 0 > 0, and for each ε ∈ (0, δ 0 /2), there exists T = T (ε) > 0 such that any solution Z(t) of (4) which Therefore, Then, for t 0 = τ ≥ 0 and t = τ + T (ε) we have Since the trivial solution of (4) or, so much the more, Now, for t ≥ s ≥ 0, there exists n ∈ N such that s + nT (ε) ≤ t < s + (n + 1)T (ε). Then, we can write (to simplify the writing, we put T instead of T (ε)) Since n ≤ t−s T < n + 1, it follows that n ln θ ≥ t−s T ln θ > (n + 1) ln θ and then n ln θ <  (4) there exist the constants K > 0 and α > 0 such that From Definition 2.1 we see that the trivial solution of (4) is Ψ−uniformly stable on R + . Now, for δ 0 = 1 and T (ε) = −α −1 ln ε K (with ε ∈ (0, K) ), for any solution Z(t) of (4) which satisfies the inequality | Ψ(t 0 )Z(t 0 ) Hence, the trivial solution of (4) is Ψ−uniformly asymptotically stable on R + . Remark 3.1 1. In the same manner as in classical stability for systems of linear differential equations, we can speak about Ψ−uniform asymptotic stability of a linear matrix differential equation (4). 2. In the same manner as in classical stability for systems of linear differential equations, it is easy to see that for linear matrix equation (4), Ψ−uniform asymptotic stability and Ψ−exponential stability are equivalent.
Theorem 3.1 contains as a particular case a result concerning Ψ−uniform asymptotic stability of systems of differential equations of the form z = A(t)z.

Indeed, consider in equation (4)
where z = (z 1 , z 2 , ...z n ) T . Now, the definitions and conditions for Ψ−uniform asymptotic stability on R + for solutions of this system and (4) are the same. Thus, we have Corollary 3.1 (Theorem 1, [5]) Let X(t) be a fundamental matrix for system z = A(t)z. Then, the trivial solution of the system above is Ψ−uniformly asymptotically stable on R + if and only if there exist the constants K > 0 and α > 0 such that The next result shows that Ψ−uniform asymptotic stability of (4) is preserved under small or absolutely integrable perturbations of the coefficient matrix A(t).
Proposition 3.1 Suppose that: 1). A(t) is a continuous matrix function on R + and the matrix differential equation (4) is Ψ−uniformly asymptotically stable on R + . 2). A 1 (t) is a continuous matrix function on R + such that satisfies one of the following conditions: Then, the linear matrix differential equation Proof. It is similar with the proof of the Theorem 4.1 below, in particular case F(t, Z) = A 1 (t)Z. Corollary 3.2 Suppose that: 1). For matrix function Ψ, there exist the constants K > 0 and α > 0 such that 2). A(t) is a continuous matrix function on R + such that satisfies one of the following conditions: Then, the linear matrix differential equation (4) is Ψ−uniformly asymptotically stable on R + .
Proof. Indeed, this follows from the Proposition 3.1.
The next result is similar. (4) is Ψ−uniformly asymptotically stable on R + .   Then, the linear matrix differential equation (6) is Ψ−uniformly asymptotically stable on R + .

Proposition 3.2 Suppose that: 1). A(t) is a continuous matrix function on R + and the matrix differential equation
Proof. It is similar with the proof of the Theorem 4.1 below, in particular case F(t, Z) = ZB(t). (4) is Ψ−uniformly asymptotically stable on R + , then the linear Lyapunov matrix differential equation (6) is Ψ−uniformly asymptotically stable on R + .

Remark 3.3 Proposition may be interpreted as saying that if the matrix differential equation
2). B(t) is a continuous matrix function on R + such that satisfies one of the following conditions: Then, the linear matrix differential equation (5) is Ψ−uniformly asymptotically stable on R + .
Proof. Indeed, this follows from the Proposition 3.2.
Theorem 3.2 Let X(t) and Y(t) be a fundamental matrices for the equations (4) and (5) respectively. Then, the linear Lyapunov matrix differential equation (6) is Ψ−uniformly asymptotically stable on R + if and only if there exist K > 0 and α > 0 such that Proof. From Lemma 2.6 one know that the equation (6) is Ψ−uniformly asymptotically stable on R + if and only if the corresponding Kronecker product system associated with equation (6), i.e. the system is I ⊗ Ψ−uniformly asymptotically stable on R + . From Lemma 2.7 one know that the matrix U(t) = Y T (t) ⊗ X(t) is a fundamental matrix for the linear differential system (12). A short computation shows that, for t ≥ s ≥ 0, Now, from Theorem 3.1, we have that the system (12) is I ⊗ Ψ−uniformly asymptotically stable on R + . Once again, from Lemma 2.6, we have that the equation (6) is Ψ−uniformly asymptotically stable on R + .

Corollary 3.4 Let Y(t) be a fundamental matrix for the equation (5).
Then, the linear Lyapunov matrix differential equation (5) is Ψ−uniformly asymptotically stable on R + if and only if there exist K > 0 and α > 0 such that Proof. Indeed, this follows from the Theorem 3.2.
Corollary 3.5 Suppose that is satisfied one of the following conditions: i). the equation (4) is Ψ−uniformly asymptotically stable on R + and the equation (5) is uniformly stable on R + ; ii). the equation (4) is Ψ−uniformly stable on R + and the equation (5) is uniformly asymptotically stable on R + ; Then, the equation (6)

Ψ−uniform asymptotic stability of a nonlinear Lyapunov matrix differential equation
The purpose of this section is to provide sufficient conditions for Ψ−uniform asymptotic stability of trivial solution of nonlinear Lyapunov matrix differential equations (1) -(3).
Theorem 4.1 Suppose that: 1). The hypothesis (H 1 ) is satisfied; 2). The equation (4) is Ψ−uniformly asymptotically stable on R + ; 3). The matrix function F : R + × M n×n → M n×n satisfies the inequality for all t ∈ R + and Z ∈ M n×n , where γ : R + → R + is a continuous function that satisfies one of the following conditions: Then, the trivial solution of the nonlinear matrix differential equation (1) is Ψ−uniformly asymptotically stable on R + .
Proof. Let X(t) be a fundamental matrix for linear equation (4). From hypothesis 2) and Theorem 3.1, there exist K > 0 and α > 0 such that If Z(t) is the solution of (1) with Z(t 0 ) = Z 0 , by variation of constants formula (see [3], Ch II, s 2(8)), and then, from hypothesis 3), Thus, the scalar function w(t) = e α(t−t 0 ) | Ψ (t) Z(t) | satisfies the inequality By Gronwall's inequality (see [3], Ch I, Lemma 3), this implies In what follows, we have three cases: In the case i), from (13) one obtain In the case ii), from (13) one obtain In the case iii), there exist the constants N > 0 and t * > t 0 such that The theorem contains as a particular case a result concerning Ψ−uniform asymptotic stability of systems of differential equations of the form as follows (See Corollary 3.1): Theorem 4.2 Suppose that: 1). The hypothesis (H 1 ) adapted for systems of differential equations (16) is satisfied; 2). The equation z = A(t)z is Ψ−uniformly asymptotically stable on R + ; 3). The matrix function f : R + × R n → R n satisfies the inequality for all t ∈ R + and z ∈ R n , where γ : R + → R + is a continuous function that satisfies one of the following conditions: Then, the trivial solution of the nonlinear system of differential equations (16) is Ψ−uniformly asymptotically stable on R + .
for all t ∈ R + and Z ∈ M n×n , where γ : R + → R + is a continuous function that satisfies one of the following conditions: Then, the trivial solution of the nonlinear matrix differential equation (2) is Ψ−uniformly asymptotically stable on R + .
Theorem 4.4 Suppose that: 1). The hypothesis (H 2 ) is satisfied; 2). The Lyapunov matrix differential equation (6) is Ψ−uniformly asymptotically stable on R + ; 3). The matrix function F : R + × M n×n → M n×n satisfies the inequality for all t ∈ R + and Z ∈ M n×n , where γ : R + → R + is a continuous function that satisfies one of the following conditions: i). M = sup t≥0 γ(t) is a sufficiently small number; Then, the trivial solution of the nonlinear Lyapunov matrix differential equation (2) is Ψ−uniformly asymptotically stable on R + .
Proof. From Lemma 2.6 one know that the trivial solution of the nonlinear Lyapunov matrix differential equation (2) is Ψ−uniformly asymptotically stable on R + if and only if the trivial solution of corresponding Kronecker product system associated with equation (2), i.e. the system is I ⊗ Ψ−uniformly asymptotically stable on R + , where z = Vec(Z) and f (t, z) = Vec (F(t, Z)) . Using Theorem 4.2, we will show that the trivial solution of the differential system (17) Thus, the hypothesis (3) of Theorem 4.2 is satisfied. Now, from Theorem 4.2, the trivial solution of the system (17) is I ⊗ Ψ−uniformly asymptotically stable on R + . From Lemma 2.6, it results that the trivial solution of the nonlinear Lyapunov matrix differential equation (2) is Ψ−uniformly asymptotically stable on R + .
The following Example illustrates the Theorem.
The fundamental matrices for the equations (4) and (5) are We have where τ = t − s and λ = e −2s −e −4t 2 . It follows that | Ω(t, s) |≤ 2e −(t−s) , for t ≥ s ≥ 0. From this and Theorem 3.2 it follows that the Lyapunov matrix differential equation (6) is Ψ−uniformly asymptotically stable on R + . Further, the matrix function F(t, Z) satisfies the inequality | Ψ(t)F(t, Z) |≤ γ(t) | Ψ(t)Z | for all t ∈ R + and Z ∈ M 2×2 , where γ : R + → R + is the continuous function that satisfies the condition L = ∞ 0 γ(t)dt < +∞. From these, it is easy to see that the function F satisfies all the hypotheses of Theorem 4.4. It is easy to see that the hypothesis (H 2 ) is satisfied; Thus, the trivial solution of the nonlinear Lyapunov matrix differential equation (2) considered is Ψ−uniformly asymptotically stable on R + . Theorem 4.5 Suppose that: 1). The hypothesis (H 2 ) is satisfied; 2). The equation (5) is Ψ−uniformly asymptotically stable on R + ; 3). The matrix function A : R + → M n×n is continuous on R + ; 4). The matrix function F : R + × M n×n → M n×n satisfies the inequality for all t ∈ R + and Z ∈ M n×n , where γ : R + → R + is a continuous function that satisfies one of the following conditions: i). M = sup t≥0 γ(t)+ | Ψ(t)A(t)Ψ −1 (t) | is a sufficiently small number; Then, the trivial solution of the nonlinear Lyapunov matrix differential equation (2) is Ψ−uniformly asymptotically stable on R + .
Proof. We see the nonlinear Lyapunov matrix differential equation (2) in the form Z = ZB(t) + A(t)Z + +F(t, Z) and the proof goes through as for above Theorem. Then, the trivial solution of the nonlinear Lyapunov matrix differential equation (2) is Ψ−uniformly asymptotically stable on R + .
Proof. The proof is similar in spirit to that of above Theorems.
Proof. From Lemma 2.6 we know that the trivial solution of equation (3) is Ψ−uniformly asymptotically stable on R + if and only if the trivial solution of corresponding Kronecker product system associated with equation (3), i.e. the system z = I ⊗ A(t) + B T (t) ⊗ I z + Thus, the hypothesis (2) of Lemma 2.8 is satisfied, with I ⊗ Ψ in role of Ψ and U(t) in role of X.
Thus, the hypothesis (3) of Lemma 2.8 is satisfied, with I ⊗ Ψ in role of Ψ, γ(t, s, z) in role of f (t, s, x) and ng in role of k, etc. Thus, applying the Lemma 2.8, it follows that the trivial solution of the differential system (18) is I ⊗ Ψ−uniformly asymptotically stable on R + . Now, from Lemma 2.6, we have that the trivial solution of the nonlinear Lyapunov matrix differential equation (3) is Ψ−uniformly asymptotically stable on R + . Remark 4.3 Theorem 4.7 generalizes Theorem 5.1, [7] and Theorem 4, [5].