On the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

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1 DOI: /awutm Analele Universităţii de Vest, Timişoara Seria Matematică Informatică LI, 2, (213), 7 28 On the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations Aurel Diamandescu Abstract. This paper deals with (necessary and) sufficient conditions for Ψ-exponential asymptotic stability of the trivial solution of nonlinear Lyapunov matrix differential equations. AMS Subject Classification (21). 34D2;34D5. Keywords. Ψ - exponential asymptotic stability, Lyapunov matrix differential equation. 1 Introduction The Lyapunov matrix differential equations occur in many branches of applied mathematics. Recent results for Ψ boundedness, Ψ stability, Ψ instability, dichotomy and conditioning for Lyapunov matrix differential equations have been given in many papers. See, for example, [4], [5], [6], [8], [9], [11]. The purpose of present paper is to prove (necessary and) sufficient conditions for Ψ exponential asymptotic stability of trivial solution of the nonlinear Lyapunov matrix differential equation Z = A(t)Z + ZB(t) + F (t, Z) (1)

2 8 An. U.V.T. and the linear Lyapunov matrix differential equation Z = [A(t) + A 1 (t)]z + Z[B(t) + B 1 (t)], (2) which can be seen as a perturbed equations of the linear equation Z = A(t)Z + ZB(t). (3) We investigate conditions on the fundamental matrices of the equations X = A(t)X, (4) Y = Y B(t) (5) and on the functions A 1, B 1 and F under which the trivial solutions of the equations (1), (2) and (3) are Ψ exponentially 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 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, group theory and particle physics. See, for example, the above cited papers and the references cited therein. 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-dimensional 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 n n, we define the norm A by A = sup Ax. It is well-known that A = max { n a ij }. X 1 1 i n j=1 Definition 2.1. ([2]) 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 11 B a 12 B a 1n B a 21 B a 22 B a 2n B A B =..... a m1 B a m2 B a mn B

3 Vol. LI (213) 9 Obviously, A B M mp nq. The important rules of calculation of the Kronecker product there are in Lemma 1, [5]. Definition 2.2. 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. The important properties and rules of calculation of the vectorization operator there are in Lemmas 2 4, [5]. In the equation (1) we assume that A and B are continuous n n matrices on R + = [, ) and F : R + M n n M n n is a continuous n n matrix function 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. Let Ψ i : R + (, ), i = 1, 2,..., n, be continuous functions and Ψ= diag[ψ 1, Ψ 2, Ψ n ]. Now, we recall the definitions of Ψ strong stability and Ψ exponential asymptotic stability for a vector differential equation x = f(t, x), (6) where x R n and f : R + R n R n is a continuous function. Definition 2.3. ([6]) The solution x(t) of (6) is said to be Ψ strongly stable on R + if for each ε >, there is a corresponding δ = δ(ε) > such that any solution x(t) of the equation which satisfies the inequality Ψ(t )( x(t ) x(t )) < δ for some t, exists and satisfies the inequality Ψ(t)( x(t) x(t)) < ε for all t. Definition 2.4. ([7]) The solution x(t) of (6) is said to be Ψ exponentially asymptotically stable on R + if there exists λ > and, for any ε >, there is a δ(ε) > such that any solution x(t) of the equation which satisfies the inequality Ψ(t )( x(t ) x(t )) < δ(ε) for some t, exists and satisfies the inequality Ψ(t)( x(t) x(t)) < εe λ(t t ) for all t t. Remark 2.1. The definitions of various types of Ψ stability on R + for a vector differential equation (6) was given in [5], [6], [7]. These definitions generalize the classical definitions of various types of stability (see [3], [12]).

4 1 An. U.V.T. Remark 2.2. In the same manner as in classical stability, we can speak about Ψ strong stability or Ψ exponential asymptotic stability on R + of a linear vector differential equation x = A(t)x (see [3], [12]). Now, we extend these definitions for a matrix differential equation X = F (t, X) (7) where X M n n and F : R + M n n M n n is a continuous function. Definition 2.5. ([6]) The solution X(t) of (7) is said to be Ψ strongly stable on R + if for each ε > there is a corresponding δ = δ(ε) > such that any solution X(t) of the equation which satisfies the inequality Ψ(t )( X(t ) X(t )) < δ for some t, exists and satisfies the inequality Ψ(t)( X(t) X(t)) < ε for all t Definition 2.6. The solution X(t) of (7) is said to be Ψ exponentially asymptotically stable on R + if there exists a positive number λ and, for any ε >, there is a δ(ε) > such that any solution X(t) of the equation which satisfies the inequality Ψ(t )( X(t ) X(t )) < δ(ε) for some t, exists and satisfies the inequality Ψ(t)( X(t) X(t)) < εe λ(t t ) for all t t. Remark 2.3. The definitions of various types of Ψ stability on R + for a matrix differential equation (7) was given in [5], [6]. Remark 2.4. In the same manner as in classical stability, we can speak about Ψ strong stability or Ψ exponential asymptotic stability on R + of a linear Lyapunov matrix differential equation (3). The following lemmas play a vital role in the proofs of main results of present paper. Lemma 2.1. The trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R + if and only if the trivial solution of the system ) z = (I n A(t) + B T (t) I n z + f(t, z) (8) where f(t, z) = Vec(F (t, Z)), is I n Ψ exponentially asymptotically stable on R +. Proof. It is similar with the proof of Lemma 2.7, [6]. Definition 2.7. The above system (8) is called corresponding Kronecker product system associated with (1).

5 Vol. LI (213) 11 We will use Lemma 5, [5], Lemma 7, [4], and the inequality 1 n Ψ(t)M(t) (I n Ψ(t)) Vec(M(t)) R n 2 Ψ(t)M(t), t. (9) for every matrix function M : R + M n n (see Lemma 6, [5]). Lemma 2.2. ([5]) Let X(t) and Y(t) be a fundamental matrices for the equations (4) and (5) respectively. Then, the matrix Z(t) = Y T (t) X(t) is a fundamental matrix for the corresponding Kronecker product system associated with (3), i.e. for the differential system ) Z = (I n A(t) + B T (t) I n Z (1) Proof. See Lemma 8, [5]. Remark 2.5. The above result is Lemma 1.1, [11]. Because the proof is incomplete, we presented it with a complete proof. 3 Ψ exponential asymptotic stability of the linear Lyapunov matrix differential equations The purpose of this section is to study conditions for Ψ exponential asymptotic stability of the linear Lyapunov matrix differential equations (2) and (3) (and (4) and (5), as a particular cases). These conditions can be expressed in terms of a fundamental matrices for the equations (4) and (5). Necessary and sufficient conditions for Ψ exponential asymptotic stability of equations (3) and (4) and are given by the next theorems. Theorem 3.1. Let U(t) be a fundamental matrix for (4). Statements (i), (ii) and (iii) are equivalent: (i) The trivial solution of (4) is Ψ exponentially asymptotically stable on R + ; (ii) The trivial solution of the differential system x = A(t)x is Ψ exponentially asymptotically stable on R + ; (iii) There exist the positive constants K and λ such that Ψ(t)U(t)U 1 (s)ψ 1 (s) Ke λ(t s), for all t s. (11)

6 12 An. U.V.T. Proof. At first, we show that statements (i) and (iii) are equivalent. Let X(t) = U(t)U 1 (t )X(t ), t, be any solution of (4). Suppose that the trivial solution of (4) is Ψ exponentially asymptotically stable on R +. From Definition 2.6, there exists the constants λ > and δ 1 > such that any solution X(t) of (4) which satisfies the inequality Ψ(t )X(t ) < δ 1 for some t, exists and satisfies the inequality or Ψ(t)X(t) < e λ(t t ), for all t t, (Ψ(t)U(t)U 1 (t )Ψ 1 (t ))(Ψ(t )X(t )) < e λ(t t ), for all t t. Let W M n n be such that W 1. If we take X(t ) = 1 2 δ 1Ψ 1 (t )W, we have Ψ(t )X(t ) < δ 1 and then Therefore, 1 2 δ 1(Ψ(t)U(t)U 1 (t )Ψ 1 (t ))W < e λ(t t ), for all t t. Ψ(t)U(t)U 1 (t )Ψ 1 (t ) 2 δ 1 e λ(t t ), for all t t. Thus, (11) holds with K = 2 δ 1. Now, suppose that (11) holds, K and λ being positive constants. For any solution X(t) of (4) which satisfies the inequality Ψ(t )X(t ) < ε for some t K, we have Ψ(t)X(t) = Ψ(t)U(t)U 1 (t )X(t ) = = (Ψ(t)U(t)U 1 (t )Ψ 1 (t ))(Ψ(t )X(t )) Ψ(t)U(t)U 1 (t )Ψ 1 (t ) Ψ(t )X(t ) εe λ(t t ), for all t t. Hence, the trivial solution of (4) is Ψ exponentially asymptotically stable on R +. Similarly, by using the form of any solution of the system x = A(t)x, i.e. x(t) = U(t)U 1 (t )x(t ), t, we can show that the statements (ii) and (iii) are equivalent. The proof is now complete. Theorem 3.2. Let X(t) and Y (t) be a fundamental matrices for the equations (4) and (5) respectively. Then, the equation (3) is Ψ exponentially asymptotically stable on R + if and only if there exist the positive constants K and λ such that (Y T (t)(y T ) 1 (s)) (Ψ(t)X(t)X 1 (s)ψ 1 (s)) Ke λ(t s), for all t s.

7 Vol. LI (213) 13 Proof. From Lemma 2.1, we know that the equation (3) is Ψ exponentially asymptotically stable on R + if and only if the corresponding Kronecker product system associated with (3), i.e. the system (1), is I n Ψ exponentially asymptotically stable on R +. From Theorem 3.1, we know that the system (1) is I n Ψ exponentially asymptotically stable on R + if and only if for the fundamental matrix U(t) = Y T (t) X(t) for (1) (see Lemma 2.2) there exist two positive constants K and λ such that (I n Ψ(t))(Y T (t) X(t))(Y T (s) X(s)) 1 (I n Ψ(s)) 1 Ke λ(t s), for all t s. After computation (see Lemma 1, [5]), we obtain the desired result. The proof is now complete. Remark 3.1. Theorems 3.1 and 3.2 generalize a result in connection with the classical exponential asymptotic stability (see [3], Chapter III, Section 2, Theorem 1). Example 1. Consider the linear matrix differential equation X = AX where A is an n n real constant matrix. It is well-known (see [3], page 56) that if every characteristic root of matrix A has real part less than α, there exists a constant K > such that the fundamental matrix X(t) = e ta for equation is such that X(t)X 1 (s) Ke α(t s), for all t s. We note that if α <, then the equation is exponentially asymptotically stable on R + (see [3], Chapter III, Section 2, Theorem 2). Further on, consider the linear matrix differential equation Y = Y B(t) where B(t) is an n n continuous periodic matrix function of period τ. From the Floquet s Theorem (see [3], Chapter II, page 47), it follows that a fundamental matrix Y (t) for the above periodic linear equation can be represented in the form Y (t) = P (t)e tl, where L is a constant matrix and P (t) is a periodic matrix with period τ.

8 14 An. U.V.T. It is well-known (see [3], page 57) that if every characteristic root of matrix L has real part less than β, there exists a constant M > such that the fundamental matrix Y (t) for the above equation is such that Y T (t)(y T ) 1 (s) Me β(t s), for all t s. We note that if β <, then the equation is exponentially asymptotically stable on R + (see [3], page 57 and Theorem 1, page 54). Now, let λ > α + β be and Ψ(t) = e λt I n. Then, we have (Y T (t)(y T ) 1 (s)) (Ψ(t)X(t)X 1 (s)ψ 1 (s)) KMe (λ α β)(t s), for all t s. Therefore, the linear Lyapunov matrix differential equation Z = AZ + ZB(t) is Ψ exponentially asymptotically stable on R +. Corollary 3.3. Suppose that is satisfied one of the following conditions: i). the equation (4) is Ψ exponentially 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 exponentially asymptotically stable on R +. Then, the equation (3) is Ψ exponentially asymptotically stable on R +. Proof. It results from the above Theorem 3.1 and Theorem 3.2, from Theorem 1, [4] and from the inequality A B A B, A, B M n n. Sufficient conditions for Ψ exponential asymptotic stability of equations (3) and (4) are given by the next theorems. Theorem 3.4. Suppose that the fundamental matrix U(t) of (4) satisfies one of the following conditions: i). there exists M 1 > such that, for all t, Ψ(t)U(t)U 1 (s)ψ 1 (s) ds M 1, U 1 (t)ψ 1 (t)ψ(s)u(s) ds M 1. ii). there exists M 2 > such that, for all t, Ψ(t)U(t)U 1 (s)ψ 1 (s) ds M 2, Ψ(s)U(s)U 1 (t)ψ 1 (t) ds M 2.

9 Vol. LI (213) 15 iii). there exists M 3 > such that, for all t, U 1 (t)ψ 1 (t)ψ(s)u(s) ds M 3, U 1 (s)ψ 1 (s)ψ(t)u(t) ds M 3. iv). there exists M 4 > such that, for all t, U 1 (s)ψ 1 (s)ψ(t)u(t) ds M 4 Ψ(s)U(s)U 1 (t)ψ 1 (t) ds M 4. Then, the equation (4) and the linear differential system x = A(t)x are Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. First of all, we consider the case i. Let h(t) = Ψ(t)U(t) 1 be for t. From the identity ( h(s)ds)ψ(t)u(t) = (Ψ(t)U(t)U 1 (s)ψ 1 (s))(h(s)ψ(s)u(s))ds, t, it follows that ( h(s)ds) Ψ(t)U(t) Ψ(t)U(t)U 1 (s)ψ 1 (s) h(s) Ψ(s)U(s) ds, t, or Ψ(t)U(t) h(s)ds M 1, t. Let H(t) = h(s)ds, t. Then, we have Because we have that Hence, Ψ(t)U(t) M 1 H(t), t >. H (t) = h(t) = 1 Ψ(t)U(t) H(t) M 1, t >, H(t) H(1)e M 1 1 (t 1), for t 1. 1 M Ψ(t)U(t) h 1 e 1 t, for t 1,

10 16 An. U.V.T. where h 1 = M 1 1 em1. H(1) Because Ψ(t)U(t) is a continuous function on R +, it follows that there exists a positive constant h 1 such that Ψ(t)U(t) h 1 M1 1e t, for t. (12) Now, let g(t) = U 1 (t)ψ 1 (t) 1 be for t. From the identity it follows that or ( g(s)ds)u 1 (t)ψ 1 (t) = (U 1 (t)ψ 1 (t)ψ(s)u(s))(g(s)u 1 (s)ψ 1 (s))ds, t, ( g(s)ds) U 1 (t)ψ 1 (t) U 1 (t)ψ 1 (t)ψ(s)u(s) g(s) U 1 (s)ψ 1 (s) ds, t, U 1 (t)ψ 1 (t) g(s)ds M 1, t. Let G(t) = g(s)ds, t. Then, we have Because we have that Hence, U 1 (t)ψ 1 (t) M 1 G(t), t >. G (t) = g(t) = 1 U 1 (t)ψ 1 (t) G(t) M 1, t >, G(t) G(1)e M 1 1 (t 1), for t 1. U 1 (t)ψ 1 1 M (t) g 1 e 1 t, for t 1, where g 1 = M 1 1 em1. G(1) Because U 1 (t)ψ 1 (t) is a continuous function on R +, it follows that there exists a positive constant g 1 such that U 1 (t)ψ 1 (t) g 1e 1 M1 t, for t. (13)

11 Vol. LI (213) 17 From (12) and (13), we have Ψ(t)U(t)U 1 (s)ψ 1 1 M (s) K 1 e 1 t s, for all t, s, where K 1 = h 1g 1 >. From this and from Theorem 3.1 (this paper) and Theorem 3.1 [6], one obtain the desired result. Finally, in the cases ii iv, the proof is similarly. The proof is now complete. Theorem 3.5. Suppose that the fundamental matrices X(t) and Y(t) for the equations (4) and (5) respectively satisfy one of the following conditions: i). there exists M 1 > such that, for all t, (Y T (t)(y T ) 1 (s)) (Ψ(t)X(t)X 1 (s)ψ 1 (s)) ds M 1, ((Y T ) 1 (t)y T (s)) (X 1 (t)ψ 1 (t)ψ(s)x(s)) ds M 1 ; ii). there exists M 2 > such that, for all t, (Y T (t)(y T ) 1 (s)) (Ψ(t)X(t)X 1 (s)ψ 1 (s)) ds M 2, (Y T (s)(y T ) 1 (t)) (Ψ(s)X(s)X 1 (t)ψ 1 (t)) ds M 2 ; iii). there exists M 3 > such that, for all t, ((Y T ) 1 (t)y T (s)) (X 1 (t)ψ 1 (t)ψ(s)x(s)) ds M 3, ((Y T ) 1 (s)y T (t)) (X 1 (s)ψ 1 (s)ψ(t)x(t)) ds M 3 ; iv). there exists M 4 > such that, for all t, ((Y T ) 1 (s)y T (t)) (X 1 (s)ψ 1 (s)ψ(t)x(t)) ds M 4, (Y T (s)(y T ) 1 (t)) (Ψ(s)X(s)X 1 (t)ψ 1 (t)) ds M 4. Then, the equation (3) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. From Lemma 2.2, we know that U(t) = Y T (t) X(t) is a fundamental matrix for the corresponding Kronecker product system associated with (3), i.e. for the differential system (1). From hypotheses, it results that U(t) satisfies the conditions of Theorem 3.4, with I n Ψ(t) instead of Ψ(t). It follows that the system (1) is I n Ψ exponentially asymptotically stable on R + and I n Ψ strongly stable on R +. Now, from Lemma 2.1 and Lemma 2.7, [6], it results that the equation (3) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. The proof is now complete.

12 18 An. U.V.T. Remark 3.2. Theorems 3.4 and 3.5 extend a results in connection with the classical asymptotic stability, uniform asymptotic stability and strong stability (see [3], Chapter III, Sections 2, 3, Theorems 1 and 8). Sufficient conditions for Ψ exponential asymptotic stability of equation (2) and its particular form Z = (A(t) + A 1 (t))z are given by the next theorems. Theorem 3.6. Suppose that: 1). The linear equation (4) is Ψ exponentially asymptotically stable on R + ; 2). A 1 (t) is a continuous n n matrix function on R + and satisfies one of the following conditions: i). M = sup t Ψ(t)A 1 (t)ψ 1 (t) is a sufficiently small number; ii). L = Ψ(s)A 1 (s)ψ 1 (s) ds < + ; iii). lim t Ψ(t)A 1 (t)ψ 1 (t) =. Then, the linear system and the linear equation z = (A(t) + A 1 (t))z Z = (A(t) + A 1 (t))z are Ψ exponentially asymptotically stable on R +. Proof. In the case of linear system z = (A(t) + A 1 (t))z, the Theorem is Theorem 3, [7]. In the case of linear equation Z = (A(t) + A 1 (t))z, the proof is similar to the proof of the above mentioned theorem. Corollary 3.7. Suppose that: 1). The linear equation (4) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R + ; 2). A 1 (t) is a continuous n n matrix function on R + and satisfies the condition Then, the linear system Ψ(s)A 1 (s)ψ 1 (s) ds < +. z = (A(t) + A 1 (t))z

13 Vol. LI (213) 19 and the linear equation Z = (A(t) + A 1 (t))z are Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. We apply the above Theorem and Theorem 3.5, [6]. Theorem 3.8. Suppose that: 1). The linear Lyapunov matrix differential equation (3) is Ψ exponentially asymptotically stable on R + ; 2). A 1 (t) and B 1 (t) are continuous n n matrices on R + and satisfy one of the following conditions: i). M = sup I n (Ψ(t)A 1 (t)ψ 1 (t)) + B1 T (t) I n is a sufficiently small t number; ii). L = I n (Ψ(s)A 1 (s)ψ 1 (s)) + B1 T (s) I n ds < + ; iii). lim I n (Ψ(t)A 1 (t)ψ 1 (t)) + B1 T (t) I n =. t Then, the linear Lyapunov matrix differential equation (2) is Ψ exponentially asymptotically stable on R +. Proof. From Lemma 2.1, we know that the equation (2) is Ψ exponentially asymptotically stable on R + if and only if the corresponding Kronecker product system associated with (2), i.e. the system z = (I n (A(t) + A 1 (t)) + (B T (t) + B T 1 (t)) I n)z (14) is I n Ψ exponentially asymptotically stable on R +. The system (14) can be written in the form z = (I n A(t) + B T (t) I n )z + (I n A 1 (t) + B T 1 (t) I n)z, i.e. as a perturbed system of the corresponding Kronecker product system associated with (3), z = (I n A(t) + B T (t) I n )z. (15) From hypothesis 1) and Lemma 2.1 again, the system (15) is I n Ψ exponentially asymptotically stable on R +. In addition, the matrix I n A 1 (t)+ B T 1 (t) I n satisfies the hypothesis 2) of Theorem 3.6. Now, from this Theorem, it follows that the system (14) is I n Ψ exponentially asymptotically stable on R +. The proof is now complete.

14 2 An. U.V.T. Remark 3.3. Theorems 3.6 and 3.8 generalize a result of J. L. Massera [1] and two results in connection with exponential asymptotic stability, [3]. Corollary 3.9. Suppose that: 1). The linear Lyapunov matrix differential equation (3) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R + ; 2). A 1 (t) and B 1 (t) are continuous n n matrices on R + and satisfy the condition I n (Ψ(s)A 1 (s)ψ 1 (s)) + B T 1 (s) I n ds < +. Then, the linear Lyapunov matrix differential equation (2) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. We apply the above Theorem and Theorem 3.6, [6]. Corollary 3.1. Suppose that: 1). The equation (4) is Ψ exponentially asymptotically stable on R + ; 2). The matrix B(t) satisfies one of the following conditions: i). M = sup t B(t) is a sufficiently small number; ii). L = B(s) ds < + ; iii). lim t B(t) =. Then, the linear Lyapunov matrix differential equation (3) is Ψ exponentially asymptotically stable on R +. Proof. In Theorem 3.8, we consider the equation (4) instead of the equation (3) and the equation (3) instead of the equation (2). Corollary Suppose that: 1). The equation (4) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R + ; 2). The matrix B(t) satisfies the condition B(s) ds < +. Then, the linear Lyapunov matrix differential equation (3) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. We apply the above Corollary and Corollary 3.6.1, [6].

15 Vol. LI (213) 21 Corollary Suppose that: 1). The equation (5) is Ψ exponentially asymptotically stable on R + ; 2). The matrix A(t) satisfies one of the following conditions: i). M = sup t Ψ(t)A(t)Ψ 1 (t) is a sufficiently small number; ii). L = Ψ(s)A(s)Ψ 1 (s) ds < + ; iii). lim t Ψ(t)A(t)Ψ 1 (t) =. Then, the linear Lyapunov matrix differential equation (3) is Ψ exponentially asymptotically stable on R +. Proof. In Theorem 3.8, we consider the equation (5) instead of the equation (3) and the equation (3) instead of the equation (2). Corollary Suppose that: 1). The equation (5) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R + ; 2). The matrix A(t) satisfies the condition Ψ(s)A(s)Ψ 1 (s) ds < + ; Then, the linear Lyapunov matrix differential equation (3) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. We apply the above Corollary and Corollary 3.6.2, [6]. Remark 3.4. Corollaries 3.7, 3.9, 3.11, 3.13 extend more results in connection with the classical uniform asymptotic stability and strong stability (see [1], [3], [1]). 4 Ψ exponential asymptotic stability of non-linear Lyapunov matrix differential equations The purpose of this section is to study the Ψ exponential asymptotic stability of trivial solution of the equation (1), where A, B and F are n n 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, Z ) of the domain of definition of F, but it does not guarantie that the solution is unique or that it can be continued for all large values of t. Thus, we state the following hypothesis:

16 22 An. U.V.T. (H): For all t R + and Z M n n, there exists a unique solution Z(t) of the equation (1), such that Z(t ) = Z. Theorem 4.1. Suppose that: 1). The hypothesis (H) is satisfied; 2). The equation (3) is Ψ exponentially asymptotically stable on R + ; 3). The matrix function F : R + M n n satisfies the inequality Ψ(t)F (t, Z) γ(t) Ψ(t)Z 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 = supγ(t) is a sufficiently small number; t ii). L = γ(t)dt < + ; iii). lim γ(t) =. t Then, the trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R + Proof. From Lemma 2.1, we know that the trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R + if and only if the trivial solution of the corresponding Kronecker product system associated with (1), i.e. the system (8), is I n Ψ exponentially asymptotically stable on R +. We will show that the trivial solution af the system (8) is I n Ψ exponentially asymptotically stable on R +. For this, let z(t) be a solution of (8) with the initial condition z(t ) = z R n2, t. From hypothesis 1) and Lemma 5, [5], the solution z(t) is unique and is defined on an interval [t, t + ), t < t + +. If U(t) is a fundamental matrix for the homogeneous system associated with (8), by the variation of constant formula ([3], Chapter II, section 2(8)), z(t) = U(t)U 1 (t )z + t U(t)U 1 (s)f(s, z(s))ds, t [t, t + ). From Lemma 2.2, we replace U(t) = Y T (t) X(t), X(t) and Y (t) being fundamental matrices for the equations (4) and (5) respectively. After computation, it follows that z(t) = (Y T (t)(y T ) 1 (t ) X(t)X 1 (t ))z + + (Y T (t)(y T ) 1 (s) X(t)X 1 (s))f(s, z(s))ds, t [t, t + ). t (16)

17 Vol. LI (213) 23 From hypothesis 2) and Theorem 3.2, it follows that there exist the positive constants K and λ such that, for all t s, (Y T (t)(y T ) 1 (s)) (Ψ(t)X(t)X 1 (s)ψ 1 (s)) Ke λ(t s). (17) From (16) and (17), for t [t, t + ), it follows that (I n Ψ(t))z(t) = (Y T (t)(y T ) 1 (t ) Ψ(t)X(t)X 1 (t )Ψ 1 (t ))(I n Ψ(t ))z + + t (Y T (t)(y T ) 1 (s) Ψ(t)X(t)X 1 (s)ψ 1 (s))(i n Ψ(s))f(s, z(s))ds and then, (I n Ψ(t))z(t) R n 2 Ke λ(t t ) (I n Ψ(t ))z R n 2 + +K t e λ(t s) (I n Ψ(s))f(s, z(s)) R n 2 ds, t [t, t +). (18) From hypothesis 3) and Lemma 6, [5], 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 n2. From this, (18) becomes (I n Ψ(t))z(t) R n 2 Ke λ(t t ) (I n Ψ(t ))z R n 2 + +Kn e λ(t s) γ(s) (I n Ψ(s))z(s) R n 2 ds, t [t, t +). t (19) Thus, the function satisfies the inequality v(t) = (I n Ψ(t))z(t) R n 2 eλ(t t ), t [t, t + ), v(t) K (I n Ψ(t ))z R n 2 +Kn t γ(s)v(s)ds, t [t, t + ). (2) In case i), we have γ(t) M, t and Gronwall s inequality gives v(t) K (I n Ψ(t ))z R n 2 enkm(t t ), t [t, t + ).

18 24 An. U.V.T. It follows that (I n Ψ(t))z(t) R n 2 K (I n Ψ(t ))z R n2 e (λ nkm)(t t ), (21) for all t [t, t + ). We suppose that M < λ. nk The inequality (21) shows that t + = + and hence, the solution z is defined on R +. Thereafter, it follows from (21) that the trivial solution of the equation (8) is I n Ψ exponentially asymptotically stable on R +. In case ii), Gronwall s inequality gives It follows that v(t) K (I n Ψ(t ))z R n 2 enkl, t [t, t + ). (I n Ψ(t))z(t) R n 2 KenKL (I n Ψ(t ))z R n 2 e λ(t t ), t [t, t + ). (22) The inequality (22) shows that t + = + and hence, the solution z is defined on R +. Thereafter, it follows from (22) that the trivial solution of the equation (8) is I n Ψ exponentially asymptotically stable on R +. In case iii), there exists T > t such that γ(t) < λ, for all t T 2nK. Let M > be such that γ(t) < M, for all t. From (2), by Gronwall s inequality, v(t) K (I n Ψ(t ))z R n 2 enk t γ(s)ds, t [t, t + ), and then, (I n Ψ(t))z(t) R n 2 K (I n Ψ(t ))z R n 2 [t, t + ). e λ(t t )+nk t γ(s)ds, t We have the following two cases further: 1). T < t +. j). For t [t, T ], we have λ(t t ) + nk t γ(s)ds λ(t t ) + nkm(t t ) λ 2 (t t ) + nkmt.

19 Vol. LI (213) 25 jj). For t (T, t + ), we have λ(t t ) + nk t γ(s)ds = λ(t t )+ +nk T t γ(s)ds + nk T γ(s)ds λ(t t ) + nkm(t t ) +nk λ (t T 2nK ) λ(t t ) + nkm(t t ) + λ(t t 2 ) = = λ(t t 2 ) + nkm(t t ) λ(t t 2 ) + nkmt. 2). T t +. For t (t, t + ), we have λ(t t ) + nk t γ(s)ds λ(t t ) + nkm(t t ) λ(t t 2 ) + nkmt Thus, from the above results, we obtain, for t [t, t + ), (I n Ψ(t))z(t) R n 2 KenKMT (I n Ψ(t ))z R n 2 e λ 2 (t t ). (23) The inequality (23) shows that t + = + and hence, the solution z is defined on R +. Thereafter, it follows from (23) that the trivial solution of the equation (8) is I n Ψ exponentially asymptotically stable on R +. The proof is now complete. Example 2. Consider the equation (1) with ( ) ( ) A(t) =, B(t) =, ( ) sin 2 z 1 sin z 2 1+t F (t, Z) = 4 (1+t 2 ) 2, (1 cos z 3 ) e 4t e t4 arctg z 4 ( ) z1 z where Z = 2. z 3 z 4 The fundamental matrices for the equations (4) and (5) are ( ) ( cos 2t sin 2t cos t sin t X(t) = e 3t, Y (t) = sin 2t cos 2t sin t cos t ), respectively. Consider Ψ(t) = We have ( e 4t e 4t ). Ψ(t)X(t)X 1 (s)ψ 1 (s) = e (t s) ( cos 2(t s) sin 2(t s) sin 2(t s) cos 2(t s) ) and

20 26 An. U.V.T. Y T (t) ( Y T ) 1 (s) = ( cos(t s) sin(t s) sin(t s) cos(t s) for t s. From the Corollary 3.3, it follows that the equation (3) is Ψ exponentially asymptotically stable on R +. Further, the function F satisfies a Lipschitz condition and Ψ(t)F (t, Z) 1 1+t 4 Ψ(t)Z, for all t R + and for all Z M 2 2. From these, it is easy to see that the function F satisfies all the hypotheses of Theorem 4.1. Thus, the trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R +. Corollary 4.2. Suppose that: 1). The hypothesis (H) is satisfied; 2). The equation (4) is Ψ exponentially asymptotically stable on R + ; 3). The matrix function F : R + M n n satisfies the inequality Ψ(t)F (t, Z) γ(t) Ψ(t)Z 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(γ(t)+ B(t) ) is a sufficiently small number; t ii). L = (γ(t)+ B(t) )dt < + ; iii). lim (γ(t)+ B(t) ) =. t Then, the trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R +. Proof. In Theorem 4.1, we consider the equation (4) instead of the equation (3). Corollary 4.3. Suppose that: 1). The hypothesis (H) is satisfied; 2). The equation (4) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R + ; 3). The matrix function F : R + M n n satisfies the inequality Ψ(t)F (t, Z) γ(t) Ψ(t)Z ),

21 Vol. LI (213) 27 for all t R + and Z M n n, where γ : R + R + is a continuous function that satisfies the condition (γ(t)+ B(t) )dt < +. Then, the trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. We apply the above Corollary and Theorem 4.1, [6]. Corollary 4.4. Suppose that: 1). The hypothesis (H) is satisfied; 2). The equation (5) is Ψ exponentially asymptotically stable on R + ; 3). The matrix function F : R + M n n satisfies the inequality Ψ(t)F (t, Z) γ(t) Ψ(t)Z 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(γ(t)+ Ψ(t)A(t)Ψ 1 (t) ) is a sufficiently small number; t ii). L = (γ(t)+ Ψ(t)A(t)Ψ 1 (t) )dt < + ; iii). lim (γ(t)+ Ψ(t)A(t)Ψ 1 (t) ) =. t Then, the trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R +. Proof. In Theorem 4.1, we consider the equation (5) instead of the equation (3). Remark 4.1. Theorem 4.1 and Corollaries 4.2 and 4.4 generalize Theorem 9, [3], Chapter III, Section 3. Corollary 4.5. Suppose that: 1). The hypothesis (H) is satisfied; 2). The equation (5) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R + ; 3). The matrix function F : R + M n n satisfies the inequality Ψ(t)F (t, Z) γ(t) Ψ(t)Z for all t R + and Z M n n, where γ : R + R + is a continuous function that satisfies the condition (γ(t)+ Ψ(t)A(t)Ψ 1 (t) )dt < +. Then, the trivial solution of the equation (1) is Ψ exponentially asymptotically stable on R + and Ψ strongly stable on R +. Proof. We apply the above Corollary and Theorem 4.1, [6]. Remark 4.2. Corollaries 4.3 and 4.5 extend Theorem 4.1, [6].

22 28 An. U.V.T. References [1] G. Ascoli, Osservazioni sopre alcune questioni di stabilita, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 8 (9), (195), [2] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill Book Company, Inc. New York (translated in Romanian), 196 [3] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Company, Boston, 1965 [4] A. Diamandescu, On Ψ stability of a nonlinear Lyapunov matrix differential equations, Electronic Journal of Qualitative Theory Differential Equations, 54, (29), 1-18 [5] A. Diamandescu, On Ψ asymptotic stability of nonlinear Lyapunov matrix differential equations, Analele Universităţii de Vest, Timişoara, Seria Matematică - Informatică, L (1), (212), 3-25 [6] A. Diamandescu, On the Ψ strong stability of nonlinear Lyapunov matrix differential equations, to appear [7] A. Diamandescu, On the Ψ uniform asymptotic stability of a nonlinear Volterra integro-differential system, Analele Universităţii din Timişoara, Seria Matematică - Informatică, XXXIX (2), (21), [8] A. Diamandescu, On the Ψ instability of nonlinear Lyapunov matrix differential equations, Analele Universităţii de Vest, Timişoara, Seria Matematică - Informatică, XLIX (1), (211), [9] A. Diamandescu, Note on the existence of a Ψ bounded solution for a Lyapunov matrix differential equation, Demonstratio Mathematica, XLV (3), (212), [1] J.L. Massera, Contributions to stability theory, Ann. of Math., 64, (1956), [11] M.S.N. Murty and G. Suresh Kumar, On dichotomy and conditioning for twopoint boundary value problems associated with first order matrix Lyapunov systems, J. Korean Math. Soc. 45, 5, (28), [12] T. Yoshizawa, Stability Theory by Lyapunov s Second Method, The Mathematical Society of Japan, 1966 Aurel Diamandescu University of Craiova, Department of Applied Mathematics, 13, Al. I. Cuza st., 2585 Craiova, Romania. adiamandescu@central.ucv.ro Received: Accepted: