Ψ-asymptotic stability of non-linear matrix Lyapunov systems

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1 Available online at wwwtjnsacom J Nonlinear Sci Appl 5 (22), 5 25 Research Article Ψ-asymptotic stability of non-linear matrix Lyapunov systems MSNMurty a,, GSuresh Kumar b a Department of Applied Mathematics, Acharya Nagarjuna University-Nuzvid Campus, Nuzvid-522, Andhra Pradesh, India b Koneru Lakshmaiah University, Department of Mathematics, Vaddeswaram, Guntur, Andra Prdesh, India This paper is dedicated to Professor Ljubomir Ćirić Communicated by Professor V Berinde Abstract In this paper, first we convert the non-linear matrix Lyapunov system into a Kronecker product matrix system with the help of Kronecker product of matrices Then, we obtain sufficient conditions for Ψ- asymptotic stability and Ψ-uniform stability of the trivial solutions of the corresponding Kronecker product system c 22 NGA All rights reserved Keywords: Matrix Lyapunov system, Kronecker product, Fundamental matrix, Ψ-asymptotic stability, Ψ-(uniform) stability 2 MSC: Primary 34D5; Secondary 49K5, 34C Introduction The importance of Matrix Lyapunov systems, which arise in a number of areas of control engineering problems, dynamical systems, and feedback systems are well known In this paper we focus our attention to the first order non-linear matrix Lyapunov systems of the form X (t) = A(t)X(t) X(t)B(t) F (t, X(t)), () where A(t), B(t) are square matrices of order n, whose elements a ij, b ij, are real valued continuous functions of t on the interval R = [, ), and F (t, X(t)) is a continuous square matrix of order n defined on (R R n n ), such that F (t, O) = O, where R n n denote the space of all n n real valued matrices Corresponding author addresses: drmsn22@gmailcom (MSNMurty ), drgsk6@kluniversityin (GSuresh Kumar ) Received 2-4-

2 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Akinyele [ introduced the notion of Ψ-stability, and this concept was extended to solutions of ordinary differential equations by Constantin [2 Later Morchalo [6 introduced the concepts of Ψ-(uniform) stability, Ψ-asymptotic stability of trivial solutions of linear and non-linear systems of differential equations Further, these concepts are extended to non-linear volterra integro-differential equations by Diamandescu [[3, [4 Recently, Murty and Suresh Kumar [[7, [8 extended the concepts of Ψ-boundedness, Ψ-stability and Ψ- instability to matrix Lyapunov systems The purpose of our paper is to provide sufficient conditions for Ψ-asymptotic and Ψ-uniform stability of trivial solutions of the Kronecker product system associated with the non-linear matrix Lyapunov system () Here, we extend the concept of Ψ-stability in [7 to Ψ-asymptotic stability for matrix Lyapunov systems The paper is well organized as follows In section 2 we present some basic definitions and notations relating to Ψ-(uniform) stability, Ψ-asymptotic stability and Kronecker products First, we convert the nonlinear matrix Lyapunov system () into an equivalent Kronecker product system and obtain its general solution In section 3 we obtain sufficient conditions for Ψ- asymptotic stability of trivial solutions of the corresponding linear Kronecker product system In section 4 we study Ψ-asymptotic stability and Ψ-uniform stability of trivial solutions of non-linear Kronecker product system The main results of this paper are illustrated with suitable examples This paper extends some of the results of Ψ-asymptotic stability of trivial solutions of linear equations (Theorem and Theorem 2)in Diamandescu [4 to matrix Lyapunov systems 2 Preliminaries In this section we present some basic definitions and results which are useful for later discussion Let R n be the Euclidean n-dimensional space Elements in this space are column vectors, denoted by u = (u, u 2, u n ) T ( T denotes transpose) and their norm defined by For a n n real matrix, we define the norm u = max{ u, u 2, u n } A = sup Ax x Let Ψ k : R (, ), k =, 2, n, n 2,be continuous functions, and let Ψ = diag[ψ, Ψ 2, Ψ n 2 Then the matrix Ψ(t) is an invertible square matrix of order n 2,for each t Definition 2 [5 Let A R m n and B R p q then the Kronecker product of A and B written A B is defined to be the partitioned matrix a B a 2 B a n B A B = a 2 B a 22 B a 2n B a m B a m2 B a mn B is an mp nq matrix and is in R mp nq Definition 22 [5 Let A = [a ij R m n, we denote A A 2 Â = V eca =, where A a 2j j = ( j n) A n a mj a j

3 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Regarding properties and rules for Kronecker product of matrices we refer to Graham [5 Now by applying the Vec operator to the non-linear matrix Lyapunov system () and using the above properties, we have ˆX (t) = H(t) ˆX(t) G(t, ˆX(t)), (2) where H(t) = (B T I n ) (I n A) is a n 2 n 2 matrix and G(t, ˆX(t)) = V ecf (t, X(t)) is a column matrix of order n 2 The corresponding linear homogeneous system of (2) is ˆX (t) = H(t) ˆX(t) (22) Definition 23 The trivial solution of (2) is said to be Ψ-stable on R if for every ε > and every t in R, there exists δ = δ(ε, t ) > such that any solution ˆX(t) of (2) which satisfies the inequality Ψ(t ) ˆX(t ) < δ, also satisfies the inequality Ψ(t) ˆX(t) < ε for all t t Definition 24 The trivial solution of (2) is said to be Ψ-uniformly stable on R, if δ(ε, t ) in Definition 23 can be chosen independent of t Definition 25 The trivial solution of (2) is said to be Ψ-asymptotically stable on R, if it is Ψ-stable on R and in addition, for any t R, there exists a δ = δ (t ) > such that any solution ˆX(t) of (2) which satisfies the inequality Ψ(t ) ˆX(t ) < δ, satisfies the condition lim Ψ(t) ˆX(t) = The following example illustrates the difference between the Ψ-stability and Ψ-asymptotic stability Example 2 Consider the non-linear matrix Lyapunov system () with Then the solution of (2) is Consider for all t, we have A(t) = [ t t 2 2t [, B(t) = e t t t 2 and [ t(x 2 3x ) e F (t, X(t)) = t x t 2 tx 2 t 2 x 2 tx 4 t 2 2tx 3 x 3 x 2 4 sec t x 4 tan t 2tx 4 e t x 3 ˆX(t) = Ψ(t) = Ψ(t) ˆX(t) = () t 2 e t e t cos t t t e t e t t t 2 cos t It is easily seen from the Definitions 23 and 25, the trivial solution of the system (2) is Ψ-stable on R, but, it is not Ψ-asymptotically stable on R

4 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Lemma 2 Let Y (t) and Z(t) be the fundamental matrices for the systems and X (t) = A(t)X(t), (23) [X T (t) = B T (t)x T (t) (24) respectively Then the matrix Z(t) Y (t) is a fundamental matrix of (22) and every solution of (22) is of the form ˆX(t) = (Z(t) Y (t))c, where c is a n 2 -column vector Proof For proof, we refer to Lemma of [7 Theorem 2 Let Y (t) and Z(t) be the fundamental matrices for the systems (23) and (24), then any solution of (2), satisfying the initial condition ˆX(t ) = ˆX, is given by ˆX(t) = (Z(t) Y (t))(z (t ) Y (t )) ˆX t (Z(t) Y (t))(z (s) Y (s))g(s, ˆX(s))ds (25) Proof First, we show that any solution of (2) is of the form ˆX(t) = (Z(t) Y (t))c X(t), where X(t) is a particular solution of (2) and is given by X(t) = t (Z(t) Y (t))(z (s) Y (s))g(s, ˆX(s))ds Here we observe that, ˆX(t ) = (Z(t ) Y (t ))c = ˆX, c = (Z (t ) Y (t )) ˆX Let u(t) be any other solution of (2), write w(t) = u(t) X(t), then w satisfies (22), hence w = (Z(t) Y (t))c, u(t) = (Z(t) Y (t))c X(t) Next, we consider the vector X(t) = (Z(t) Y (t))v(t), where v(t) is an arbitrary vector to be determined, so as to satisfy equation (2) Consider X (t) = (Z(t) Y (t)) v(t) (Z(t) Y (t))v (t) H(t) X(t) G(t, ˆX(t)) = H(t)(Z(t) Y (t))v(t) (Z(t) Y (t))v (t) (Z(t) Y (t))v (t) = G(t, ˆX(t)) v (t) = (Z (t) Y (t))g(t, ˆX(t)) v(t) = t (Z (s) Y (s))g(s, ˆX(s))ds Hence the desired expression follows immediately 3 Ψ-asymptotic stability of linear systems In this section we study the Ψ-asymptotic stability of trivial solutions of linear system (22) Theorem 3 Let Y (t) and Z(t) be the fundamental matrices of (23) and (24) Then the trivial solution of (22) is Ψ-asymptotically stable on R if and only if lim Ψ(t)(Z(t) Y (t)) =

5 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Proof The solution of (22) with the initial point at t is ˆX(t) = (Z(t) Y (t))(z (t ) Y (t )) ˆX(t ), for t First, we suppose that the trivial solution of (22) is Ψ-asymptotically stable on R Then, the trivial solution of (22) is Ψ-stable on R and for any t R, there exists a δ = δ(t ) > such that any solution ˆX(t) of (22) which satisfies the inequality Ψ(t ) ˆX(t ) < δ, and satisfies the condition lim Ψ(t) ˆX(t) = Therefore, for any ɛ > and t, there exists a δ > such that Ψ(t ) ˆX(t ) < δ and also satisfies Ψ(t)(Z(t) Y (t))(z (t ) Y (t ))Ψ (t )Ψ(t ) ˆX(t ) < ɛ for all t t ɛ,t Let v R n2 be such that v For ˆX(t ) = δ 2 Ψ (t )v, we have Ψ(t ) ˆX(t ) < δ and hence, Ψ(t)(Z(t) Y (t))(z (t ) Y (t ) δ 2 Ψ (t )v < ɛ Ψ(t)(Z(t) Y (t))(z (t ) Y (t ))Ψ (t ) < 2ɛ Ψ(t)(Z(t) Y (t)) δ 2ɛ δ (Z (t ) Y (t ))Ψ (t ), for t t ɛ,t Therfore, lim Ψ(t)(Z(t) Y (t)) = Conversely, suppose that lim Ψ(t)(Z(t) Y (t)) = Then, there exists M > such that Ψ(t)(Z(t) Y (t)) M for t From (i) of Theorem 3 [7, it follows that the trivial solution of (22) is Ψ-stable on R For any ˆX(t ) R n2, we have lim Ψ(t) ˆX(t) = lim Ψ(t)(Z(t) Y (t))(z (t ) Y (t )) ˆX(t ) = Thus, the trivial solution of (22) is Ψ-asymptotically stable on R The above Theorem 3 is illustrated by the following example Example 3 Consider the linear homogeneous matrix Lyapunov system corresponding to () with [ [ A(t) =, B(t) = 2 Then the fundamental matrices of (23), (24) are [ t Y (t) = [ e t, Z(t) = e 2t Now the fundamental matrix of (22) is e t (t ) e t Z(t) Y (t) = (t )e 2t e 2t Consider Ψ(t) = e 2t e t e 2t () 2 e 2t

6 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), for all t, we have Ψ(t)(Z(t) Y (t)) = e t () 2 () 3 2 It is easily seen from Theorem 3, the system (22) is Ψ-asymptotically stable on R Remark 3 Ψ-asymptotic stability need not imply classical asymptotic stability The Remark 3 is illustrated by the following example Example 32 Consider the linear homogeneous matrix Lyapunov system corresponding to () with A(t) = [ [, B(t) = Then the fundamental matrices of (23), (24) are [ e Y (t) = t sin t e t cos t e t cos t e t sin t [ e t, Z(t) = e t Now the fundamental matrix of (22) is Z(t) Y (t) = sin t cos t cos t sin t sin t cos t cos t sin t Clearly the system (22) is stable, but it is not asymptotically stable on R Consider Ψ(t) = for all t, we have Ψ(t)(Z(t) Y (t)) = sin t cos t cos t sin t sin t cos t cos t sin t Thus, from Theorem 3 the system (22) is Ψ-asymptotically stable on R Theorem 32 Let Y (t), Z(t) be the fundamental matrices of (22), (24) If there exists a continuous function φ : R (, ) such that φ(s)ds =, and a positive constant N satisfying φ(s) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) ds N, for all t then, the linear system (22) is Ψ-asymptotically stable on R

7 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Proof Let b(t) = Ψ(t)(Z(t) Y (t)) for t From the identity ( = it follows that ( ) φ(s)b(s) ds Ψ(t)(Z(t) Y (t)) φ(s)ψ(t)(z(t) Y (t))(z (s) Y (s))ψ (s)ψ(s)(z(s) Y (s))b(s) ds, ) φ(s)b(s) ds Ψ(t)(Z(t) Y (t)) φ(s) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) Ψ(s)(Z(s) Y (s)) b(s) ds Thus, the scalar function a(t) = φ(s)b(s)ds satisfies the inequality a(t)b (t) N, for t We have a (t) = φ(t)b(t) N φ(t)a(t) for t It follows that and hence a(t) a(t )e N t φ(s) ds, for t t > Ψ(t)(Z(t) Y (t)) = b (t) Na (t )e N t φ(s) ds, for t t > Since Ψ(t)(Z(t) Y (t)) is a continuous function on the compact interval [, t, there exists a positive constant M such that Ψ(t)(Z(t) Y (t)) M for t Therefore, the trivial solution of (22) is Ψ-stable on R, and also from φ(s)ds =, it follws that lim Ψ(t)(Z(t) Y (t)) = Hence by using Theorem 3, system (22) is Ψ-asymptotically stable 4 Ψ-asymptotic stability of non-linear systems In this section we obtain sufficient conditions for Ψ-asymptotic stability and Ψ-uniform stability of trivial solutions of non-linear system (2) Theorem 4 Suppose that (i) The fundamental matrices Y (t) and Z(t) of (23), (24) are satisfying the condition φ(s) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) ds N, for all t where N is a positive constant and φ is a continuous positive function on R such that φ(s)ds = (ii) The function G satisfies the condition Ψ(t)G(t, ˆX(t)) α(t) Ψ(t) ˆX(t) for every vector valued continuous function ˆX : R R n2, where α is a continuous non-negative function on R such that α(t) q = sup t φ(t) < N

8 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Then, the trivial solution of equation (2) is Ψ-asymptotically stable on R Proof From the first assumption of the theorem, Theorems 3 and 32, we have hence there exists a positive constant M such that From the second assumption of the theorem, we have lim Ψ(t)(Z(t) Y (t)) =, Ψ(t)(Z(t) Y (t)) M, for all t For a given ɛ > and t, we choose δ = min{ɛ, Ψ(t ) ˆX ) < δ For τ > t and t [t, τ Consider Therefore, α(t) φ(t) sup α(t) t φ(t) = q < N ( qn)ɛ M (Z (t ) Y (t ))Ψ (t ) } Let ˆX R n2 Ψ(t) ˆX(t) Ψ(t)(Z(t) Y (t))(z (t ) Y (t ))Ψ (t )Ψ(t ) ˆX(t ) Ψ(t)(Z(t) Y (t))(z (s) Y (s)) Ψ(s)G(s, ˆX(s)) ds t Ψ(t)(Z(t) Y (t)) (Z (t ) Y (t ))Ψ (t ) Ψ(t ) ˆX φ(s) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) α(s) Ψ(s) ˆX(s) ds t φ(s) < M (Z (t ) Y (t ))Ψ (t ) δ Nq sup Ψ(t) ˆX(t) t t τ sup Ψ(t) ˆX(t) ( Nq) M (Z (t ) Y (t ))Ψ (t ) δ < ɛ t t τ such that It follows that the trivial solution of equation (2) is Ψ-stable on R To prove, the trivial solution of (2) is Ψ-asymptotically stable, we must show further that lim Ψ(t) ˆX(t)) = Suppose that lim sup Ψ(t) ˆX(t)) = λ > Let be such that qn < <, then there exists t t such that Ψ(t) ˆX(t) < λ for all t t Thus for t > t, we have Ψ(t) ˆX(t) Ψ(t)(Z(t) Y (t)) (Z (t ) Y (t ))Ψ (t ) Ψ(t ) ˆX(t ) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) Ψ(s)G(s, ˆX(s)) ds t < Ψ(t)(Z(t) Y (t)) (Z (t ) Y (t ))Ψ (t ) δ t Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) α(s) Ψ(s) ˆX(s) ds t φ(s) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) α(s) φ(s) < Ψ(t)(Z(t) Y (t)) (Z (t ) Y (t ))Ψ (t ) δ Ψ(s) ˆX(s) ds

9 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), t Ψ(t)(Z(t) Y (t)) (Z (s) Y (s))ψ (s) α(s) Ψ(s) ˆX(s) ds t φ(s) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) qλ ds Ψ(t)(Z(t) Y (t)) { (Z (t ) Y (t ))Ψ (t ) δ t (Z (s) Y (s))ψ (s) α(s) Ψ(s) ˆX(s) ds} Mqλ From lim Ψ(t)(Z(t) Y (t)) =, it follows that there exists T >, sufficiently large, such that Ψ(t)(Z(t) Y (t)) < λ Mqλ 2Q for all t T, where Q = (Z (t ) Y (t ))Ψ (t ) δ Thus, for t T we have It follows from the definition of which is a contradiction Therefore t (Z (s) Y (s))ψ (s) α(s) Ψ(s) ˆX(s) ds Ψ(t) ˆX(t) < λ Mqλ 2 < λ Mqλ 2 λ λ Mqλ 2 Mqλ < λ lim Ψ(t) ˆX(t) = Thus, the trivial solution of (2) is Ψ-asymptotically stable on R Example 4 Consider the non-linear matrix Lyapunov system () with A(t) = [ [, B(t) = [ sin(x ) 4(), and F (t, X(t)) = x 2 2() x 3 8() sin(x 4 ) 6() The fundamental matrices of (23), (24) are Y (t) = [ t [ e t, Z(t) = e t Therefore, the fundamental matrix of (22) is e t (t ) e t Z(t) Y (t) = (t )e t e t

10 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Consider for all t, then we have Ψ(t) = e t () 2 e t e t () 2 e t Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) = Taking φ(t) =, for all t Clearly φ(t) is continuous on R and φ(s) Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) ds = ( ) s I 4 t φ(s)ds = Also t, for all t t Further, the matrix G satisfies condition (ii), with α(t) = 2(), α(t) is a continuous non-negative function on R and satisfies α(t) q = sup t φ(t) = 2 < N = Thus, from Theorem 4, the trivial solution of non-linear system (2) is Ψ-asymptotically stable on R Theorem 42 Let Y (t), Z(t) be the fundamental matrices of (23), (24) respectively satisfying the condition Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) L, for all s t <, where L is a positive number Assume that the function G satisfies Ψ(t)G(t, ˆX(t)) α(t) Ψ(t) ˆX(t)), t < and for every ˆX R n2, where α(t) is a continuous non-negative function such that β = α(s)ds < Then, the trivial solution of (2) is Ψ-uniformly stable on R Proof Let ɛ > and δ(ɛ) = ɛ 2L e Lβ For t and ˆX R n2 be such that Ψ(t ) ˆX < δ(ɛ), we have Ψ(t) ˆX(t) Ψ(t)(Z(t) Y (t))(z (t ) Y (t ))Ψ (t )Ψ(t ) ˆX(t ) t Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s)ψ(s)g(s, ˆX(s)) ds Ψ(t)(Z(t) Y (t))(z (t ) Y (t ))Ψ (t ) Ψ(t ) ˆX ) By Gronwall s inequality t Ψ(t)(Z(t) Y (t))(z (s) Y (s))ψ (s) Ψ(s)G(s, ˆX(s)) ds L Ψ(t ) ˆX L t α(s) Ψ(s) ˆX(s) ds Ψ(t) ˆX(t) L Ψ(t ) ˆX e L t α(s)ds Lδ(ɛ)e Lβ < ɛ, for all t t This proves that the trivial solution of (2) is Ψ-uniformly stable on R

11 MSNMurty, GSuresh Kumar, J Nonlinear Sci Appl 5 (22), Example 42 In Example 4, taking F (t, X(t)) = [ x sin(x 3 ) () 2 () 2 sin(x 2 ) x 4 () 2 () 2 Then the conditions of Theorem 42 are satisfied with L = and α(t) = Clearly, α(t) is continuous () 2 non-negative function and α(s)ds = Therefore, from Theorem 42 the trivial solution of (2) is Ψ- uniformly stable on R References [ O Akinyele, On partial stability and boundedness of degree k, Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur, (8), 65 (978), [2 A Constantin,Asymptotic properties of solutions of differential equations, Analele Universităţii din Timişoara, Seria Ştiin ţe Matematice, vol XXX, fasc 2-3 (992), [3 A Diamandescu, On the Ψ - stability of Nonlinear Volterra Integro-differential System, Electronic Journal of Differential Equations, 25 (56) (25), -4 [4 A Diamandescu, On the Ψ - asymptotic stability of Nonlinear Volterra Integro-differential System, Bull Math Sc MathRoumanie, Tome 46(94) (-2) (23), 39-6 [5 A Graham, Kronecker Products and Matrix Calculus: with applications, Ellis Horwood Series in Mathematics and its Applications Ellis Horwood Ltd, Chichester; Halsted Press [John Wiley & Sons, Inc, New York, 98 2, 22, 2 [6 J Morchalo, On Ψ L p-stability of nonlinear systems of differential equations, Analele Ştiinţifice ale Universităţii Al I Cuza Iaşi, Tomul XXXVI, s I - a, Matematicăf 4(99), [7 MSN Murty and G Suresh Kumar, On Ψ-Boundedness and Ψ-Stability of Matrix Lyapunov Systems, Journal of Applied Mathematics and Computing, Springer, 26 (28), 67-84, 2, 3 [8 MSN Murty, GS Kumar, PN Lakshmi and D Anjaneyulu, On Ψ-instability of Non-linear Matrix Lyapunov Systems, Demonstrtio Mathematica, 42 (4) (29),