Synchronization Stability in Weighted Complex Networks with Coupling Delays

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1 Commun. Theor. Phys. (Beijing China) 51 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 51 No. 4 April Synchronization Stability in Weighted Complex Networks with Coupling Delays WANG Qing-Yun 12 DUAN Zhi-Sheng 2 CHEN Guan-Rong 23 and LU Qi-Shao 4 1 School of Statistics and Mathematics Inner Mongolia Finance and Economics College Huhhot China 2 State Key Laboratory for Turbulence and Complex Systems Department of Mechanics and Aerospace Engineering College of Engineering Peking University Beijing China 3 Department of Electronic Engineering City University of Hong Kong Hong Kong SAR China 4 School of Science Beijing University of Aeronautics and Astronautics Beijing China (Received June ) Abstract Realistic networks display not only a complex topological structure but also a heterogeneous distribution of weights in connection strengths. In addition the information spreading through a complex network is often associated with time delays due to the finite speed of signal transmission over a distance. Hence the weighted complex network with coupling delays have meaningful implications in real world and resultantly gains increasing attention in various fields of science and engineering. Based on the theory of asymptotic stability of linear time-delay systems synchronization stability of the weighted complex dynamical network with coupling delays is investigated and simple criteria are obtained for both delay-independent and delay-dependent stabilities of synchronization states. The obtained criteria in this paper encompass the established results in the literature as special cases. Some examples are given to illustrate the theoretical results. PACS numbers: Xt i b Key words: weighted complex networks coupling delays synchronization stability 1 Introduction Many social biological and physical systems can be casted into the form of complex dynamical networks. [1 3] Dynamics of the complex dynamical networks in various fields has been extensively investigated to explore their complexity including pattern formation spreading processes and synchronization. Especially synchronization is an important topic in the study of dynamical performance of complex networks and is also considered as one of the mechanisms to transmit and code information in complex systems. There are a lot of existing research reports on synchronization of the unweighted complex networks. Synchronization in a network of small-world coupled Chua s circuits was investigated in Ref. [4]. It was shown that for any given coupling strength and a sufficiently large number of cells the small-world dynamical network can synchronize even if the original nearestneighbor coupled network cannot achieve synchronization under the same condition. For the neural network with symmetric connections it was noted that the critical values depend on specific network structure when neurons achieve complete synchronization. [5] It was numerically observed that a small fraction of phase-repulsive links can enhance synchronization in a complex network of dynamical units. [6] Since the finite speed of signal transmission over a distance gives rise to a finite time delay it is well known that the information flow in complex network is not instantaneous in general. In view of the time-delay phenomenon which is frequently encountered in practical situations the complex network is further extended to include coupling delays among its nodes and synchronization conditions of these networks have been investigated analytically. For example synchronization in oscillator networks with delayed coupling was studied in Ref. [7]. They derived a stability criterion for the synchronization state in networks of identical phase oscillators with delayed coupling. Synchronization of small-world networks with coupling delays was investigated. [8] Results showed that the stability of the synchronized state is independent of the network topology. Li and Chen have discussed continuous complex dynamical networks with coupling delays in whole networks and the stability theorem of synchronization is established by constructing a Lyapunov Krasovskii functional which is often difficult to be found. [9] In terms of the linear matrix inequality or stability theory of the delay systems some new criteria of synchronization stability in symmetric networks with coupling delays were obtained for both delay-independent and delay-dependent cases. [10 12] However most studies on the synchronizability of networks are focused on unweighted and symmetrical net- The project supported by National Natural Science Foundation of China under Nos and China Post-doctoral Special Science Foundation No and the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant No the research also supported in part by the Project of Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences Corresponding author nmqingyun@163.com

2 No. 4 Synchronization Stability in Weighted Complex Networks with Coupling Delays 685 works. It is well known that many real-world networks are weighted and asymmetrical. Very recently scholars began to investigate the dynamics of weighted complex networks. Synchronization in weighted complex networks was studied. [13] Results show that the synchronizability of random networks with a large minimum degree is determined by two leading parameters: the mean degree and the heterogeneity of the distribution of nodes intensity. The synchronizability of weighted aging scale-free networks with non-normalized asymmetrical coupling matrices has been studied in detail. [14] It is shown that the synchronizability of such weighted networks can be dramatically affected by the asymmetrical parameter and it can be improved when the couplings from older nodes to younger nodes become dominant. Importantly thus synchronization stability of weighted complex networks with coupling delays has not yet been analytically investigated. In the present paper we study the synchronization of weighted complex dynamical networks with coupling delays. Based on the theory of asymptotic stability of linear time-delay systems the stability criteria of the synchronization state of the weighted complex networks with coupling delays are derived for both delay-independent and delay-dependent cases. The rest of the paper is organized as follows. In Sec. 2 some preliminaries are given. Stability criteria of the synchronization for weighted complex dynamical networks with coupling delays are established in Sec. 3. Numerical examples are illustrated in Sec. 4 and conclusion is presented in Sec Preliminaries Consider a linear time-delay dynamical system: ẋ = Ax + Bx(t τ) (1) where x R n A B R n n and τ > 0 is a time delay. The fundamental results which give the condition for asymptotic stability of the system are summarized as follows respectively. Lemma 2.1 [15] If the condition µ(a) + B < 0 holds then zero solution of the system is asymptotically stable. Lemma 2.2 [16] Let L 1 0. Stability of zero solution of the system is achieved if the following condition is satisfied Reλ i (A + Bexp( τs)) < 0 (i = N) (2) where s takes the value in ranges given by s = jω 0 ω L 2 (3) s = L 1 + jω 0 ω L 2 (4) s = r + jl 2 0 r L 1 (5) where Reλ i (X) is the real part of eigenvalue λ i of a matrix X. L 1 = µ(a) + B L 2 = µ( ja) + B and j 2 = 1. µ(x) is the matrix measure for X R N N which can be defined by ( I + ɛx 1 ) µ(x) = lim. ɛ 0 ɛ X denotes the norm of the matrix X. The computation of the matrix measure in this paper can be described as follows µ(x) = 1 2 max (λ i (X + X)) (6) i where X = (x ij ) N N is a square matrix and denotes the conjugate transpose symbol. Lemma 2.2 which includes information of the delay is referred to as the delay-dependent stability criterion for system (1). It is noticed that condition (2) is the eigenvalue problem of the complex matrix. Hence we can resort to the following criterion to explore the stability of the linear time-delay system by using Lemma 2.2. Lemma 2.3 [17] Reλ i (B) < 0 (B is a complex matrix) for all λ i if and only if for the characteristic equation f(λ) = λ n + (a R n 1 + ja In 1 )λn (a R 1 + ja I1 )λ + (a R0 + ja I0 ) = 0 (7) all north-westerly minors D 1 D 2... D n of even order of the Bilharz matrix B are positive where 1 a In 1 a Rn 2 a In 3 a Rn 4 a In 5 a Rn 6 a In a Rn 1 a In 2 a Rn 3 a In 4 a Rn 5 a In 6 a Rn a In 1 a Rn 2 a In 3 a Rn 4 a In 5 a Rn 6... B = 0 0 a Rn 1 a In 2 a Rn 3 a In 4 a Rn 5 a In a In 1 a Rn 2 a In 3 a Rn 4 a InN a Rn 1 a In 2 a Rn 3 a In 4 a Rn a In 1 a Rn 2 a In 3 ( ) 1 D 1 ain 1 = D 2 0 a Rn 1 a In 2 a Rn 3 = 0 a Rn a In 1 a Rn a Rn 1 a In 2

3 686 WANG Qing-Yun DUAN Zhi-Sheng CHEN Guan-Rong and LU Qi-Shao Vol a In 1 a Rn 2 a In 3 a Rn 4 a In 5 0 a Rn 1 a In 2 a Rn 3 a In 4 a Rn 5 D a In 1 a Rn 2 a In 3 a Rn 4 = 0 0 a Rn 1 a In 2 a Rn 3 a In a In 1 a Rn 2 a In a Rn 1 a In 2 a Rn 3 3 Criteria of Synchronization Stability for Weighted Complex Networks with Coupling delays In this section based on the theory for asymptotic stability of linear time-delay systems we will develop some criteria of synchronization stability in weighted complex networks with coupling delays for both delay-independent and delay-dependent cases. The dynamics of a general weighted network of N coupled identical oscillators with coupling delays are described by ẋ i = F(x i ) + σ G ij H(x j (t τ)) i = N (8) where F: R n R n is continuously differentiable x i = (x i1 x i2...x in ) T R n are the state variables of node i σ is the coupling strength H: R m R m is an arbitrary function of each node s variables that is used in the coupling and G = (G ij ) N N is the coupling configuration matrix of the network in which G ij satisfies G ii = G ij (i = N). (9) i The above assumptions can ensure that the completely synchronized state M = {x i = s i ṡ = F(s)} (10) is an invariant manifold of Eq. (8). Let x = (x 1 x 2...x N ) F(x) = (F(x 1 ) F(x 2 )...F(x N )) H(x) = (H(x 1 ) H(x 2 )... H(x N )) we can rewrite system (8) in a compact form as follows ẋ = F(x) + σg H(x τ ) (11) where is the direct product and H(x τ ) = (H(x 1 (t τ)) H(x 2 (t τ)...h(x N (t τ)). It is noted that the matrix G is not necessarily symmetric which represents the coupling configuration of the weighted network. In what follows we suppose that the matrix G is diagonalizable namely there exists a nonsingular matrix Φ = (φ 1 φ 2... φ N ) such that Gφ k = λ k φ k (k = N) where λ k (k = N) are the eigenvalues of G. Lemma 3.1 Consider the weighted dynamical network with coupling delays (8). If the following N 1 pieces of n-dimensional linear time-varying delayed differential equations are asymptotically stable about their zero solutions: ẇ(t) = DF(s(t))w + σλ i DH(s τ )w τ i = 2...N (12) where DF(s(t)) is Jacobian of F(x(t)) at s(t) s τ = s(t τ) and w τ = w(t τ) then the synchronized states (10) of the system (8) are asymptotically stable. Proof To investigate the stability of the synchronized states let x i = s(t) + η i (t) (13) and then we can get the variational equation of Eq. (11) η = 1 N DF(s)η + σg DH(s τ )η τ (14) where η = (η 1 η 2...η N ) and η τ = (η1 τ ητ 2...ητ N) with ηk τ = η k(t τ) k = N. By diagonalizing G this leaves us with a block diagonalized variational equation with each block having the form η k = DF(s)η k + σλ k DH(s τ )ηk τ (15) where λ k is an eigenvalue of G k = N 1. It is clear that we have transformed the stability problem of synchronized states to the stability problem of the N pieces of n-dimensional linear time-varying delayed differential equations (15). Since λ 0 = 0 corresponds to the synchronizing state s(t) the synchronized states are asymptotically stable when the N 1 pieces of n-dimensional linear time-varying delayed differential equations are asymptotically stable about their zero solutions: ẇ(t) = DF(s(t))w +σλ i DH(s τ )w τ i = 2... N (16) The proof is thus completed. In what follows we will formulate main results based on the above discussions. Firstly we consider the case when all eigenvalues of the matrix G are real. Letting L λ i 1 = µ(df(s(t))) + σ λ i DH(s τ ) L 2 = µ( jdf(s(t))) + σ λ i DH(s τ ). In terms of Lemmas and 3.1 we can get the following stability criteria of synchronization for the weighted complex networks with coupling delays. Theorem 3.2 (Delay-independent criteria) If L λ max 1 < 0 holds for all s(t) then the synchronization

4 No. 4 Synchronization Stability in Weighted Complex Networks with Coupling Delays 687 state (10) is asymptotically stable for any delay where λ max = max i=2...n ( λ i ). Proof For linear time-delay system (12) we have L λ i 1 = µ(a) + B = µ(df(s(t))) + σ λ i DH(s τ ). According to Lemma 2.1 it is inferred that if L i 1 < 0 for any λ i and all s(t) then zero solution of linear time-delay system (12) is asymptotically stable. Since λ max = max i=2...n ( λ i ) we only need L λ max 1 < 0. Theorem 3.3 (Delay dependent criteria) If L λ i 1 0 for a certain s(t) or λ i and Reλ i (DF(s(t)) + σλ i DH(s τ )exp( τs)) < 0 holds for any λ i and s(t) then synchronization state (10) is asymptotically stable where s takes the value in ranges given by s = jω 0 ω L 2 (17) s = L λ i 1 + jω 0 ω L 2 (18) s = r + jl 2 0 r L λ i 1. (19) Theorem 3.3 can immediately be proved by means of Lemma 2.2 and 3.1. Obviously the obtained stability criterion includes information on the length of the delay therefore it can be a delay dependent criterion of synchronization stability. It is a more general result for testing the synchronization of weighted complex networks with the coupling delays. Furthermore the obtained criteria encompass the established results in Ref. [11] as special cases in which the connection of the complex networks is symmetric. Lemma 3.4 [16] For any complex square matrix X we have Reλ i (X) µ(x). It is clear that we can get following corollary in term of Theorem 3.3 and Lemma 3.4 Corollary If L λ i 1 0 for a certain s(t) or λ i and µ(df(s(t)) + σλ i DH(s τ )exp( τs)) < 0 for any λ i and s(t) then synchronization state (10) is asymptotically stable. However if the eigenvalue of matrix G is complex then Eq. (12) can be rewritten as follows η = DF(s)η + σ(α + jβ)dh(s τ )η τ (20) where λ = α + jβ α β R. Furthermore separating η into real part η r and imaginary part η i we have η r = DF(s)η r + σαdh(s τ )ηr τ σβdh(s τ )ηi τ (21) η i = DF(s)η i + σαdh(s τ )ηi τ + σβdh(s τ )ηr τ (22) where ηr τ = η r (t τ) and ηi τ = η i(t τ). Hence stability of zero solutions of Eq. (12) can be estimated by means of Eqs. (21) (22). Now let [ ] [ ηr η τ ] η = η τ = r η i ηi τ then we can combine Eqs. (21) (22) into the following form η = [ ] DF(s) 0 η 0 DF(s) + [ ] σαdh(sτ ) σβdh(s τ ) η τ. (23) σβdh(s τ ) σαdh(s τ ) Consequently according to Lemmas 2.1 and 2.2 similar criteria of stability to Theorems 3.2 and 3.3 can be given by Eq. (23). On the other hand the dynamics of a delayed complex network without the overall self-delayed interaction can be written as follows: i = F(x i ) + σ i a ij (H(x j (t τ)) H(x i )) i = N. (24) Let G ii = i a ij G ij = a ij (i j) F 1 (x i ) = F(x i ) + σg ii H(x i ) F 2 (x i (t τ)) = σg ii H(x i (t τ)) and then Eq. (24) can be rewritten as follows: i = F 1 (x i ) + F 2 (x i (t τ)) + σ G ij H(x j (t τ)) i = N. (25) Hence the synchronized state is ṡ = F 1 (s) + F 2 (s(t τ)). (26) Consequently we can get a similar linearized equation at synchronization manifold (26) to Eq. (12) ẇ(t) = DF 1 (s(t))w + (DF 2 (s τ ) + σλ i DH(s τ ))w τ i = 2...N. (27) It is concluded that the stability of the synchronized state can be estimated from Eq. (27). Similar to Theorem 3.2 and 3.3 we can get the stability criteria of synchronization state (26) for the delay-independent and delaydependent cases respectively by means of Eq. (27). 4 Some Illustrative Examples Example 1 The above synchronization conditions can be applied to networks with different topologies and different sizes. In order to illustrate the main results of the above theoretical analysis we firstly consider a lowerdimensional weighted network model with 5 nodes in which each node is a simple three-dimensional stable linear system described by where i = F(x i ) + σ 5 G ij H(x j (t τ)) i = (28) F = ( x 1i 2x 2i 3x 3i ) T x i = (x 1i x 2i x 3i ) T R 3

5 688 WANG Qing-Yun DUAN Zhi-Sheng CHEN Guan-Rong and LU Qi-Shao Vol. 51 are the state variables of node i. It is clear that its Jacobian matrix is DF(s(t)) = Assume that the H(x i ) = diag{x 1i x 2i x 3i } namely DH = diag{1 1 1}. The weighted matrix G is given as follows G = All eigenvalues of matrix G are and Based on Theorem 3.2 if L λ max 1 = µ(df(s(t))) + σλ max DH(s τ ) = 1 + σλ max < 0 then the synchronization of the complex network can be achieved for any delay. By simple calculations it is concluded that when σ < 1/ the synchronization of the complex network can be achieved for any delay. For clearer visions we take the coupling strength σ = 0.16 < and time delay τ = 2 20 respectively. It is obvious in Fig. 1 that the networks can eventually get synchronization state at s(t) = 0 irrespectively of the size of time delay. Fig. 1 (a) Time series of the first variable x 1i of node i for time delay τ = 2; (b) Time series of the first variable x 1i of node i for time delay τ = 20. Here the coupling strength σ = However if L λ i 1 = 1 + σ λ i 0 for a certain λ i we can resort to Theorem 3.3 to estimate regions on parameter plane (σ τ) where the stability of synchronization can be realized. Hence we need to make the following calculations 1 + σλ i exp( τs) 0 0 H = (DF(s(t)) + σλ i DH(s τ ) exp( τs)) = σλ i exp( τs) σλ i exp( τs) where s takes the value in ranges given by s = jω 0 ω σ λ i (29) s = 1 + σ λ i + jω 0 ω σ λ i (30) s = r + j(σ λ i ) 0 r 1 + σ λ i. (31) If the real part Reλ i (H ) of any eigenvalue of the matrix H is less than zero then synchronization state of system is stable. All eigenvalues of matrix H are where 1 + σλ i exp( τs) 2 + σλ i exp( τs) 3 + σλ i exp( τs) 1 + σλ i exp( τs) > 2 + σλ i exp( τs) > 3 + σλ i exp( τs). Hence the condition of Theorem 3.3 can be satisfied as long as Re ( 1 + σλ i exp( τs)) < 0. For the convenience of calculations we assume that s = r + jω. Hence we have Re( 1 + σλ i exp( τs)) = 1 + σλ i exp( τr)cosτω < 0. By some simple calculations it is shown that when τ = 0.5 and σ = 0.5 Re( 1+σλ i exp( τs)) < 0 for any λ i. On the other hand when τ = 0.8 and σ = 0.5 Re( 1 + σλ i exp( τs)) > 0 for λ i = r = 0 and ω = σ λ i. The corresponding numerical results are shown in Figs. 2(a) and 2(b) respectively. It can be seen clearly that when τ = 0.5 and σ = 0.5 the time series of x 1i (i = ) can eventually get the synchronization state s(t) = 0. On the contrary when τ = 0.8 and σ = 0.5 networks can never

6 No. 4 Synchronization Stability in Weighted Complex Networks with Coupling Delays 689 achieve synchronization. These coincide with the above theoretical analysis. Assume H(x i ) = (x 1i 0) T then DH = diag{1 0}. The weighted matrix G is given as follows G = with all eigenvalues being and According to Theorem 3.3 we can get (DF(s(t)) + cλ i Γexp( τs)) [ f ] (s(t)) + σλ i exp( τs) 1 = 1 b where s takes the value in ranges given by s = jω 0 ω 1 + σ λ i (35) s = µ(df(s(t))) + σ λ i + jω 0 ω 1 + σ λ i (36) s = r + j(1 + σ λ i ) 0 r µ(df(s(t))) + σ λ i. (37) In what follows we can test the stability of the following characteristic polynomials to study the stability of the synchronization state in the coupled BVP network P(λ) = λ 2 + (b f (s(t)) σλ i exp( τs))λ + 1 b(f (s(t)) + σλ i exp( τs)) = 0. (38) Fig. 2 (a) Time series of the first variable x 1i of node i for time delay τ = 0.5; (b) Time series of the first variable x 1i of node i for time delay τ = 0.8. Here the coupling strength σ = 0.5. Example 2 The next example is a weighted complex network with dynamics of its every node being governed by the Bonhöffer van der Pol (BVP) oscillator 5 i = F(x i ) + σ G ij H(x j (t τ)) i = (32) where F = ( x 3 1i /3 + ax 1i x 2i x 1i bx 2i ) T x i = (x 1i x 2i ) T R 2 are the state variables of node i. a and b are positive parameters. When we take a = 1 and b = 0.75 such that the model can exhibit a limit cycle. The Jacobian of the system at synchronization state s(t) is [ ccf ] (s(t)) 1 DF(s(t)) = 1 b and µ(df(s(t))) = { f (s(t)) if f (s(t)) > b b if f (s(t)) b (33) µ( jdf(s(t))) = 1. (34) where f(x) = x 3 /3 + ax. For the convenience of calculation we assume that s = r + jω. Hence we have P(λ) = λ 2 + (b f (s(t)) σλ i exp( τr) cos(τω) + j(σλ i exp( τr) sin(τω)))λ + 1 b(f (s(t)) + σλ i exp( τr) cos(τω)) + j(bσλ i exp( τr) sin(τω)) = 0. (39) Then applying Lemma 2.3 the necessary and sufficient stability condition for Eq. (39) is D 1 = r 21 = b f (s(t)) σλ i exp( τr)cos(τω) > 0. (40) It is shown that when F(τ ω r σ) = σλ i exp( τr) cos(τω) > a b + s 2 (t) > a b (41) the stability condition for Eq. (42) can be realized. Note λ min = min j=2...5 (λ i ) it is inferred that if F(τ ω r σ) = σλ min exp( τr) cos(τω) > a b then the stability of the synchronization state can be achieved. Based on the above rigorous analysis we have the following result. Theorem 4.1 The synchronization state in the weighted BVP network is stable if there exist τ and σ such that F(ω r τ σ) = σλ min exp( τr) cos(τω) is always more than a b. Here 0 r µ(df(s(t))) + σ λ min and 0 ω 1 + σ λ min.

7 690 WANG Qing-Yun DUAN Zhi-Sheng CHEN Guan-Rong and LU Qi-Shao Vol. 51 To understand this theorem clearly when we choose τ = 0.2 and σ = 0.8 numerical result shows in Fig. 3 that F (ω r τ σ) > a b = 0.25 which satisfies the condition of Theorem 4.1. Hence it is concluded that the synchronization of the weighted BVP network can be achieved as shown in Fig Conclusion Fig. 3 Variations of F (ω r τ σ) in plane (r ω). Here τ = 0.2 and σ = 0.8. Fig. 4 Time series of xi in five coupled BVP network which shows synchronization is eventually achieved with temporal evolution. References [1] X.F. Wang and G.R. Chen IEEE Circuits and Systems Magazine 3(1) (2003) 6. [2] S.H. Strogatz Nature (London) 410 (2001) 268. [3] S.H. Dorogovtsev and J.F.F. Mendes Evolution of Networks: From Biological Nets to the Internet and WWW Oxford University Press Oxford (2003). [4] X.F. Wang and G.R. Chen Int. J. Bifur. Chaos 12 (2002) 187. [5] Q.Y. Wang Q.S. Lu G.R. Chen and D.H. Guo Phys. Lett. A 356 (2006) 17. [6] I. Leyva I. Sendin -Nadal J.A. Almendral and M.A.F. Sanjua n Phys. Rev. E 74 (2006) [7] M.G. Earl and S.H. Strogatz Phys. Rev. E 67 (2003) [8] C. Li H. Xu X. Liao and J. Yu Phys. A 335 (3-4) (2004) 359. [9] C.G. Li and G.R. Chen Phys. A 343 (2004) 263. In this paper based on the stability theory of the linear time-delay system we have obtained general stability criteria of synchronization state in the weighted complex dynamical networks with coupling delays for both delay-independent and delay-dependent cases. It is noted that the similar stability criteria of synchronization can be established for the delayed complex weighted networks with self-feedback delay and without self-feedback delay. By means of these criteria we avoided constructing the Lyapunov Krasovskii function which is generally difficult to be found to investigate the stability of the synchronization state. More importantly these synchronization conditions are applicable to networks with different topologies and different sizes. Finally two examples are given to confirm the correctness of the theoretical analysis. Acknowledgments We hope to thank referees for valuable comments. [10] C.P. Li W.G. Sun and J. Kurths Phys. A 361 (2006) 24. [11] Q.Y. Wang G.R. Chen Q.S. Lu and F. Hao Phys. A 378 (2007) 527. [12] H.J. Gao J. Lam and G.R. Chen Phys. Lett. A 360 (2006) 263. [13] C.S. Zhou A.E. Motter and J. Kurths Phys. Rev. Lett. 96 (2006) [14] Y.L. Zou J. Zhu and G.R. Chen Phys. Rev. E 74 (2006) [15] T. Mori IEEE Transactions on Automatic Control 30(2) (1985) 158. [16] T. Mori and H. Kokame IEEE Transactions on Automatic Control 34(4) (1989) 460. [17] P.C. Parks and V. Hahn Stability Theory Prentice Hall International Series in Systems and Control Engineering Prentice Hall (1992).