Applying Lemma 2.1 and from (2) and (3), we get 2x T (t)ΔA T i (t)P i x(t) ≤ ε −1 4i x T (t)H T 4i H 4i x(t)+ε 4i x T (t)P i E T 4i E 4i P i x(t), 2x T (t −h(t))ΔB T i (t)P i x(t) ≤ ε −1 5i x T (t −h(t))H T 5i H 5i x(t −h(t)) + ε 5i x T (t)P i E T 5i E 5i P i x(t). Next, by taking derivative of V 2,i (x t ), V 3,i (x t ) and V 4,i (x t ), respectively, along the system trajectories yields ˙ V 2,i (x t ) ≤ x T (t)Q i x(t) −(1 −μ)e −2βh(t) x T (t −h(t))Q i x(t −h(t)) − 2βV 2,i (x t ), ˙ V 3,i (x t ) ≤ h M x T (t)R i x(t) −  t t −h(t) e 2β(s−t) x T (s)R i x(s)ds −2βV 3,i (x t ), ˙ V 4,i (x t ) ≤ h M  x (t) x(t − h(t))  T  S 11,i S 12,i S T 12,i S 22,i  x (t) x(t − h(t))  −  t t −h(t) e 2β(s−t)  x (s) x(s −h(s))  T  S 11,i S 12,i S T 12,i S 22,i  x (s) x(s −h(s))  ds −2βV 4,i (x t ). Then, the derivative of V i (x t ) along any trajectory of solution of (1) is estimated by ˙ V i (x t ) ≤ N ∑ i=1 λ i (t)  x (t) x(t −h(t))  T Θ i  x (t) x(t −h(t))  −2βV 2,i (x t ) −  t t −h(t) e 2β(s−t) x T (s)R i x(s)ds −2βV 3,i (x t ) −  t t −h(t) e 2β(s−t)  x (s) x(s −h(s))  T  S 11,i S 12,i S T 12,i S 22,i  x (s) x(s − h(s))  ds −2βV 4,i (x t ). (40) For i ∈ S u , it follows from (40) that ˙ V i (x t ) ≤ N ∑ i=1 λ i (t)  x (t) x(t − h(t))  T Θ i  x (t) x(t − h(t))  . (41) Similar to Theorem 3.1, from (33) and (41), we get V i (x t ) ≤ N ∑ i=1 λ i (t)  V i (x t 0 )  e ξ i (t−t 0 ) , t ≥ t 0 . (42) where ξ i = 2 max i {λ M (Θ i )} min i {λ m (P i )} . 89 Exponential Stability of Uncertain Switched System with Time-Varying Delay For i ∈ S s , from (13), (14) and (40), we have ˙ V i (x t ) ≤ N ∑ i=1 λ i (t)  x (t) x(t −h(t))  T Θ i  x (t) x(t −h(t))  −2βV 2,i (x t ) −( 2β + 1 h M )(V 3,i (x t )+V 4,i (x t )). (43) Similar to Theorem 3.1, from (34) and (43), we get V i (x t ) ≤ N ∑ i=1 λ i (t)  V i (x t 0 )  e −ζ i (t−t 0 ) , t ≥ t 0 . (44) where ζ i = min{ min i {λ m (−Θ i )} max i {λ M (P i )} ,2β}. In general, from (39), (42) and (44), with the same argument as in the proof of Theorem 3.1, we get V i (x t ) ≤ l(t) ∏ m=1 ψe λ + (t m −t m−1 ) × N(t)−1 ∏ n=l(t)+1 ψe ζ i n h M e −λ − (t n −t n−1 ) ×V i 0 (x t 0 )  e −λ − (t−t N(t)−1 ) , t ≥ t 0 . Using (35), we have V i (x t ) ≤ l(t) ∏ m=1 ψ × N(t)−1 ∏ n=l(t)+1 ψe ζ i n h M ×V i 0 (x t 0 )  e −λ ∗ (t−t 0 ) , t ≥ t 0 . By (36) and (37), we get V i (x t ) ≤ V i 0 (x t 0 )  e −(λ ∗ −ν)(t−t 0 ) , t ≥ t 0 . Thus, by (38), we have  x(t) ≤  α 3 α 1  x t 0  e − 1 2 (λ ∗ −ν)(t−t 0 ) , t ≥ t 0 , which concludes the proof of the Theorem 3.3.  4. Numerical examples Example 4.1 Consider linear switched system (1) with time-varying delay but without matrix uncertainties and without nonlinear perturbations. Let N = 2, S u = {1}, S s = {2}.Let the delay function be h (t)=0.51 sin 2 t.Wehaveh M = 0.51, μ = 1.02, λ(A 1 + B 1 )= 0.0046, −0.0399, λ(A 2 )=−0.2156, 0.0007. Let β = 0.5. Since one of the eigenvalues of A 1 + B 1 is negative and one of eigenvalues of A 2 is positive, we can’t use results in (Alan & Lib, 2008) to consider stability of switched system (1). By using the LMI toolbox in Matlab, we have matrix solutions of (5) for unstable subsystems and (6) for stable subsystems as the following: For unstable subsystems, we get 90 Time-Delay Systems P 1 =  41.6819 0.0001 0.0001 41.5691  , Q 1 =  24.7813 −0.0002 −0.0002 24.7848  , R 1 =  33.1027 −0.0001 −0.0001 33.1044  , S 11,1 =  33.1027 −0.0001 −0.0001 33.1044  , S 12,1 =  −0.0372 −0.0023 −0.0023 0.7075  , S 22,1 =  50.0412 0.0001 0.0001 50.0115  , T 1 =  41.7637 −0.0001 −0.0001 41.7920  . For stable subsystems, we get P 2 =  71.8776 2.3932 2.3932 110.8889  , Q 2 =  7.2590 −0.3265 −0.3265 0.8745  , R 2 =  10.4001 −0.4667 −0.4667 1.2806  , S 11,2 =  12.7990 −0.4854 −0.4854 3.5031  , S 12,2 =  −3.1787 0.0240 0.0240 −2.8307  , S 22,2 =  4.6346 −0.0289 −0.0289 4.0835  , T 2 =  16.9964 0.0394 0.0394 17.7152  , X 11,2 =  17.2639 −0.1536 −0.1536 14.2310  , X 12,2 =  −9.6485 −0.1466 −0.1466 −12.5573  , X 22,2 =  16.9716 −0.1635 −0.1635 13.8095  , Y 2 =  −3.4666 −0.1525 −0.1525 −6.3485  , Z 2 =  6.8776 −0.0574 −0.0574 5.7924  . By straight forward calculation, the growth rate is λ + = ξ = 2.8291, the decay rate is λ − = ζ = 0.0063, λ(Ω 1,1 )=25.8187, 25.8188, 58.7463, 58.8011, λ(Ω 2,2 )=−10.1108, −3.7678, −2.0403, −0.7032 and λ(Ω 3,2 )=1.4217, 4.2448, 5.4006, 9.1514, 29.3526, 30.0607. Thus, we may take λ ∗ = 0.0001 and ν = 0.00001. Thus, from inequality (7),wehaveT − ≥ 456.3226 T + .By choosing T + = 0.1, we get T − ≥ 45.63226. We choose the following switching rules: (i) for t ∈ [0, 0.1) ∪ [50, 50.1) ∪ [100, 100.1) ∪ [150, 150.1) ∪ , subsystem i = 1 is activated. (ii) for t ∈ [0.1, 50) ∪[50.1, 100) ∪[100.1, 150) ∪ [150.1, 200) ∪ , subsystem i = 2 is activated. Then, by Theorem 3.1, the switching system (1) is exponentially stable. Moreover, the solution x (t) of the system satisfies  x(t) ≤ 11.8915e −0.000045t , t ∈ [0, ∞). The trajectories of solution of switched system switching between the subsystems i = 1and i = 2 are shown in Figure 1, Figure 2 and Figure 3, respectively. 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time x1,x2 x1 x2 Fig. 1. The trajectories of solution of linear switched system. 91 Exponential Stability of Uncertain Switched System with Time-Varying Delay 0 10 20 30 40 50 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 time x1,x2 x1 x2 Fig. 2. The trajectories of solution of subsystem i = 1. 0 50 100 150 200 −0.05 0 0.05 0.1 0.15 0.2 time x1,x2 x1 x2 Fig. 3. The trajectories of solution of subsystem i = 2. Example 4.2 Consider uncertain switched system (1) with time-varying delay and nonlinear perturbation. Let N = 2, S u = {1}, S s = {2} where A 1 =  0.1130 0.00013 0.00015 −0.0033  , B 1 =  0.0002 0.0012 0.0014 −0.5002  , A 2 =  −5.5200 1.0002 1.0003 −6.5500  , B 2 =  0.0245 0.0001 0.0001 0.0237  , E 1i = E 2i =  0.2000 0.0000 0.0000 0.2000  , H 1i = H 2i =  0.1000 0.0000 0.0000 0.1000  , i = 1, 2, F 1i = F 2i =  sin t 0 0sint  , i = 1, 2, 92 Time-Delay Systems f 1 (t, x(t), x(t − h(t))) =  0.1x 1 (t) sin(x 1 (t)) 0.1x 2 (t −h(t)) cos( x 2 (t))  , f 2 (t, x(t), x(t − h(t))) =  0.5x 1 (t) sin(x 1 (t)) 0.5x 2 (t −h(t)) cos( x 2 (t))  . From  f 1 (t, x(t), x(t − h(t)))  2 =[0.1x 1 (t) sin(x 1 (t))] 2 +[0.1x 2 (t − h(t)) cos(x 2 (t))] 2 ≤ 0.01x 2 1 (t)+0.01x 2 2 (t −h(t)) ≤ 0.01  x(t)  2 +0.01  x (t − h (t))  2 ≤ 0.01[ x(t)  +  x(t −h(t)) ] 2 , we obtain  f 1 (t, x(t), x(t −h(t))) ≤ 0.1  x(t)  +0.1  x(t −h(t))  . The delay function is chosen as h (t)=0.25 sin 2 t.From  f 2 (t, x(t), x(t − h(t)))  2 =[0.5x 1 (t) sin(x 1 (t))] 2 +[0.5x 2 (t − h(t)) cos(x 2 (t))] 2 ≤ 0.25x 2 1 (t)+0.25x 2 2 (t −h(t)) ≤ 0.25  x(t)  2 +0.25  x (t − h (t))  2 ≤ 0.25[ x(t)  +  x(t −h(t)) ] 2 , we obtain  f 2 (t, x(t), x(t −h(t))) ≤ 0.5  x(t)  +0.5  x(t −h(t))  . We may take h M = 0.25, and from (4), we take γ 1 = 0.1, δ 1 = 0.1, γ 2 = 0.5, δ 2 = 0.5. Note that λ (A 1 )=0.11300016, −0.00330016. Let β = 0.5, μ = 0.5. Since one of the eigenvalues of A 1 is negative, we can’t use results in (Alan & Lib, 2008) to consider stability of switched system (1). From Lemma 2.4 , we have the matrix solutions of (33) for unstable subsystems and of (34) for stable subsystems by using the LMI toolbox in Matlab as the following: For unstable subsystems, we get ε 31 = 0.8901, ε 41 = 0.8901, ε 51 = 0.8901, P 1 =  0.2745 −0.0000 −0.0000 0.2818  , Q 1 =  0.4818 −0.0000 −0.0000 0.5097  , R 1 =  0.8649 −0.0000 −0.0000 0.8729  , S 11,1 =  0.8649 −0.0000 −0.0000 0.8729  , S 12,1 = 10 −4 ×  −0.1291 −0.8517 −0.8517 0.1326  , S 22,1 =  1.0877 −0.0000 −0.0000 1.0902  . For stable subsystems, we get ε 32 = 2.0180, ε 42 = 2.0180, ε 52 = 2.0180, P 2 =  0.2741 0.0407 0.0407 0.2323  , Q 2 =  1.3330 −0.0069 −0.0069 1.3330  , R 2 =  1.0210 −0.0002 −0.0002 1.0210  , S 11,2 =  1.0210 −0.0002 −0.0002 1.0210  , S 12,2 =  −0.0016 −0.0002 −0.0002 −0.0016  , S 22,2 =  0.8236 −0.0006 −0.0006 0.8236  . By straight forward calculation, the growth rate is λ + = ξ = 8.5413, the decay 93 Exponential Stability of Uncertain Switched System with Time-Varying Delay rate is λ − = ζ = 0.1967, λ(Θ 1 )=0.1976, 0.2079, 1.1443, 1.1723 and λ(Θ 2 )= − 0.7682, −0.6494, −0.0646, −0.0588. Thus, we may take λ ∗ = 0.0001 and ν = 0.00001. Thus, from inequality (35),wehaveT − ≥ 43.4456 T + . By choosing T + = 0.1, we get T − ≥ 4.34456. We choose the following switching rules: (i) for t ∈ [0, 0.1) ∪ [5.0, 5.1) ∪ [10.0, 10.1) ∪ [15.0, 15.1) ∪ , system i = 1 is activated. (ii) for t ∈ [0.1, 5.0) ∪ [5.1, 10.0) ∪ [10.1, 15.0) ∪ [15.1, 20.0) ∪ , system i = 2 is activated. Then, by theorem 3.3.1, the switched system (1) is exponentially stable. Moreover, the solution x (t) of the system satisfies  x(t) ≤ 1.8770e −0.000045t , t ∈ [0, ∞). The trajectories of solution of switched system switching between the subsystems i = 1and i = 2 are shown in Figure 4, Figure 5 and Figure 6, respectively. 0 5 10 15 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time x1,x2 x1 x2 Fig. 4. The trajectories of solution of switched system with nonlinear perturbations 5. Conclusion In this paper, we have studied the exponential stability of uncertain switched system with time varying delay and nonlinear perturbations. We allow switched system to contain stable and unstable subsystems. By using a new Lyapunov functional, we obtain the conditions for robust exponential stability for switched system in terms of linear matrix inequalities (LMIs) which may be solved by various algorithms. Numerical examples are given to illustrate the effectiveness of our theoretical results. 6. Acknowledgments This work is supported by Center of Excellence in Mathematics and the Commission on Higher Education, Thailand. We also wish to thank the National Research University Project under Thailand’s Office of the Higher Education Commission for financial support. 94 Time-Delay Systems 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time x1,x2 x1 x2 Fig. 5. The trajectories of solution of system i = 1 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time x1,x2 x1 x2 Fig. 6. The trajectories of solution of system i = 2 7. References Alan, M.S. & Lib, X. (2008). On stability of linear and weakly nonlinear switched systems with time delay, Math. Comput. Modelling, 48, 1150-1157. Alan, M.S. & Lib, X. (2009). Stability of singularly perturbed switched system with time delay and impulsive effects, Nonlinear Anal., 71, 4297-4308. Alan, M.S., Lib, X. & Ingalls, B. (2008). Exponential stability of singularly perturbed switched systems with time delay, Nonlinear Analysis:Hybrid systems, 2, 913-921. Boyd, S. et al. (1994). 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Exponential stability of recurrent neural networks with time-varying discrete and distributed delays, Nonlinear A nalysis. Real World Appl., 10, 2581-2589. Li, P., Zhong, S.M. & Cui, J.Z. (2009). Stability analysis of linear switching systems with time delays, Chaos Solitons Fractals, 40, 474-480. Lien C.H. et al. (2009). Exponential stability analysis for uncertain switched neutral systems with interval-time-varying state delay, Nonlinear Analysis:Hybrid sys- tems,3, 334-342. Lib, J., Lib, X. & Xie, W.C. (2008). Delay-dependent robust control for uncertain switched systems with time delay, Nonlinear Analysis:Hybrid systems, 2, 81-95. Niamsup, P. (2008). Controllability approach to H ∞ control problem of linear time- varying switched systems, Nonlinear Analysis:Hybrid systems, 2, 875-886. Phat, V.N., Botmart, T. & Niamsup, P. (2009). Switching design for exponential stability of a class of nonlinear hybrid time-delay systems, Nonlinear An alysis:Hybrid systems,3,1-10. Wu, M. et al. (2004). Delay-dependent criteria for robust stability of time varying delay systems, Automatica J. IFAC, 40, 1435-1439. Xie, G. & Wang, L. (2006). Quadratic stability and stabilization of discrete time switched systems with state delay, Proceedings of American Control Conf., Minnesota, USA, pp.1539- 1543. Xu, S. et al. (2005). Delay-dependent exponential stability for a class of neural networks with time delays, J. Comput. Appl. Math., 183, 16-28. Zhang, Y., Lib, X. & Shen X. (2007). Stability of switched systems with time delay, Nonlinear Analysis:Hybrid systems, 1, 44-58. 96 Time-Delay Systems Alexander Stepanov Synopsys GmbH, St Petersburg representative office Russia 1. Introduction Problems of stabilization and determining of stablility characteristics of steady-state regimes are among the central in a control theory. Especial difficulties can be met when dealing with the systems containing nonlinearities which are nonanalytic function of phase. Different models describing nonlinear effects in real control systems (e.g. servomechanisms, such as servo drives, autopilots, stabilizers etc.) are just concern this type, numerous works are devoted to the analysis of problem of stable oscillations presence in such systems. Time delays appear in control systems frequently and are important due to significant impact on them. They affect substantially on stability properties and configuration of steady state solutions. An accurate simultaneous account of nonlinear effects and time delays allows to receive adequate models of real control systems. This work contains some results concerning to a stability problem for periodic solutions of nonlinear controlled system containing time delay. It corresponds further development of an article: Kamachkin & Stepanov (2009). Main results obtained below might generally be put in connection with classical results of V.I. Zubov’s control theory school (see Zubov (1999), Zubov & Zubov (1996)) and based generally on work Zubov & Zubov (1996). Note that all examples presented here are purely illustrative; some examples concerning to similar systems can be found in Petrov & Gordeev (1979), Varigonda & Georgiou (2001). 2. Models under consideration Consider a system ˙ x = Ax + cu (t − τ),(1) here x = x(t) ∈ E n , t ≥ t 0 ≥ τ, A is real n × n matrix, c ∈ E n ,vectorx(t), t ∈ [t 0 − τ, t 0 ],is considered to be known. Quantity τ > 0 describes time delay of actuator or observer. Control statement u is defined in the following way: u (t − τ)= f ( σ(t − τ) ) , σ(t − τ)=γ  x(t − τ), γ ∈ E n , γ = 0; nonlinearity f can, for example, describe a nonideal two-position relay with hysteresis: f (σ)=  m 1 , σ < l 2 , m 2 , σ > l 1 , (2) On Stable Periodic Solutions of One Time Delay System Containing Some Nonideal Relay Nonlinearities 5 here l 1 < l 2 , m 1 < m 2 ;andf (σ(t)) = f − = f (σ(t − 0)) if σ ∈ [l 1 ; l 2 ]. In addition to the nonlinearity (2) a three-position relay with hysteresis will be considered: f (σ)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0,  | σ | ≤ l 0 , | σ | ∈ ( l 0 ; l ] , f − = 0; m 1 ,  σ ∈ [ −l; −l 0 ) , f − = m 1 , σ < −l; m 2 ,  σ ∈ ( l 0 ; l ] , f − = m 2 , σ > l; (3) (here m 1 < m < m 2 ,0< l 0 < l); Suppose that hysteresis loops for the nonlinearities are walked around in counterclockwise direction. 3. Stability of periodic solutions Denote x(t − t 0 , x 0 , u) solution of the system (1) for unchanging control law u and initial conditions (t 0 , x 0 ). Let the system (1), (3) has a periodic solution with four switching points ˆ s i such as ˆ s 1 = x ( T 4 , ˆ s 4 , m 2 ) , ˆ s 2 = x ( T 1 , ˆ s 1 ,0 ) , ˆ s 3 = x ( T 2 , ˆ s 2 , m 1 ) , ˆ s 4 = x ( T 3 , ˆ s 3 ,0 ) . Let s i , i = 1, 4 are points of this solution (preceding to the corresponding ˆ s i )suchas γ  s 1 = l 0 , γ  s 2 = −l, γ  s 3 = −l 0 , γ  s 4 = l, (let us name them ˇ Tpre-switching points ˇ T, for example), and ˆ s 1 = x ( τ, s 1 , m 2 ) , ˆ s 2 = x ( τ, s 2 ,0 ) , ˆ s 3 = x ( τ, s 3 , m 1 ) , ˆ s 4 = x ( τ, s 4 ,0 ) , or ˆ s i+1 = x ( T i , ˆ s i , u i ) , ˆ s i = x ( τ, s i , u i−1 ) , where u 1 = 0, u 2 = m 1 , u 3 = 0, u 4 = m 2 (hereafter suppose that indices are cyclic, i.e. for i = 1, m one have i + 1 = 1ifi = m and i − 1 = m if i = 1). Denote v i = As i+1 + cu i , k i = γ  v i . Theorem 1. Let k i = 0 and  M  < 1,where M = 1 ∏ i=4 M i , M i =  I − k −1 i v i γ   e AT i , then the periodic solution under consideration is orbitally asymptotically stable. 98 Time-Delay Systems [...]... 1.08072 ˆ ˆ ˆ s4 ≈ ⎝ −1.95858 ⎠ , s5 ≈ ⎝ −0.87355 ⎠ , s6 ≈ 3.43423 2.93 260 ⎛ ⎞ 0.72238 ⎝ −1.05935 ⎠ , 2.95759 ⎛ ⎞ 0.11909 ⎝ −1.4 465 0 ⎠ , 2.0 563 5 1 06 Time- Delay Systems T1 ≈ 1.8724, T2 ≈ 0.4018, T3 ≈ 6. 8301, T4 ≈ 0.4019, T5 ≈ 1 .60 87, T6 ≈ 0.4084 Let Γ = e− Aτ1 γ1 + e− Aτ2 τ1 ˆ l 1 = l 1 + γ1 0 ˆ l 2 = l 2 + γ1 then γ2 ≈ 0.55 260 7 −1.1444 96 −0.5849 56 , τ1 0 Denote 0 e− At cm1 dt + γ2 ˆ ˆ ˆ Γ s1 = Γ s3... f (−0. 565 x3 (t) − 1.11x2 (t − 0.015) + 0.54x1 (t − 0.1)) , where f is given by the (2) I.e τ1 = 0, γ1 = 0 0 −0. 565 , τ2 = 0.015, τ3 = 0.1, γ2 = 0 −1.11 0 , γ3 = 0.54 0 0 In such a case the system has a periodic solution with four switching points ˆ s1 ≈ 1.1250 −1. 066 2 3.3411 , ˆ s2 ≈ 0.18 06 −1.3848 2.0040 , ˆ s3 ≈ 0.7081 −0 .63 17 2. 067 2 , ˆ s4 ≈ 0.5502 −2.1717 3.9 062 , T1 ≈ 1. 566 8, T2 ≈ 0.38 46, T3 ≈... solution with four switching points; the pre-switching points are: ⎛ ⎛ ⎞ ⎞ 0. 468 349 0.0051 76 s1 ≈ ⎝ 0.497302 ⎠ , s2 ≈ ⎝ −0.00 063 3 ⎠ , s3 = − s1 , s4 = − s2 ; −0.0 063 07 0.5010 36 and T1 ≈ 53 .63 54, T2 ≈ 0.7973, T3 = T1 , T4 = T2 As M ≈ 0.0078 < 1, then the periodic solution is orbitally asymptotically stable 100 Time- Delay Systems ˆ Similarly, the system (1), (3) may have a periodic solution with a pair... Consider the system (6) , (2) Let τ1 = 0.013, τ2 = 0.015, ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ −0.25 −1 −0.25 1 0.5 36 0⎠, A = ⎝ 0.75 1 0.75 ⎠ , c = ⎝1⎠ , γ1 = ⎝ 0.25 −7 −3.75 1 0 m1,2 = ∓1, l1 = −0.1, ⎛ ⎞ 0 γ2 = ⎝ −1.108 ⎠ , −0. 567 l2 = 0.5 System (1), (2) has periodic solution with six switching points: ⎛ ⎛ ⎞ ⎞ 0 .69 484 0. 062 26 ˆ ˆ ˆ s1 ≈ ⎝ −0 .64 902 ⎠ , s2 ≈ ⎝ −1.91945 ⎠ , s3 ≈ 2.128 76 2.92801 ⎛ ⎛ ⎞ ⎞ 0.517 06 1.08072 ˆ ˆ ˆ s4... a case the system (4), (3) has periodic solution with pre-switching points ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 0 .68 19 0.1127 −0.0534 −0 .60 96 s1 ≈ ⎝ 0.5383 ⎠ , s2 ≈ ⎝ −0.0073 ⎠ , s3 ≈ ⎝ −0 .63 96 ⎠ , s4 ≈ ⎝ −0. 066 4 ⎠ , 0.0328 −0.0979 −0.5557 0.5070 T1 ≈ 42.2723, One can verify that k i = 0, and T2 ≈ 0.8977, T3 ≈ 33.5405, T4 ≈ 0.8 969 M ≈ 0.8223 < 1 So, the solution under consideration is orbitally asymptotically stable Note... Similar computations can be observed in case of nonlinearity (3) 104 Time- Delay Systems 5 Stability in case of multiple delays In more general case the system under consideration can also contain several nonlinearities or several positive delays τi (i = 1, k) in control loop: ˙ x (t) = Ax (t) + c f k ∑ γi x(t − τi ) , i =1 γi ∈ E n , γi = 0 (6) ˆ Let, for example, k = 2, τ1 = 0, τ2 = τ, denote γ = γ1 , γ... γ, ˆ ˆ li = li − γ τ 0 e Ai−1 ( τ −t) ci−1 dt 108 Time- Delay Systems Denote v i = A i s i +1 + c i , Mi = I − k−1 vi Γ i+1 e Ai Ti +( Ai−1 − Ai ) τ i k i = Γ i +1 v i , Theorem 5 Let k i = 0, i = 1, 4, and 1 ∏ Mi < 1, i =4 (9) then the periodic solution is orbitally asymptotically stable Let us skip the proof, it is similar to the above one Example 6 Let A, c, l1,2 , m1,2 are the same as in the example... can calculate l1,2 : ˆ l1 ≈ 1.110552, And, finally, ˆ l2 ≈ 1.087485 t1 = t2 ≈ 4. 460 6 06 The system under consideration has a T-periodic solution, T = ti Let s1 = 1 0 , then s2 ≈ 0.19809 1.02122 , T1 = T3 ≈ 1.07715, s3 = − s1 , T2 = T4 ≈ 1.15315; and k k ds1+1 = Mds1 , So, as s1,1 = 1, then ds1,1 = 0, s4 = − s4 , M= 00 1.1 362 1 k+ k ds1,2 1 = ds1,2 , and the periodic solution under consideration cannot... = 0.75, ⎛ ⎞ 1 c = ⎝1⎠ , 1 l = 1, ⎛ ⎞ 0.1 γ = ⎝ 0⎠, −1 m1,2 = ∓1 Then the system (1), (3) has a periodic solution with pre-switching points ⎛ ⎛ ⎞ ⎞ 0.2727 0 s1 = ⎝ 0.28 86 ⎠ , s2 = ⎝ 0 ⎠ , T1 = 149 .60 21, T2 = 0.7847, −0.7227 −1 M ≈ 0.92 86 < 1, and the solution is orbitally asymptotically stable 4 Some extensions (bilinear system, multiple control etc.) Consider a bilinear system ˙ x = Ax + (Cx + c) u... system (4), (2) has at least one periodic solution By the analogy with the system (1), a system with multiple controls can be observed: ˙ x = Ax + c1 u1 (σ1 (t − τ1 )) + c2 u2 (σ2 (t − τ2 )) (5) 102 Time- Delay Systems Suppose for simplicity that u i are simple hysteresis nonlinearities given by (2): u i (σ) = u (σ) = σi < l2 , σi > l1 , m1 , m2 , σi = γi x, i = 1, 2 Denote x (t − t0 , x0 , u1 , u2 ) solution . S 22,2 =  4 .63 46 −0.0289 −0.0289 4.0835  , T 2 =  16. 9 964 0.0394 0.0394 17.7152  , X 11,2 =  17. 263 9 −0.15 36 −0.15 36 14.2310  , X 12,2 =  −9 .64 85 −0.1 466 −0.1 466 −12.5573  , X 22,2 =  16. 97 16 −0. 163 5 −0. 163 5. 50.0115  , T 1 =  41. 763 7 −0.0001 −0.0001 41.7920  . For stable subsystems, we get P 2 =  71.87 76 2.3932 2.3932 110.8889  , Q 2 =  7.2590 −0.3 265 −0.3 265 0.8745  , R 2 =  10.4001 −0. 466 7 −0. 466 7 1.28 06  , S 11,2 =  12.7990 −0.4854 −0.4854. with time delays, J. Comput. Appl. Math., 183, 16- 28. Zhang, Y., Lib, X. & Shen X. (2007). Stability of switched systems with time delay, Nonlinear Analysis:Hybrid systems, 1, 44-58. 96 Time- Delay