finite time boundedness analysis for a class of switched linear systems with time varying delay

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 982414, pages http://dx.doi.org/10.1155/2014/982414 Research Article Finite-Time Boundedness Analysis for a Class of Switched Linear Systems with Time-Varying Delay Yanke Zhong and Tefang Chen School of Information Science and Engineering, Central South University, Changsha 410075, China Correspondence should be addressed to Yanke Zhong; zhongyanke1981@163.com Received 21 October 2013; Accepted January 2014; Published 13 February 2014 Academic Editor: Valery Y Glizer Copyright © 2014 Y Zhong and T Chen This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The problem of finite-time boundedness for a class of switched linear systems with time-varying delay and external disturbance is investigated First of all, the multiply Lyapunov function of the system is constructed Then, based on the Jensen inequality approach and the average dwell time method, the sufficient conditions which guarantee the system is finite-time bounded are given Finally, an example is employed to verify the validity of the proposed method Introduction The switched system is a special kind of hybrid dynamic system, composed of a family of subsystems and a switching law specifying the switches between subsystems [1, 2] The fact that the structure and working mechanism of the switched system are more complex than general systems leads to that the switched system possesses much richer dynamic characteristics The switched systems are widely applied in engineering practice, such as power system control, robot control, network control, and so forth [3–9] In practice, switched systems are commonly subjected to time-delay and external disturbance Due to their significant impact on the performances of switched systems, many scholars have been attracted to investigate the problem Sun et al analyzed the asymptotic stability of the switched linear system with time-delay perturbation by using common Lyapunov function and multiple Lyapunov function [3] Lu and Zhao also investigated the asymptotic stability for switched linear systems with time-delay and proposed an effective method which can direct researchers to choose an appropriate switching law to make sure the system is asymptotic stable [10] Zhao and Zhang studied the stability of the switched system with time-varying delays based on the average dwell time and time-delay decomposition approaches [11] For switched systems with time-varying delay, Lian et al utilized the Lyapunov-Krasovskii function method to design H infinity filter [12] For switched systems affected by the nonlinear impact and disturbance, Sun used transfer matrix estimation and Gronwall inequality methods to design a feedback law stabilizing system [13] For the switched system with fixed time-delay and norm bounded disturbance, Lin et al proposed the finite-time boundedness concept and a method to judge whether the system is finite-time bounded [14] Up to now, to the best of the authors’ knowledge, there are a few papers concerning the finite-time boundedness problem of switched system For switched systems with timevarying delay and external disturbance, the problem has not yet been discussed by any literature However, in practical engineering, the time-delays are generally changeable over time, not fixed In addition, many practical systems are just required that their state trajectories are bounded over a fixed interval In other words, those systems may be unstable On the contrary, although some systems are asymptotically stable, they cannot meet the application requirements because of their large transient state amplitudes Considering the wide application of switched systems with time-varying delay and the requirements for transient behaviors in engineering fields, it is a significant task to investigate finite-time boundedness for switched linear systems with time-varying delay and external disturbance The main contributions in this paper are Abstract and Applied Analysis listed as follows (1) For the convenience of processing, a concise definition on the finite-time boundedness is proposed for the switched system (2) Sufficient conditions of finite-time boundedness for switched linear systems with time-varying delay and external disturbance are given in a limited time interval which implies that the frequency of switching signal is not infinite Definition (see [15]) For 𝑇 ≥ 𝑡 ≥ 0, let 𝑁𝜎 (𝑡, 𝑇) denote the switching number of 𝜎(𝑡) over (𝑡, 𝑇] If 𝑁𝜎 (𝑡, 𝑇) ≤ 𝑁0 + Preliminaries and Problem Formulation Consider the following switched linear system with timevarying delay and external disturbance: 𝑥̇ (𝑡) = 𝐴 𝜎(𝑡) 𝑥 (𝑡) + 𝐵𝜎(𝑡) 𝑥 (𝑡 − ℎ (𝑡)) + 𝐺𝜎(𝑡) 𝑤 (𝑡) , 𝑥 (𝑡) = 𝜑 (𝑡) , ℎ (t) ≥ 0, 𝑡 ≥ 0, 󵄨󵄨 ̇ 󵄨󵄨 max 󵄨󵄨𝜑 (𝑡)󵄨󵄨 ≤ 𝜌, 𝜌 ≥ 0, 𝑡 ∈ [−𝑑, 0) , 𝑑 ≥ ℎ (0) , (1) 𝑤𝑇 (𝑡) 𝑤 (𝑡) 𝑑𝑡 ≤ 𝛾, 𝛾 ≥ 0, ∀𝑡 > (2) Assumption For the time-varying delay, the following inequalities hold: ℎ (𝑡) ≥ 0, ℎ̇ (𝑡) ≤ 𝑘, 𝑘 < 1, ℎ (𝑡) ≤ ℎmax , holds for 𝜏𝑎 ≥ and an integer 𝑁0 ≥ 0, then 𝜏𝑎 is called average dwell time 𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 ) ≤ 𝑐1 󳨐⇒ 𝑥𝑇 (𝑡) 𝑥 (𝑡) < 𝑐2 , 𝑐1 < 𝑐2 , ∀𝑡 ∈ [0, 𝑇𝑓 ] , Assumption (see [14]) The value of external disturbance changes over time, but it satisfies +∞ (4) Definition For a given four positive constants 𝑐1 , 𝑐2 , 𝑇𝑓 , 𝛾, and a switching signal 𝜎(𝑡), if where 𝑥(𝑡) is state variable and 𝜎(𝑡) is the switching law which is a piecewise continuous function with 𝜎(𝑡) ∈ 𝑀 = [1, 2, , 𝑚] which means the switched system is consisted of 𝑚 subsystems The 𝑖th subsystem is activated when 𝜎(𝑡) = 𝑖 ⋅ 𝐴 𝜎(𝑡) , 𝐵𝜎(𝑡) , and 𝐺𝜎(𝑡) are constant matrices ℎ(𝑡) represents time-varying delay 𝑤(𝑡) stands for external disturbance 𝜑(𝑡) is the continuous vector-valued initial function on ̇ denotes the derivative of 𝜑(𝑡) ⋅ 𝜌 is a positive 𝑡 ∈ [−𝑑, 0) ⋅ 𝜑(𝑡) constant For the convenience of subsequent processing, assume that the system (1) satisfies the following assumptions ∫ 𝑇−𝑡 𝜏𝑎 (5) 𝑇𝑓 ∀𝑤 (𝑡) : ∫ 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑡 ≤ 𝛾, then the system (1) is said to be finite-time bounded Where 𝑥𝑇 (𝑡0 )𝑥(𝑡0 ) = sup−𝑑≤𝑡≤0 {𝑥𝑇 (𝑡)𝑥(𝑡)}, without loss of generality, specify 𝑐1 = sup−𝑑≤𝑡≤0 {𝑥𝑇 (𝑡)𝑥(𝑡)} Remark Definition implies that if the system (1) is finitetime bounded, the state remains within the prescribed bound in the fixed interval Notice that finite-time boundedness is different from asymptotic stability The system which is finite-time bounded may not be asymptotically stable while a system is asymptotically stable does not mean it is finitetime bounded either In a word, there is no necessary relation between them Remark The definition of finite-time boundedness in this paper is much more concise than that in [14] However, they are consistent in essence By using the definition in this paper, some complex matrix transformations can be avoided in the subsequent mathematical processing (3) where 𝑘 and ℎmax are positive constants Assumption The system state variable does not “jump” at switching instant, that is to say the state trajectory is continuous In addition, the switching number of 𝜎(𝑡) is finite Main Result Theorem For system (1), for all 𝑖 ∈ 𝑀 and for all 𝑡 ∈ [0, 𝑇𝑓 ], assume there exists symmetric positive matrixes 𝑃𝑖 , 𝑅𝑖1 , 𝑅𝑖2 , 𝑄𝑖 , 𝑍𝑖1 , 𝑍𝑖2 , and 𝐻 and positive constants 𝛼, 𝛽 ≥ such that 𝛼𝑑/2 𝑑 𝑑 𝑃𝑖 𝐵𝑖 + 𝐴𝑇𝑖 𝑍𝑖 𝐵𝑖 𝑃𝑖 𝐺𝑖 + 𝐴𝑇𝑖 𝑍𝑖 𝐺𝑖 𝑒 𝑍𝑖,1 𝜉 [ 11 ] 𝑑 [ ] 𝑑 𝑇 [ ∗ 𝑑 𝐵𝑇 𝑍 𝐵 − − 𝑘) 𝑄 ] 0 𝑍 𝐺 𝐵 (1 [ ] [ ] 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 [ ] 𝛼𝑑 𝛼𝑑/2 𝛼𝑑/2 [∗ ] < ∗ 𝑒 𝑅 − 𝑍 𝑍 𝑒 𝑒 𝑖 𝑖 𝑖,2 [ ] 𝑑 𝑑 [ ] [ ] 𝛼𝑑 𝛼𝑑 [∗ ] ∗ ∗ −𝑒 𝑅𝑖,2 − 𝑒 𝑍𝑖,2 [ ] 𝑑 [ ] 𝑑 𝑇 ∗ ∗ ∗ ∗ (𝐺𝑖 𝑍𝑖 𝐺𝑖 − 𝐻) [ ] (6) Abstract and Applied Analysis If the average dwell time satisfies 𝜏𝑎 < 𝑉𝑖,3 (𝑡) = ∫ 𝑡−ℎ(𝑡) 𝑇𝑓 ln 𝛽 ln 𝐶1 + ln 𝜆 − ln (𝜂1 𝐶1 + 𝜂2 ) − 𝛼𝑇𝑓 , (7) 𝑉𝑖,4 (𝑡) = ∫ 𝑡 −𝑑/2 𝑡+𝜃 +∫ 𝜉11 = 𝐴𝑇𝑖 𝑃𝑖 + 𝑃𝑖 𝐴 𝑖 + 𝑅𝑖,1 + 𝑄𝑖 −𝑑 𝑑 + 𝐴𝑇𝑖 (𝑍𝑖,1 + 𝑍𝑖,2 ) 𝐴 𝑖 − 𝑒𝛼𝑑/2 − 𝛼𝑃𝑖 , 𝑑 𝑅𝑖 = 𝑅𝑖,1 + 𝑅𝑖,2 , 𝑅𝑖,1 ≤ 𝛽𝑅𝑗,1 , 𝑍𝑖,1 ≤ 𝛽𝑍𝑗,1 , 𝑅𝑖,2 ≤ 𝛽𝑅𝑗,2 , 𝑍𝑖,2 ≤ 𝛽𝑍𝑗,2 , 𝜆 = max {𝜆 max (𝑃𝑖 )} , 𝑃𝑖 ≤ 𝛽𝑃𝑗 , 𝜆 = max {𝜆 max (𝑄𝑖 )} , 𝜆 = max {𝜆 max (𝑍𝑖,1 )} , 𝜆 = max {𝜆 max (𝑍𝑖,2 )} , 𝜂1 = 𝜆 + (10) Furthermore, it follows that 𝑖∈𝑀 ̇ (𝑡) − 𝛼𝑉𝑖,1 = 𝑥𝑇 (𝑡) [𝐴𝑇 𝑃𝑖 + 𝑃𝑖 𝐴 𝑖 ] 𝑥 (𝑡) 𝑉𝑖,1 𝑖 𝜆 = {𝜆 (𝑃𝑖 )} , 𝑖∈𝑀 + 𝑥𝑇 (𝑡 − ℎ (𝑡)) 𝐵𝑖𝑇 𝑃𝑖 𝑥 (𝑡) + 𝑤𝑇 (𝑡) 𝐺𝑖𝑇 𝑃𝑖 𝑥 (𝑡) + 𝑥𝑇 (𝑡) 𝑃𝑖 𝐵𝑖 𝑥 (𝑡 − ℎ (𝑡)) + 𝑥𝑇 (𝑡) 𝑃𝑖 𝐺𝑖 𝑤 (𝑡) 𝑑2 𝑑2 𝑑 𝜂2 = 𝜌2 𝜆 𝑒𝛼𝑑/2 + 𝜌2 𝜆 𝑒𝛼𝑑 + 𝜆 𝛾, 2 − 𝛼𝑥𝑇 (𝑡) 𝑃𝑖 𝑥 (𝑡) , 𝑇 𝐶1 = sup {𝑥 (𝑡0 ) 𝑥 (𝑡0 )} , −𝑑≤𝑡0 ≤0 + 𝑥𝑇 (𝑡 − 𝑑2 𝛼𝑑/2 𝑑2 𝛼𝑑 + 𝜌 𝜆6𝑒 𝜌 𝜆5𝑒 The left of inequality (6) is a symmetric matrix Thus, the symmetric terms are denoted by “∗” 𝜆 max (𝑃𝑖 ) represents the maximum eigenvalue of 𝑃𝑖 Proof Construct the multiply Lyapunov function as follows: 𝑉𝑖,1 (𝑡) = 𝑥𝑇 (𝑡) 𝑃𝑖 𝑥 (𝑡) , 𝑉𝑖,2 (𝑡) = ∫ 𝑡−(𝑑/2) +∫ 𝑥 (𝑠) 𝑒 𝑡−(𝑑/2) 𝑡−𝑑 −𝛼(𝑠−𝑡) 𝑅𝑖,1 𝑥 (𝑠) 𝑑𝑠 𝑥𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑅𝑖,2 𝑥 (𝑠) 𝑑𝑠, (12) ̇ (𝑡) = 𝛼𝑉𝑖,3 (𝑡) + 𝑥𝑇 (𝑡) 𝑄𝑖 𝑥 (𝑡) 𝑉𝑖,3 (8) 𝑉 (𝑡) = 𝑉𝑖 (𝑡) = 𝑉𝑖,1 (𝑡) + 𝑉𝑖,2 (𝑡) + 𝑉𝑖,3 (𝑡) + 𝑉𝑖,4 (𝑡) , 𝑑 𝛼𝑑/2 𝑑 [𝑅𝑖,2 − 𝑅𝑖,1 ] 𝑥 (𝑡 − ) )𝑒 2 − 𝑥𝑇 (𝑡 − 𝑑) 𝑒𝛼𝑑 𝑅𝑖,2 𝑥 (𝑡 − 𝑑) , 𝑑 −1 + 𝑒𝛼𝑇𝑓 𝜆 𝛾)) (𝜆 ) 𝑇 (11) ̇ (𝑡) = 𝛼𝑉𝑖,2 (𝑡) + 𝑥𝑇 (𝑡) 𝑅𝑖,1 𝑥 (𝑡) 𝑉𝑖,2 𝐶2 = (𝛽𝑁𝑒𝛼𝑇𝑓 𝜂1 𝐶1 + 𝛽𝑁𝑒𝛼𝑇𝑓 𝑡 (9) + 𝑥𝑇 (𝑡) 𝑃𝑖 𝐵𝑖 𝑥 (𝑡 − ℎ (𝑡)) + 𝑥𝑇 (𝑡) 𝑃𝑖 𝐺𝑖 𝑤 (𝑡) 𝑑 𝛼𝑑/2 𝑑 𝑒 𝜆 + 𝑒𝛼𝑑 𝜆 + ℎmax 𝑒𝛼ℎmax 𝜆 , 2 ×( 𝑥̇𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑍𝑖,2 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝑖∈𝑀 𝑖∈𝑀 𝜆 = 𝜆 max (𝐻) , 𝑡+𝜃 + 𝑥𝑇 (𝑡 − ℎ (𝑡)) 𝐵𝑖𝑇 𝑃𝑖 𝑥 (𝑡) + 𝑤𝑇 (𝑡) 𝐺𝑖𝑇 𝑃𝑖 𝑥 (𝑡) 𝑖, 𝑗 ∈ [1, 2, , 𝑚] , 𝑖∈𝑀 𝑖∈𝑀 𝑡 ̇ (𝑡) = 𝑥𝑇 (𝑡) [𝐴𝑇 𝑃𝑖 + 𝑃𝑖 𝐴 𝑖 ] 𝑥 (𝑡) 𝑉𝑖,1 𝑖 𝑄𝑖 ≤ 𝛽𝑄𝑗 𝜆 = max {𝜆 max (𝑅𝑖,2 )} , ∫ Calculate the derivatives of 𝑉𝑖,1 (𝑡), 𝑉𝑖,2 (𝑡), 𝑉𝑖,3 (𝑡), and 𝑉𝑖,4 (𝑡) as 𝜆 = max {𝜆 max (𝑅𝑖,1 )} , 𝑖∈𝑀 −𝑑/2 𝑥𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑄𝑖 𝑥 (𝑠) 𝑑𝑠, 𝑥̇𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑍𝑖,1 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 ∫ then system (1) is finite-time bounded, where 𝑍𝑖 = 𝑍𝑖,1 + 𝑍𝑖,2 , 𝑡 − 𝑥𝑇 (𝑡 − ℎ (𝑡)) (1 − ℎ̇ (𝑡)) × 𝑒𝛼ℎ(𝑡) 𝑄𝑖 𝑥 (𝑡 − ℎ (𝑡)) ≤ 𝛼𝑉𝑖,3 (𝑡) + 𝑥𝑇 (𝑡) 𝑄𝑖 𝑥 (𝑡) − 𝑥𝑇 (𝑡 − ℎ (𝑡)) (1 − 𝑘) 𝑒𝛼ℎ(𝑡) 𝑄𝑖 𝑥 (𝑡 − ℎ (𝑡)) ≤ 𝛼𝑉𝑖,3 (𝑡) + 𝑥𝑇 (𝑡) 𝑄𝑖 𝑥 (𝑡) − 𝑥𝑇 (𝑡 − ℎ (𝑡)) (1 − 𝑘) 𝑄𝑖 𝑥 (𝑡 − ℎ (𝑡)) , (13) ̇ (𝑡) = 𝛼𝑉𝑖,4 (𝑡) + 𝑑 𝑥̇𝑇 (𝑡) [𝑍𝑖,1 + 𝑍𝑖,2 ] 𝑥̇ (𝑡) 𝑉𝑖,4 −∫ 𝑡 𝑡−(𝑑/2) 𝑥̇𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑍𝑖,1 𝑥̇ (𝑠) 𝑑𝑠 Abstract and Applied Analysis 𝑡−(𝑑/2) 𝑥𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑍𝑖,2 𝑥 (𝑠) 𝑑𝑠 −∫ 𝑡−𝑑 = 𝛼𝑉𝑖,4 (𝑡) + ≥ 𝑑 [𝐴 𝑥 (𝑡) + 𝐵𝑖 𝑥 𝑖 𝑇 𝑑 [𝑥 (𝑡 − ) − 𝑥 (𝑡 − 𝑑)] 𝑑 × 𝑒𝛼𝑑 𝑍𝑖,2 [𝑥 (𝑡 − × (𝑡 − ℎ (𝑡)) + 𝐺𝑖 𝑤 (𝑡)] 𝑑 ) − 𝑥 (𝑡 − 𝑑)] 𝑇 × [𝑍𝑖,1 + 𝑍𝑖,2 ] (15) By (14) and (15), we obtain × [𝐴 𝑖 𝑥 (𝑡) + 𝐵𝑖 𝑥 (𝑡 − ℎ (𝑡)) + 𝐺𝑖 𝑤 (𝑡)] 𝑡 −∫ 𝑡−(𝑑/2) 𝑡−(𝑑/2) −∫ 𝑡−𝑑 𝑇 −𝛼(𝑠−𝑡) 𝑥 (𝑠) 𝑒 𝑡 𝑡−(𝑑/2) ≥ (14) 𝑡−(𝑑/2) 𝑡−𝑑 − − 𝑑 𝑇 [𝑥 (𝑡) − 𝑥 (𝑡 − )] 𝑑 𝑑 )] , 𝑥𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑍𝑖,2 𝑥 (𝑠) 𝑑𝑠 𝑥 (𝑡) [𝑥 (𝑡 − ℎ (𝑡))] ] [ ] [ 𝑑 ] [ 𝑉𝑖̇ (𝑡) − 𝛼𝑉𝑖 (𝑡) ≤ [ 𝑥 (𝑡 − ) ] [ ] ] [ [ 𝑥 (𝑡 − 𝑑) ] [ 𝑤 (𝑡) ] 𝑑 𝑇 [𝑥 (𝑡) − 𝑥 (𝑡 − )] 𝑑 × 𝑒𝛼𝑑/2 𝑍𝑖,1 [𝑥 (𝑡) − 𝑥 (𝑡 − 𝑥̇𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑍𝑖,1 𝑥̇ (𝑠) 𝑑𝑠 × 𝑒𝛼𝑑/2 𝑍𝑖,1 [𝑥 (𝑡) − 𝑥 (𝑡 − ∫ × [𝑍𝑖,1 + 𝑍𝑖,2 ] [𝐴 𝑖 𝑥 (𝑡) + 𝐵𝑖 𝑥 (𝑡 − ℎ (𝑡)) + 𝐺𝑖 𝑤 (𝑡)] 𝑍𝑖,2 𝑥 (𝑠) 𝑑𝑠 Due to the Jensen inequality, inequality (15) holds ∫ ̇ (𝑡) ≤ 𝛼𝑉𝑖,4 (𝑡) + 𝑑 [𝐴 𝑖 𝑥 (𝑡) + 𝐵𝑖 𝑥 (𝑡 − ℎ (𝑡)) + 𝐺𝑖 𝑤 (𝑡)]𝑇 𝑉𝑖,4 𝑥̇𝑇 (𝑠) 𝑒−𝛼(𝑠−𝑡) 𝑍𝑖,1 𝑥̇ (𝑠) 𝑑𝑠 𝑑 )] 𝑇 𝑑 [𝑥 (𝑡 − ) − 𝑥 (𝑡 − 𝑑)] 𝑑 × 𝑒𝛼𝑑 𝑍𝑖,2 [𝑥 (𝑡 − 𝑑 ) − 𝑥 (𝑡 − 𝑑)] (16) From (11), (12), (13), and (16), it is easy to get 𝑇 𝛼𝑑/2 𝑑 𝑑 𝑃𝑖 𝐵𝑖 + 𝐴𝑇𝑖 𝑍𝑖 𝐵𝑖 𝑃𝑖 𝐺𝑖 + 𝐴𝑇𝑖 𝑍𝑖 𝐺𝑖 𝑒 𝑍𝑖,1 𝜉 [ 11 ] 𝑑 [ ] 𝑑 𝑇 [ ∗ 𝑑 𝐵𝑇 𝑍 𝐵 − (1 − 𝑘) 𝑄 0 𝐵𝑖 𝑍𝑖 𝐺𝑖 ] [ ] 𝑖 𝑖 𝑖 2 [ ] 𝛼𝑑 𝛼𝑑/2 [ ] 𝛼𝑑/2 ×[∗ ] ∗ 𝑒 𝑅𝑖 − 𝑒 𝑍𝑖 𝑒 𝑍𝑖,2 [ ] 𝑑 𝑑 [ ] 𝛼𝑑 𝛼𝑑 [∗ ] 𝑅 − 𝑍 ∗ ∗ −𝑒 𝑒 𝑖,2 𝑖,2 [ ] 𝑑 [ ] 𝑑 𝑇 𝑍 𝐺 ∗ ∗ ∗ ∗ 𝐺 [ 𝑖 𝑖 𝑖 ] 𝑥 (𝑡) [𝑥 (𝑡 − ℎ (𝑡))] ] [ [ 𝑑 ] ] [ × [ 𝑥 (𝑡 − ) ] [ ] ] [ [ 𝑥 (𝑡 − 𝑑) ] [ 𝑤 (𝑡) ] (17) Abstract and Applied Analysis According to the definition of finite-time boundedness, the rest of the proof will be divided into two steps Under the given conditions, we need to prove that 𝑥𝑇 (𝑡)𝑥(𝑡) < 𝑐2 and 𝑐1 < 𝑐2 , respectively (i) We will prove that 𝑥𝑇 (𝑡)𝑥(𝑡) < 𝐶2 holds for all 𝑡 on [0, 𝑇𝑓 ] By (6) and (17), inequality (18) holds Assume the switching number of 𝜎(𝑡) over [0, 𝑇𝑓 ] is 𝑁 (24) is obtained via the iterative calculation 𝑉𝑖 (𝑡) < 𝛽𝑁𝑒𝛼𝑡 𝑉𝑖 (0) + 𝑑 𝑁 𝛼(𝑡−𝑡1 ) 𝛽 𝑒 𝑡1 × ∫ 𝑤𝑇 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 + ⋅ ⋅ ⋅ 𝑡0 𝑑 𝑉𝑖̇ (𝑡) − 𝛼𝑉𝑖 (𝑡) < 𝑤𝑇 (𝑡) 𝐻𝑤 (𝑡) (18) + 𝑑 𝛼(𝑡−𝑡𝑘 ) 𝑡𝑘 𝑇 𝛽𝑒 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 𝑡𝑘−1 Since (𝑑/𝑑𝑡)(𝑒−𝛼𝑡 𝑉𝑖 (𝑡)) = 𝑒−𝛼𝑡 [𝑉𝑖̇ (𝑡) − 𝛼𝑉𝑖 (𝑡)], inequality (18) can be transformed into + 𝑑 𝑇𝑓 𝑇 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠, 𝑡𝑘 (24) 𝑒𝛼𝑇𝑓 ≥ 𝑒𝛼𝑡 , 𝑑 𝑑 −𝛼𝑡 (𝑒 𝑉𝑖 (𝑡)) < 𝑒−𝛼𝑡 𝑤𝑇 (𝑡) 𝐻𝑤 (𝑡) 𝑑𝑡 (19) Let 𝑡𝑘 stand for the instant of the 𝐾th switching Integrating from 𝑡𝑘 to 𝑡 on both sides of (19), it follows that 𝑉𝑖 (𝑡) < 𝑒𝛼(𝑡−𝑡𝑘 ) 𝑉𝑖 (𝑡𝑘 ) + 𝑑 𝑡 𝑇 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 𝑡𝑘 𝑒𝛼𝑇𝑓 > 𝑒𝛼(𝑇𝑓 −𝑡1 ) > 𝑒𝛼(𝑇𝑓 −𝑡2 ) > ⋅ ⋅ ⋅ > 𝑒𝛼(𝑇𝑓 −𝑡𝑘 ) > for 𝑡 ∈ [0, 𝑇𝑓 ] , 𝛽𝑁 ≥ 𝛽𝑁−1 ≥ ⋅ ⋅ ⋅ ≥ 𝛽 ≥ Thus, it follows that 𝑉𝑖 (𝑡) < 𝛽𝑁𝑒𝛼𝑇𝑓 𝑉𝑖 (0) + (20) 𝑑 𝑁 𝛼𝑇𝑓 𝛽 𝑒 𝑡1 × ∫ 𝑤𝑇 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 + ⋅ ⋅ ⋅ 𝑡0 Notice that 𝑃𝑖 ≤ 𝛽𝑃𝑗 , 𝑅𝑖,1 ≤ 𝛽𝑅𝑗,1 , 𝑅𝑖,2 ≤ 𝛽𝑅𝑗,2 , 𝑄𝑖 ≤ 𝛽𝑄𝑗 , 𝑍𝑖,1 ≤ 𝛽𝑍𝑗,1 , 𝑍𝑖,2 ≤ 𝛽𝑍𝑗,2 , 𝑖, and 𝑗 ∈ [1, 2, , 𝑚] and the continuity of 𝑥(𝑡), hence (21) holds 𝑉𝑖 (𝑡) < 𝛽𝑒𝛼(𝑡−𝑡𝑘 ) 𝑉𝑖 (𝑡𝑘− ) + 𝑑 𝑡 𝑇 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠, 𝑡𝑘 (21) where 𝑡𝑘− denotes the instant just before 𝑡𝑘 It is easy to see 𝑉𝑖 (𝑡𝑘− ) < 𝑒𝛼(𝑡𝑘 −𝑡𝑘−1 ) 𝑉𝑖 (𝑡𝑘−1 ) + Then (23) is obtained 𝑉𝑖 (𝑡) < 𝛽2 𝑒𝛼(𝑡−𝑡𝑘−1 ) 𝑉𝑖 (𝑡(𝑘−1)− ) 𝑑 𝛼(𝑡−𝑡𝑘 ) 𝑡𝑘 𝑇 𝛽𝑒 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 𝑡𝑘−1 + 𝑑 𝑡 𝑇 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 𝑡𝑘 𝑡𝑘 𝑑 + 𝛽𝑁𝑒𝛼𝑇𝑓 ∫ 𝑤𝑇 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 𝑡𝑘−1 + (26) 𝑑 𝛼𝑇𝑓 𝑁 𝑡 𝑇 𝑒 𝛽 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠, 𝑡𝑘 𝑉𝑖 (𝑡) < 𝛽𝑁𝑒𝛼𝑇𝑓 𝑉𝑖 (0) + 𝑑 𝛼𝑇𝑓 𝑁 𝑡 𝑇 𝑒 𝛽 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 (27) On the other hand, since ≤ 𝑒𝛼(𝑡−𝑠) ≤ 𝑒𝛼𝑡 ≤ 𝑒𝛼𝑇𝑓 and 𝐻 ≤ 𝜆 max (𝐻), we have 𝑑 𝑡𝑘 𝑇 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 𝑡𝑘−1 (22) + (25) (23) 𝑑 𝛼𝑇𝑓 𝑁 𝑡 𝑇 𝑒 𝛽 ∫ 𝑤 (𝑠) 𝑒𝛼(𝑡−𝑠) 𝐻𝑤 (𝑠) 𝑑𝑠 ≤ 𝑡 𝑑 2𝛼𝑇𝑓 𝑁 𝛽 𝜆 max (𝐻) ∫ 𝑤𝑇 (𝑠) 𝑤 (𝑠) 𝑑𝑠 𝑒 ≤ 𝑑 2𝛼𝑡 𝑁 𝑒 𝛽 𝜆 max (𝐻) 𝛾 (28) Applying the above inequality to (27), we get 𝑉𝑖 (𝑡) < 𝛽𝑁𝑒𝛼𝑇𝑓 𝑉𝑖 (0) + 𝑑 2𝛼𝑇𝑓 𝑁 𝛽 𝜆 max (𝐻) 𝛾 𝑒 (29) Abstract and Applied Analysis With respect to 𝑉𝑖 (0) in (29), it is processed as follows: 𝑉𝑖 (0) = 𝑥𝑇 (0) 𝑃𝑖 𝑥 (0) + ∫ −𝑑/2 +∫ −𝑑/2 −𝑑 +∫ −ℎ(0) +∫ 𝑇 −𝛼𝑠 𝑥 (𝑠) 𝑒 𝑥𝑇 (𝑠) 𝑒−𝛼𝑠 𝑅𝑖,1 𝑥 (𝑠) 𝑑𝑠 +∫ −𝑑 𝑉𝑖 (𝑡) < 𝛽𝑁𝑒𝛼𝑇𝑓 × (𝜆 + 𝑅𝑖,2 𝑥 (𝑠) 𝑑𝑠 𝑑 𝛼𝑑/2 𝑑 𝑒 𝜆 + 𝑒𝛼𝑑 𝜆 + ℎmax 𝑒𝛼ℎmax 𝜆 ) 𝐶1 2 + 𝛽𝑁𝑒𝛼𝑇𝑓 ( 𝑥𝑇 (𝑠) 𝑒−𝛼𝑠 𝑄𝑖 𝑥 (𝑠) 𝑑𝑠 𝑇 −𝛼𝑠 ∫ 𝑥̇ (𝑠) 𝑒 𝜃 𝑍𝑖,2 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 𝑑2 𝛼𝑑/2 𝑑2 𝛼𝑑 + 𝜌 𝜆6𝑒 𝜌 𝜆5𝑒 + ∫ 𝑥̇𝑇 (𝑠) 𝑒−𝛼𝑠 𝑍𝑖,1 𝑥̇ (𝑠) 𝑑𝑠 𝑑𝜃 −𝑑/2 𝜃 −𝑑/2 Inequality (32) is obtained via (29) and (31) as 𝑑 𝛼𝑇𝑓 𝑒 𝜆 𝛾) = 𝛽𝑁𝑒𝛼𝑇𝑓 𝜂1 𝐶1 + 𝛽𝑁𝑒𝛼𝑇𝑓 ×( < 𝜆 max (𝑃𝑖 ) sup {𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 )} 𝑑2 𝛼𝑑/2 𝑑2 𝛼𝑑 𝑑 𝛼𝑇𝑓 + 𝜌 𝜆 𝑒 + 𝑒 𝜆 𝛾) 𝜌 𝜆5𝑒 2 (32) −𝑑≤𝑡≤0 + 𝑑 𝛼𝑑/2 𝑒 𝜆 max (𝑅𝑖,1 ) sup {𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 )} −𝑑≤𝑡≤0 + 𝑑 𝛼𝑑 𝑒 𝜆 max (𝑅𝑖,2 ) sup {𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 )} −𝑑≤𝑡≤0 According to the definition of 𝑉𝑖 (𝑡), inequality (33) holds 𝑉𝑖 (𝑡) > 𝑥𝑇 (𝑡) 𝑃𝑖 𝑥 (𝑡) ≥ 𝜆 (𝑃𝑖 ) 𝑥𝑇 (𝑡) 𝑥 (𝑡) + ℎmax 𝑒𝛼ℎmax 𝜆 max (𝑄𝑖 ) sup {𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 )} ≥ {𝜆 (𝑃𝑖 )} 𝑥𝑇 (𝑡) 𝑥 (𝑡) −𝑑≤𝑡≤0 +∫ −𝑑/2 +∫ −𝑑/2 −𝑑 −𝜃𝜌2 𝜆 max (𝑍𝑖,1 ) 𝑒−𝛼𝜃 𝑑𝜃 𝑖∈𝑀 Then the following holds based on (32) and (33): −𝜃𝜌2 𝜆 max (𝑍𝑖,2 ) 𝑒−𝛼𝜃 𝑑𝜃 < 𝜆 max (𝑃𝑖 ) sup {𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 )} 𝑥𝑇 (𝑡) 𝑥 (𝑡) < (𝛽𝑁𝑒𝛼𝑇𝑓 𝜂1 𝐶1 + 𝛽𝑁𝑒𝛼𝑇𝑓 −𝑑≤𝑡≤0 ×( 𝑑 + 𝑒𝛼𝑑/2 𝜆 max (𝑅𝑖,1 ) sup {𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 )} −𝑑≤𝑡≤0 + (33) 𝑑 𝛼𝑑 𝑒 𝜆 max (𝑅𝑖,2 ) sup {𝑥𝑇 (𝑡0 ) 𝑥 (𝑡0 )} −𝑑≤𝑡≤0 𝑑2 𝛼𝑑/2 𝑑2 𝛼𝑑 + 𝜌 𝜆6𝑒 𝜌 𝜆5𝑒 + (34) 𝑑 𝛼𝑇𝑓 −1 𝑒 𝜆 𝛾)) (𝜆 ) = 𝐶2 + ℎmax 𝑒𝛼ℎmax 𝜆 max (𝑄𝑖 ) sup {𝑥𝑇 (𝑡0 𝑡) 𝑥 (𝑡0 )} −𝑑≤𝑡≤0 + 𝑑 𝑑 𝑑 ⋅ 𝜌 𝜆 max (𝑍𝑖,1 ) 𝑒𝛼𝑑/2 + ⋅ 𝑑𝜌2 𝜆 max (𝑍𝑖,2 ) 𝑒𝛼𝑑 2 (30) (ii) Next, 𝐶1 < 𝐶2 will be demonstrated By (7), we have 𝑇𝑓 Applying known mathematical relationships to (30), (31) can be obtained as 𝑑 𝑑 𝑉𝑖 (0) < 𝜆 𝐶1 + 𝑒𝛼𝑑/2 𝜆 𝐶1 + 𝑒𝛼𝑑 𝜆 𝐶1 2 𝛼ℎmax + ℎmax 𝑒 𝑑2 𝑑2 𝜆 𝐶1 + 𝜌2 𝜆 𝑒𝛼𝑑/2 + 𝜌2 𝜆 𝑒𝛼𝑑 (31) 𝜏𝑎 𝑁> 𝑇𝑓 𝜏𝑎 > > ln 𝐶1 + ln 𝜆 − ln (𝜂1 𝐶1 + 𝜂2 ) − 𝛼𝑇𝑓 ln 𝛽 ln 𝐶1 + ln 𝜆 − ln (𝜂1 𝐶1 + 𝜂2 ) − 𝛼𝑇𝑓 ln 𝛽 ln (𝜂1 𝐶1 + 𝜂2 ) − ln 𝜆 > ln 𝐶1 − 𝑁 ln 𝛽 − 𝛼𝑇𝑓 , 𝜂1 𝐶1 + 𝜂2 𝛼𝑇𝑓 𝑁 𝑒 𝛽 > 𝐶1 𝜆8 , (35) , (36) (37) (38) Abstract and Applied Analysis On the other hand, due to 𝑒𝛼𝑇𝑓 ≥ 1, there exist the following mathematical relations: 𝑁 𝛼𝑇𝑓 𝐶2 = (𝛽 𝑒 ×( 𝜂1 𝐶1 + 𝛽 𝑒 𝑑2 𝛼𝑑/2 𝑑2 𝛼𝑑 + 𝜌 𝜆6𝑒 𝜌 𝜆5𝑒 ≥ (𝛽𝑁𝑒𝛼𝑇𝑓 𝜂1 𝐶1 + 𝛽𝑁𝑒𝛼𝑇𝑓 ℎ̇ (𝑡) ≤ 𝑘 = 0.5, 𝐶1 = 0.26 𝑇 0.5 ∗ 10 = and ∫0 𝑓 𝑤𝑇 (𝑠)𝑤(𝑠)𝑑𝑡 ≤ 𝛾 ≈ 0.022 Solving (6) leads to feasible solutions that (39) 𝑑 𝛼𝑑/2 𝑑 𝛼𝑑 + 𝜌 𝜆6𝑒 𝜌 𝜆5𝑒 0.8983 −0.0167 0.1555 𝑃1 = [−0.0167 1.0898 −0.3930] , [ 0.1555 −0.3930 0.9754 ] 0.6101 0.1828 −0.1480 𝑃2 = [ 0.1828 0.8908 −0.3026] , [−0.1480 −0.3026 0.8153 ] 𝜂1 𝐶1 + 𝜂2 𝛼𝑇𝑓 𝑁 𝑒 𝛽 𝜆8 0.7188 −0.1052 0.0283 𝑅1,1 = [−0.1052 0.7458 −0.1300] , [ 0.0283 −0.1300 0.7114 ] Combining (38) and (39), we get 𝑐2 > 𝑐1 By (i) and (ii), the system (1) satisfies the definition of finite-time boundedness under given conditions This completes the proof of Theorem Remark Notice that (6) is not a linear matrix inequality Thus, it cannot be directly solved via LMI toolbox Before solving (6), the inequality can be transformed to a linear matrix inequality by specifying the value of 𝛼 A Numerical Example An example is presented to illustrate Theorem Consider 𝑥̇ (𝑡) = 𝐴 𝜎(𝑡) 𝑥 (𝑡) + 𝐵𝜎(𝑡) 𝑥 (𝑡 − ℎ (𝑡)) + 𝐺𝜎(𝑡) 𝑤 (𝑡) , 𝑥 (𝑡) = 𝜑 (𝑡) , ∀𝑡 ∈ [−0.2, 0] , Let 𝛼 = 0.02, 𝛽 = 1.1, and 𝑇𝑓 = 10, then ℎ(𝑡) ≤ ℎmax = 𝑑 −1 + 𝜆 𝛾)) (𝜆 ) = 𝑑 = 0.2, (40) 𝑑 −1 + 𝑒𝛼𝑇𝑓 𝜆 𝛾)) (𝜆 ) ×( 𝑇 𝜑 (𝑡) ≡ [0.5 0.1 0] , 󵄨 󵄨 max 󵄨󵄨󵄨𝜑̇ (𝑡)󵄨󵄨󵄨 ≤ 𝜌 = 0, 𝑁 𝛼𝑇𝑓 ℎ (𝑡) = 0.5𝑡, 𝑡 ≥ 0, 𝑡 ∈ [−𝑑, 0) , 1.3854 −0.1336 0.0209 𝑅1,2 = [−0.1336 1.3613 −0.1532] , [ 0.0209 −0.1532 1.3368 ] 0.5289 0.0089 −0.0566 𝑅2,1 = [ 0.0089 0.6200 −0.0083] , [−0.0566 −0.0083 0.6743 ] 1.1615 0.0282 −0.0465 𝑅2,2 = [ 0.0282 1.2555 −0.0175] , [−0.0465 −0.0175 1.3518 ] 4.2184 −0.5908 −0.1106 𝑄1 = [−0.5908 4.4575 −0.3066] , [−0.1106 −0.3066 4.0548 ] 3.8150 0.0518 0.0675 𝑄2 = [0.0518 3.7399 0.0356] , [0.0675 0.0356 4.2709] −1.7 1.7 𝐴 = [ 1.3 −1 0.7 ] , [ 0.7 −0.6] −1 𝐴 = [0.7 −0.6] , [1.7 −1.7] 0.3150 −0.0492 −0.0003 𝑍1,1 = [−0.0492 0.2950 −0.0639] , [−0.0003 −0.0639 0.3082 ] 1.5 −1.7 0.1 𝐵1 = [−1.3 −0.3] , 0.6 ] [−0.7 −1 −0.3 0.1 𝐵2 = [1.3 −0.1 0.6] , [1.5 0.1 1.8] 0.4060 0.0002 0.0006 𝑍1,2 = [0.0002 0.4036 0.0013] , [0.0006 0.0013 0.3934] 0 𝐺1 = 𝐺2 = [0 0] , [0 1] 0.2124 0.0176 −0.0190 𝑍2,1 = [ 0.0176 0.2812 −0.0034] , [−0.0190 −0.0034 0.3118 ] 0.03 sin (𝑡) ], 0.02 cos (2𝑡) 𝑤 (𝑡) = [ [0.015 (sin (𝑡 + 1) + cos (𝑡 − 2))] 0.3934 0.0070 0.0039 𝑍2,2 = [0.0070 0.3920 −0.0131] , [0.0039 −0.0131 0.3961 ] Abstract and Applied Analysis System mode The figure of switching law 100 2.5 50 x(t) The figure of x(t) −50 1.5 −100 10 10 Time (s) (a) 0.5 The figure of xT (t)x(t) 8000 Time (s) 10 xT (t)x(t) Figure 1: The diagram of switching law 6000 4000 2000 12.5148 0.6115 0.1387 𝐻 = [ 0.6115 13.0302 −0.5248] , [ 0.1387 −0.5248 12.4327 ] 𝜆 = 1.4539, 𝜆 = 0.9114, 𝜆 = 1.5749, 𝜆 = 4.9735, 𝜆 = 0.2449, 𝜆 = 0.1192, 𝜆 = 13.5367, Time (s) (b) Figure 2: The diagrams of 𝑥(𝑡) and 𝑥𝑇 (𝑡)𝑥(𝑡) Acknowledgment 𝜆 = 0.5200 (41) Further, we get that 𝐶2 = 2000.6421 > 𝐶1 and 𝜏𝑎 < 1.8263 The simulation of the numerical example is performed and its results are shown in Figures and From Figure 1, one can get that 𝜏𝑎 < 1.8263 holds From Figure 2, it is easily found that the value of 𝑥𝑇 (𝑡)𝑥(𝑡) remains within 𝐶2 for 𝑡 ∈ [0, 𝑇𝑓 ] So, the system is indeed finite-time bounded over [0, 𝑇𝑓 ] Conclusion (1) For the switched linear system, a new definition on finite-time boundedness is proposed which can reduce some complex matrix calculations (2) Under given conditions, the sufficient conditions which guarantee the system is finite-time bounded are given for the switched linear system with time-varying delay and external 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