Pointwise stabilisation of a string with time delay in the observation

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1 International Journal of Control ISSN: (Print) (Online) Journal homepage: Pointwise stabilisation of a string with time delay in the observation Kun-Yi Yang & Jun-Min Wang To cite this article: Kun-Yi Yang & Jun-Min Wang (17) Pointwise stabilisation of a string with time delay in the observation, International Journal of Control, 9:11, , DOI: 1.18/ To link to this article: Accepted author version posted online: 18 Oct 16. Published online: 15 Nov 16. Submit your article to this journal Article views: 65 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at Download by: [Beijing Institute of Technology Date: 19 September 17, At: 17:9

2 INTERNATIONAL JOURNAL OF CONTROL, 17 VOL. 9, NO. 11, Pointwise stabilisation of a string with time delay in the observation Kun-Yi Yang a and Jun-Min Wang b a College of Science, North China University of Technology, Beijing, P. R. China; b School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, P. R. China Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 ABSTRACT Pointwise stabilisation is proposed in this paper for a string equation where the observation signal is subject to a time delay. Different from the boundary control, the feedback stabiliser is acting at the middle joint of the string. Well-posedness of the open-loop system and solvability of the observer are shown first. An observer system is then designed to estimate the state at the time interval when the observation is available, and a predictor system is designed to predict the state at the time interval when the observation is not available. Pointwise output feedback controller is introduced to make the closed-loop system asymptotically stable for the non-smooth initial values and exponentially stable for the smooth initial values, respectively. Simulation results demonstrate that the output feedback based on the observer and predictor effectively stabilises the pointwise control system with time delay. 1. Introduction In industrial systems, it is common that a number of actuators are combined with some joints. One of the most significant examples is pointwise control system mentioned by such references as Ammari, Tucsnak, and enrot (), Ammari, Liu, and Tucsnak (), Ammari and Tucsnak (), Chen, Delfour, Krall, and Payre (1987), Chen et al. (1989) andtucsnak(1998). A string equation with a pointwise interior actuator is considered by Tucsnak (1998) wheretheuniformdecayrate is gained by the spectrum analysis method. Later stabilisation of the same pointwise control string system with a Neumannright-handboundary condition is characterisedcompletelybyammarietal.() dependent on the positions of the actuator while the fastest decay rate is obtained with the actuator at the middle of string actuator. Stabilisation and control of N serially connected Euler Bernoulli beams are considered by Chen et al. (1987), while design and behaviour of dissipative joints for coupled beams are analysed in Chen et al. (1989). Both uniform and non-uniform energy decays of an Euler Bernoulli beam subject to a pointwise feedback force are shown in Ammari and Tucsnak (). Obviously, it is difficult to stabilise pointwise control systems whose stability properties depend on the location of the joints in an unrobust way since it may need more actuators(seeammari& Tucsnak, ). Furthermore, the ARTICLE ISTORY Received 7 October 15 Accepted 15 October 16 KEYWORDS String equation; time delay; pointwise control; estimated state feedback; stability energy decay rates for both Euler Bernoulli and Rayleigh beams with pointwise shear force and bending moment arestudiedbyammarietal.(). And it is indicated that the energy decays exponentially independently of the position of the actuator for the Euler Bernoulli beam, while the energy exponential decay depends on the position of the actuator for the Rayleigh beam. Later on, by exerting the distribute control on the two connected Rayleigh beams, the exponential stability is shown to be robust with respect to the location of the joint by the Riesz basis approach (Guo, Wang, & Zhou, 8), while well-posedness and exponential stabilisation of the hinged Reyleigh beam is obtained by the well-posed theory (Weiss & Curtain, 8). On the other hand, in a physical control system, there frequently exists a time delay between the controller and observation signal, such as the process of sampleddata control (see Karafyllis & Krstic, 13; Zhong, 6). Existenceoftimedelaymaydeformordamagestability of the originally stable system (see Gumowski & Mira, 1968). It is indicated in Datko (1997) anddatko, Lagnese, and Polis (1986) that even a small amount of time delay may make the string equation system unstable. For distributed parameter control systems, time delay in observation or control may lead to difficult mathematical challenge (see Fleming, 1988). Stabilization may be obtained by transforming the linear system with delayed control action into systems without CONTACT Kun-Yi Yang yangkunyi@ncut.edu.cn 16 Informa UK Limited, trading as Taylor & Francis Group

3 INTERNATIONAL JOURNAL OF CONTROL 395 Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 delays (Artstein, 198). Later on, for the robust control of time-delay systems, controller parameterization and design to controller implementation have been analyzed (Zhong, 6). A class of switched linear systems with time delay have been stabilized by proposing switching law (Zhao et al., 1). By the methods of Lyapunov- Krasovskii funtional approach and linear matrix inequalities, asymptotic stability of the switched systems with time delays or the neutral networks have been considered (Wang et al. 11, Wangetal.1, or Zhang et al. 15). More effectively, such designs as smith predictor are commonly available for finite-dimensional systems in order to compensate arbitrarily long input delays. Such compensation have recently been introduced for distributed parameter systems in Krstic (8). Compensations of arbitrary long delay for the input are established in the unstable reaction diffusion and wave equation, respectively (Krstic, 9, 11). Indeed stability of the reaction diffusion equation is shown to be robust with respect to long input delay after designing these kinds of compensation (see Krstic, 9). Forastringequa- tion with the output feedback loop which is subject to a time delay, the necessary and sufficient conditions have beengiventoguaranteetheexponentialstabilityofthe closed-loop system if the time delay is equal to the even multiples of the wave propagation time (Wang, Guo, & Krstic, 11). Theotherkindsoffeedbackwhereacertain delay is included as a part of the control law have been proven to make the vibrating string system exponentially stable by Gugat (1a, 1b), and Gugat and Tucsnak (11). For the distributed parameter systems where the boundary observation is suffered from an arbitrary time delay, based on the design of the observer and predictor, for instance, an output feedback law has been introducedtostabiliseaone-dimensionalwaveequation(guo, Xu, & ammouri, 1). This method of the observer predictor-based scheme has also effectively been applied to stabilise an Euler Bernoulli beam with output delay (see Guo & Yang, 9). To our knowledge, stabilisation of pointwise control systems with time delay, especially that with delayed observation, has been rarely involved in even recent research. This paper considers the system as following that the actuator is constrained in the middle point of the string: w tt (x, t) = w xx (x, t), < x < 1, t >, w(, t) = w x (1, t) =, t, w(, t) = w(, t), t, w x (, t) w x (, t) = u(t), t, y(t) = w t (, t ), t, w(x, ) = w (x), w t (x, ) = w 1 (x), x 1, (1) where x is the position, t is the time, w represents the state, u(t) iscontrol,(w, w 1 ) T is the initial value, ( > ) is a given time delay, andy(t) is observation which is suffered from the time delay. In this system, the left boundary end of the string is fixed while the right boundary end is free, and the observation signal is subject to the time delay. The system above is considered in ilbert space as follows: = { ( f, g) T 1 ((, ) (, 1)) L ((, ) (, 1)) f () = f (1) = } () with the inner-product-induced norm: E (t) = 1 (w(, t), w t (, t)) T = 1 [wx (x, t) w t (x, t)dx. (3) Based on the Salamon Weiss well-posed infinitedimensional system theory, we prove that the open-loop system is well-posed. Then we design the observer system atthetimeintervalwhentheobservationsignalisavailable, and design the predictor system at the time interval when the observation signal is not available. Naturally, we construct a control law with the estimated state by the state of the observer and predictor. We prove that the closed-loop system is asymptotically stable for nonsmooth initial values, and exponentially stable for smooth initial values. Numerical simulations are illustrated to show stabilised effectiveness of the output feedback controller. By the approach of the observer- and predictorbased scheme, we provide a unified methodology to solve stabilisation of the pointwise control distributed parameter system where observation signal is suffered from a time delay. We proceed as follows. The next section shows the well-posedness of the original open-loop system. In Section 3, we design the observer and predictor systems. The stability properties of the closed-loop system under the estimated state feedback control are then illustrated in Section 4. Section 5 gives simulation results and Section 6 concludes the paper.. Well-posedness of open-loop system We introduce a new variable z(x, t) = w t (, t x). Then the system (1)becomes

4 Downloaded by [Beijing Institute of Technology at 17:9 19 September K.-Y. YANG AND J.-M. WANG w tt (x, t) = w xx (x, t), < x < 1, t >, w(, t) = w x (1, t) =, t, w(, t) = w(, t), t, w x (, t) w x (, t) = u(t), t, z t (x, t) z x (x, t) =, < x < 1, t >, z(, t) = w t (, t), t, w(x, ) = w (x), w t (x, ) = w 1 (x), x 1, z(x, ) = z (x), x 1, y(t) = z(1, t), t, (4) where z istheinitialvalueofthenewvariablez. With the state variable (w(, t), w t (, t), z(, t)) T,we consider the system (4) in the energy state space = L (, 1). Thenormof(w(, t), w t (, t), z(, t)) T in is defined by the energy as follows: E 1 (t) = 1 (w(, t), w t (, t), z(, t)) T = 1 [w x (x, t) w t (x, t)dx 1 z (x, t)dx. (5) The input space and the output space are the same U = Y = C. Lemma.1: The system (4) is well-posed: for each u L loc (, ) and initial datum (w, w 1, z ) T, there exists a unique solution (w(, t), w t (, t), z(, t)) T C(, ; ) of (4) and for each T > there exists a constant C T > such that (w(, T ), w t (, T ), z(, T )) T y(t) dt C T [ (w, w 1, z ) T u(t) dt Proof: It is known that the following system w tt (x, t) = w xx (x, t), < x < 1, t >, w(, t) = w x (1, t) =, t, w(, t) = w(, t), t, w x (, t) w x (, t) = u(t), t, y w (t) = w t (, t), t, can be written as ( ) ( ) d w w = A Bu(t) dt w (A, B, C) : t ( ) w t w y w (t) = C = w t (, t) w t. (6) (7) (8) where A : D(A)( ) is defined as follows: A( f, g) T = (g, f ) T, ( f, g) T D(A), D(A) ={( f, g) T ( ((, ) (, 1)) 1 ((, ) (, 1))) f ( ) = f ( ), f ( ) = f ( )}, (9) and B = ( ), C = (,,δ(x) ). (1) δ(x) ere δ( ) denotes the Dirac function. Obviously, both B and C are unbounded operators. By the well-posed linear infinite-dimensional system theory (see Curtain, 1997;Guo&Zhang,5;Tucsnak& Weiss, 9), it is equivalent to showing that C is admissible for e At, B is admissible for e A t and the transfer function is bounded on some right-half complex plane. A direct computation shows that CA (ϕ, ψ) T = ϕ(), (ϕ, ψ) T, (11) which tells us that CA is bounded. Define ρ(t) = (x 1)w x (x, t)w t (x, t)dx (x 1)w x (x, t)w t (x, t)dx. (1) Then E (t) = E () and ρ(t) E (t)for t. Noticing that we have that ρ(t) = w t (, t) w x (, t) E (t), (13) w t (, t)dt (T )E (), (14) which together with (11)showthatC is admissible for e At. Actually, a straightforward computation gives that A (ϕ, ψ) T = ( ψ,ϕ ) T, (ϕ, ψ) T D(A ), D(A ) ={(ϕ, ψ) T ( ((, ) (, 1)) 1 ((, ) (, 1))) ψ( ) = ψ( ), ϕ ( ) = ϕ ( )}, (15) and B = (,,δ(x) ). Similar computation as that above shows that B is admissible for e A t.

5 INTERNATIONAL JOURNAL OF CONTROL 397 Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 Finally, the transfer function for the system (7) is found to be (s) = 1 es e s, s >, e s e s which is obviously bounded on some right-half complex plane. Therefore, system (7) is well-posed in the sense of D. Salamon (see Curtain, 1997): for any u L loc (, ) and (w, w 1 ) T, there exists a unique solution (w(, t), w t (, t)) T C(, ; ) of (8); and for any T >, there exists a constant D T > suchthat (w(, T ), w t (, T )) T y w (t) dt D T [ (w, w 1 ) T u(t) dt. (16) On the other hand, the analytic expression of the `z part equation { zt (x, t) z x (x, t) =, (17) z(, t) = w t (, t), z(x, ) = z (x), can be obtained: z (x t z(x, t) = ), x t, w t (, t x), x < t (18), by integrating along the characteristic line. Therefore, we have, for any T >, and z(x, T ) dx = y(t) dt = T/ T wt T 1 1 T/ 1 z (x)dx w t (, t)dt, T, (, t)dt, T >, z (x)dx, T, z (x)dx w t These together with (16) gives (6). (, t)dt, T >. Step 1: Constructanobservertoestimatethestate {(w(x, s), w s (x, s)) T, s [, t, t >}fromtheknown observation {y(s ), s [, t, t >}. Since the observation {y(s ), s [, t, t >} is known and {(w(x, s), w s (x, s)) T, s [, t, t >} satisfies w ss (x, s) = w xx (x, s), < x < 1, s >, w(, s) = w x (1, s) =, s, w(, s) = w(, s), s, w x (, s) w x (, s) = u(s), s, y(s )= w s (, s), s, w(x, ) = w (x), w s (x, ) = w 1 (x), x 1. (19) We can construct naturally a Luenberger observer for system (19)asfollowing,fork 1 > : ŵ ss (x, s) = ŵ xx (x, s), < x < 1, < s t,t >, ŵ(, s) = ŵ x (1, s) =, s t,t, ŵ(, s) = ŵ(, s), s t,t, ŵ x (, s) ŵ x (, s) = u(s) k 1 [ŵ s (, s) y(s ), s t,t, ŵ(x, ) = ŵ (x), ŵ s (x, ) = ŵ 1 (x), x 1, () where (ŵ, ŵ 1 ) T is the (arbitrarily assigned) initial state of observer. In order for ()tobeanobserverfor(19), we have to show its convergence. To do this, let ε(x, s) = ŵ(x, s) w(x, s), s t,t >. (1) Then by (19)and(), ϵ(x, s)satisfies ε ss (x, s) = ε xx (x, s), < x < 1, < s t,t >, ε(, s) = ε x (1, s) =, s t,t, ε(, s) = ε(, s), s t,t, ε x (, s) ε x (, s) = k 1 ε s (, s), s t,t, ε(x, ) = ŵ (x) w (x), ε s (x, ) = ŵ 1 (x) w 1 (x), x 1. () The system () can be written as follows: 3. Observer and predictor design In this section, we proceed two steps to estimate the state of (1) via the observer and predictor. ( ) ( ) d ε(, s) ε(, s) = B ds ε s (, s) ε s (, s) (3)

6 398 K.-Y.YANGANDJ.-M.WANG where the operator B : D(B)( ) is defined as follows: B ( f, g ) T ( ) = g, f T, ( f, g) T D(B), D(B) ={( f, g) T ( ((, ) (, 1)) 1 ((, ) (, 1))) f ( ) = f ( ), f ( ) f ( ) = k 1 g()}. (4) Then ϵ t (x, s)satisfies εss t (x, s) = εt xx (x, s), < x < 1, t <s t, t >, ε t (, s) = εx t (1, s) =, t s t, t, ε t (x, t )= ε(x, t ), εs t (x, t ) = ε s (x, t ), x 1, t, (3) which is a conservative system, that is, It is known that B generates an exponentially stable C - semigroup on by Ammari et al. ()andliu(1988), that is to say, for any (w, w 1 ) T and (ŵ, ŵ 1 ) T, there exists a unique solution (ϵ(, s), ϵ s (, s)) T of () which satisfies (ε t (, t), ε t s (, t))t = (ε(, t ),ε s (, t )) T. This together with (5)and(7) gives (8). Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 (ε(, s), ε s (, s)) T Me ωs (ŵ w, ŵ 1 w 1 ) T, s [, t, t >, (5) forsomepositiveconstantsm and ω. Step : Predict {(w(x, s), w s (x, s)) T, s (t, t, t > }by{(ŵ(x, s), ŵ s (x, s)) T, s [, t, t >}. This can be achieved by solving (1) with estimated initial value {(ŵ(x, t ),ŵ s (x, t )) T, t >} obtained from (19): ŵss t (x, s) = ŵt xx (x, s), < x < 1, t <s t, t >, ŵ t (, s) = ŵx t (1, s) =, t s t, t, ŵ t (, s) = ŵ t (, s), t s t, t, ŵx t (, s) ŵx t (, s) = u(s), t s t, t, ŵ t (x, t )= ŵ(x, t ), ŵs t (x, t ) = ŵ s (x, t ), x 1, t. (6) We finally get the estimated state variable by w(x, t) = ŵ t (x, t), w t (x, t) = ŵ t s (x, t), t >. (7) Theorem 3.1: For any t >,wehave w(, t) w(, t), w t (, t) w t (, t)) T Me ω(t ) (ŵ w, ŵ 1 w 1 ) T, (8) where (ŵ, ŵ 1 ) T is the initial state of observer (), (w, w 1 ) T is the initial state of original system (1),andM,ω are constants in (5). Proof: Let ε t (x, s) = ŵ t (x, s) w(x, s), x 1, t s t, t >. (9) 4. Stability for closed-loop system Since the feedback u(t) = k w t (, t)(k > ) stabilises exponentially the system (1), and we have the estimation w t (, t) of w t (, t), it is natural to design the estimated state feedback control law as follows: u (t) = {, t [,, k ŵ t s (, t), t >,k >, (31) under which, the closed-loop system becomes a system of partial differential equations (3) (34): w tt (x, t) = w xx (x, t), < x < 1, t >, w(, t) = w x (1, t) =, t, w(, t) = w(, t), t, w x (, t) w x (, t) = u (t), t, w(x, ) = w (x), w t (x, ) = w 1 (x), x 1, (3) ŵ ss (x, s) = ŵ xx (x, s), < x < 1, < s t,t >, ŵ(, s) = ŵ x (1, s) =, s t,t, ŵ(, s) = ŵ(, s), s t,t, ŵ x (, s) ŵ x (, s) = u (s) k 1 [ŵ s (, s) y(s ), s t,t, ŵ(x, ) = ŵ (x), ŵ s (x, ) = ŵ 1 (x), x 1, (33) and ŵss t (x, s) = ŵt xx (x, s), < x < 1, t <s t, t >, ŵ t (, s) = ŵx t (1, s) =, t s t, t, ŵ t (, s) = ŵ t (, s), t s t, t, (34) ŵx t (, s) ŵx t (, s) = u (s), t s t, t, ŵ t (x, t )= ŵ(x, t ), ŵs t (x, t ) = ŵ s (x, t ), x 1, t.

7 INTERNATIONAL JOURNAL OF CONTROL 399 Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 We consider system (3) (34) in the state space X = 3. It is obvious that system (3) (34) isequivalent to (35) (37)fort >provided that u L loc (, ) which will be clarified by (66) later in Lemma of the appendix: w tt (x, t) = w xx (x, t), < x < 1, t >, w(, t) = w x (1, t) =, t, w(, t) = w(, t), t, (35) w x (, t) w x (, t) = k [w t (, t) εs t (, t), t, w(x, ) = w (x), w t (x, ) = w 1 (x), x 1, ε ss (x, s) = ε xx (x, s), < x < 1, < s t,t >, ε(, s) = ε x (1, s) =, s t,t, ε(, s) = ε(, s), s t,t, ε x (, s) ε x (, s) = k 1 ε s (, s), s t,t, ε(x, ) = ŵ (x) w (x), ε s (x, ) = ŵ 1 (x) w 1 (x), x 1, (36) εss t (x, s) = εt xx (x, s), < x < 1, t <s t, t >, ε t (, s) = εx t (1, s) =, t s t, t, ε t (x, t )= ε(x, t ), εs t (x, t ) = ε s (x, t ), x 1, t, (37) where ϵ(x, s) andϵ(x, s, t) aregivenby(1) and(9), respectively. Theorem 4.1: Let k 1 >, k >, and t >. Then for any (w, w 1 ) T, (ŵ, ŵ 1 ) T, there exists a unique solution of systems (35)-(37) such that (w(, t), w t (, t)) T C(, ; ), (ϵ(, s), ϵ s (, s)) T C(, t ; ), (ε t (, s), εs t (, s))t C([t,t (, ); ). Moreover, for any t >,s [,t and q (t,t, there exists a constant C tsq > such that (w(, t), wt (, t)) T (ε(, s), εs (, s)) T (ε t (, q), ε t s (, q))t C tsq [ (w, w 1 ) T (ŵ, ŵ 1 ) T. (38) Proof: For any (w, w 1 ) T,(ŵ, ŵ 1 ) T, since B defined by (3) and(4) generates an exponentially stable C -semigroup on, thereisauniquesolution (ε(, s), ε s (, s)) T C(, t ; ) of (36)suchthat(5) holds true. Now for any given time t >,write(37) as follows: d ds ( ) ( ) ε t (, s) ε εs t = A t (, s) (, s) εs t (, s) (39) where A is defined by A ( f, g ) T ( ) = g, f T, ( f, g) T D(A) D(A) ={( f, g) T ( ((, ) (, 1)) 1 ((, ) (, 1)))}. (4) Then A is skew-adjoint in and hence generates a conservative C -semigroup on. Forany(ϵ(, t ), ε s (, t )) T determined by (36), there exists auniquesolution(ε t (, s), εs t (, s))t C([t,t (, ); ) of (37)suchthat (ε t (, s), εs t (, s))t = (ε(, t ),εs (, t )) T, s [t,t, t >. (41) Nowweshowwell-posednessof(35) whichcanbe written as that, for t >, w tt (, t) Aw(, t) k BB [ w t (, t) εs t (, t) = (4) where B = δ(x)anda: D(A)( L ) L is defined as follows: Af = f, f D(A), D(A) ={f ((, ) (, 1)), f () =, f (1) =, f ( ) = f ( ), f ( ) = f ( )}. By Corollary 1 of Guo and Luo (), (4)iswell-posed, that is, for any t >, there exists a unique solution of (4) suchthat(w(, t), w t (, t)) T C(, ; ). Moreover, there exists a positive constant C t > suchthat (w(, t), w t (, t)) T C t [ (w(,),w t (,)) T εs t (, s)ds C t [ (w, w 1 ) T εs t (, s)ds. (43) Ontheotherhand,fromLemma of the appendix, we have that for t >, k =, 1,, t εs t (,ρ)dρ (ω M ) ( 16k)ω (ŵ w, ŵ 1 w 1 ) T, where ω, M are defined in (5). This together with (41), (43)and(5) gives the required result. Now we show the asymptotic stability for non-smooth initial values.

8 4 K.-Y. YANG AND J.-M. WANG Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 Theorem 4.: Let k 1 >, k >. Thenforany(w, w 1 ) T, (ŵ, ŵ 1 ) T, the solution of (35) satisfies lim t (w(, t), w t (, t)) T =. (44) Proof: Equation (35)canbewrittenasfollows: ( ) ( ) d w(, t) w(, t) = A dt w t (, t) c k w t (, t) Bεs t (, t), (45) where A c ( f, g) T = (g, f ) T, ( f, g) T D(A c ), D(A c ) ={( f, g) T ( ((, ) (, 1)) 1 ((, ) (, 1))), f ( ) = f ( ), f ( ) f ( ) = k g()} (46) and B is defined in (1). A direct computation shows BA c (ϕ, ψ) T = ϕ(), (ϕ, ψ) T, (47) which means BA c is bounded. For the energy E (t) of thesystem(35), a simple computation gives which shows Ė (t) = k wt (, t), (48) k w t (, t) dt E (), (49) for any T >. This inequality together with (47) illustrates that B is admissible for e Act. Therefore, there exists auniquesolutionto(45)suchthat(w(, t), w t (, t)) T C(, ; ).TheadmissibleofB implies that e Ac(t s) Bεs t (, s)ds C t ε t s (, ) L (,t) C t t ε t s (, ) L (,t), forsomepositiveconstantsc t independent of εs t (, t). On the other hand, it is known that e Act is exponentially stable. Since B is admissible for e Act with the control space L loc (, ), itisalsoadmissiblewithcontrolspace (, ). By Proposition.5 of Weiss (1989), we have L loc e Ac(t s) Bεs t (, s)ds t e Ac(t s) B( εs t (, s))ds t L ε t s (, ) L (t,t) for some constants L > that is independent of εs t (, t), and { u(t), t, (u v)(t) = v(t), t >. Suppose that for some M, ω >. Then we have e A ct M e ω t (w(, t), w t (, t)) T = e A c(t ) (w(,),w t (,)) T (5) e Ac(t s) k Bεs t (, s)ds, (51) which concludes that the the following inequality (w(, t), wt (, t)) T M e ω (t ) (w, w 1 ) T e Ac(t s) k Bεs t (, s)ds e Ac(t s) k Bεs t (, s)ds t M e ω (t ) (w, w 1 ) T k M e ω e A c(t s) e A c(t t ) Bεs t (, s)ds k M e ω e Ac(t s) Bεs t (, s)ds t M e ω (t ) (w, w 1 ) T k M eω (t ) e ω t e A c(t s) Bεs t (, s)ds k M e ω e Ac(t s) Bεs t (, s)ds t M e ω (t ) (w, w 1 ) T k M eω (t ) e ω t C t t ε t s (, ) L (,t ) k M e ω L ε t s (, ) L (t,t). (5) Passing to the limit as t for the inequality (5), we finally obtain lim t (w(, t), w t (, t)) T k M e ω L ε t s (, ) L (t,t), (53) whichtogetherwiththefollowingfact lim ε t s (, ) L (t,t) = t

9 INTERNATIONAL JOURNAL OF CONTROL 41 Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 from (66) in Appendix gives the result of Theorem 4.. Next we get the exponential stability of the closed-loop system for smooth initial values as the following theorem. Theorem 4.3: If (w, w 1 ) T and (ŵ, ŵ 1 ) T satisfy (ŵ w, ŵ 1 w 1 ) T D(B), where B is defined by (3), then the system (35) decays exponentially in the sense that (w(, t), w t (, t)) T C e αt[ (w, w 1 ) T 3 k 1 Meω B(ŵ w, ŵ 1 w 1 ) T (54) for some C > independent of initial values and <α< min {ω, ω } where ω and ω are defined in (5) and (5), respectively. Proof: Now we also only consider the case of 16k <16k 1liketheEquations(73) and(74) inlemma of Appendix. If (ε(, ), ε s (, )) T D(B) where B is defined by (3), then by (5), that is, Be Bs (ε(, ), ε s (, )) T Me ωs B(ε(, ), ε s (, )) T, s, (55) ( ) 1/ [εxx (x, s) ε xs (x, s)dx ( ) 1/ Me ωs [εxx (x, ) ε xs (x, )dx, s. (56) Since ϵ( 1, s) = ϵ x (1, s) = andϵ x (, s) ϵ x (, s) = k 1 ϵ s (, s), it is easy to prove the following inequalities: ε s (16k,s) 6k 6k ε xx (x, s)dx, ε s ( 16k, s) = ε s (, s) εxs (x, s)dx, ε x ( 16k, s) 6k ε xs (x, s)dx 6k εxs (x, s)dx ε s (, s), εs (, s) = ε xs (x, s)dx εxs (x, s)dx, ε x ( 16k, s) ε x (16k,s) 6k = ε xx (x, s)dx ε x (, s) 16k ε xx (x, s)dx ε x (, s) 6k = ε xx (x, s)dx k 1 ε s (, s) 16k [ 1 ε xx (x, s)dx k 1 ε s (, s), εx (16k,s) = ε x( 16k, s) ε x (16k,s) ε x ( 16k, s) ε x ( 16k, s) ε x (16k,s) εx (6k, s), (57) whichcombineswith(68), (73) and(74) inlemma of Appendix indicate that ε t s (, t) 3 k 1 Me ω(t ) [εxx (x, ) ε xs (x, )dx = 3 k 1 Me ω(t ) B(ŵ w, ŵ 1 w 1 ) T. (58) For the system (35) which is written as(45), suppose <α<min {ω, ω }whereω and ω are defined in (5) and (5), respectively, and let Then Y (t) = e αt (w(, t), w t (, t)) T. (59) d dt Y (t) = (A c α)y (t) k B(, e αt εs t (, t))t. (6) Since A c α and B satisfy assumptions (.1) (.4) of Theorem of (Lasiecka and Triggiani,, p.653), apply ( ) of this theorem with ε = to obtain Y (t) C [ (w(,),wt (,)) T e α ε t s (, ) L (, ), t, (61)

10 4 K.-Y. YANG AND J.-M. WANG for some C >. Therefore, Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 (w(, t), w t (, t)) T C e αt [ (w(,), w t (,)) T e α ε s t (, ) L (, ), t. (6) Since by (58), e α ε s t (, ) L (, ) ( ) 1/ = e αρ εs t (,ρ) dρ 3 k 1 Meω B(ŵ w, ŵ 1 w 1 ) T ( ) 1/ e (ω α)ρ dρ (3 k 1 ) ω α Meα B(ŵ w, ŵ 1 w 1 ) T, from (6)wehavethat (w(, t), wt (, t)) T C e αt [ (w, w 1 ) T (3 k 1 ) ω α Meα (63) B(ŵ w, ŵ 1 w 1 ) T, t. (64) 5. Numerical simulation In this section, we use the backward Euler method in the time domain and the Chebyshev spectral method in the space domain to give some numerical simulation results for the closed-loop system (35) (37). ere we choose the space grid size N = 4, time step dt =.1 and time span [, 1. Parameters and coefficients, respectively, are chosen to be =.5, k 1 = k = 1. For the initial values, { w (x) = e x, x 1, { w 1 (x) = e x, x 1, (65) ε(x, ) = x, x 1, ε s (x, ) = x, x 1, we plot the displacement w(x, t) andvelocityw t (x, t) as the Figure 1 and below respectively. It is seen that the displacementofthestringisalmostatrestaftert = 5. That is to say, the predictor observer-based scheme is also useful to make the pointwise control system converge for the string equation in which observation is subjected to atimedelay. Figure 1. State of the closed-loop system. Figure. Velocity of the closed-loop system. 6. Conclusion In a conclusion, this paper stabilises a string equation with delayed observation signal, where the feedback stabiliser is acting at the the middle joint of the string. Well-posedness of the open-loop system is obtained first. An observer is designed for the available observation, while a predictor is designed for the unavailable observation. Pointwise output feedback controller-based on the observer and predictor gives the closed-loop system which is exponentially stable for the smooth initial values and asymptotically stable for the non-smooth initial values. Simulation results demonstrate the stabilised effectivenss of the controller. For the feedback stabilisation of the wave equation pointwise control system with time delay, an interesting further research problem would be the robustness against small changes of the actuators locations or the robustness against perturbations in the time delay.

11 INTERNATIONAL JOURNAL OF CONTROL 43 Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 Acknowledgment Theauthorswouldliketothankanonymousrefereesandeditors for their careful reading and helpful suggestions for the impovement of the manuscript. Disclosure statement No potential conflict of interest was reported by the authors. Funding This work was supported by the National Natural Science Foundation of China [grant number 61358, [grant number ; the Training Program for Outstanding Young Teachers of North China University of Technology [grant number XN718; the Construction Plan for Innovative Research Team of North China University of Technology [grant number XN75. References Ammari,K.,Liu,Z.Y.,&Tucsnak,M.(). Decay rates for a beam with pointwise force and moment feedback. Mathematics of Control, Signals, and Systems, 15, Ammari,K.,&Tucsnak,M.(). Stabilization of Euler- Bernoulli beams by means of a pointwise feedback force. 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12 44 K.-Y. YANG AND J.-M. WANG Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 Weiss, G. (1989). Admissibility of unbounded control operators. SIAM Journal of Control and Optimization, 7, Weiss, G., & Curtain, R.F. (8). Exponential stabilization of a Rayleigh beam using collocated control. IEEE Transaction on Automatic Control, 53, Zhang, L., Zhuang, S., Shi, P., & Zhu, Y. (15). Uniform tube based stabilization of switched linear systems with mode-dependent persistent dwell-time. IEEE Transactions on Automatic Control, 6, Zhao, X., Zhang, L., Shi, P., &, Liu, M. (1). Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Transactions on Automatic Control, 57, Zhong, Q.C. (6). Robust control of time-delay systems. Springer. Appendix In this appendix, we present a lemma concerning the estimate t εs t (,ρ)dρ which is applied in proof of Theorem 4.1. Lemma For any (w, w 1 ) T, (ŵ, ŵ 1 ) T, the solution of the system (37) satisfies the inequality as follows: t εs t (,ρ)dρ (ω M ) ( 16k)ω (ŵ w, ŵ 1 w 1 ) T, t >,k =, 1,..., (66) where ω, Maredefinedin(5) in the case of 16k < 16k1. Proof: For brevity in notation, let us consider the following wave equation: p ξξ (x,ξ) p xx (x,ξ)=, < x < 1, <ξ<, p(,ξ)= p x (1,ξ)=, ξ, p(x, ) = ε(x, t ), x 1, p ξ (x, ) = ε s (x, t ), x 1. Then Denote by (67) ε t s (, t) = p ξ (,). (68) f (x, s) = e sρ f (x,ρ)dρ (69) as Laplace transform of the classical function f(x, ξ)where x is a parameter. Take Laplace transform for the system (A)to obtain d p(x, s) x p(x, s) dx = sε(x, t ) ε ρ (x, t ), p(, s) = p x (1, s) =, whose solution is found: p(x, s) cosh(sx) cosh(s) sinh(sx) sinh(s) = s cosh(s) (7) [sε(η,t ) ε ρ (η, t )sinh(s(1 η))dη cosh(sx) sinh(s) sinh(sx) cosh(s) s cosh(s) 1 s x [sε(η,t ) ε ρ (η, t )cosh(s(1 η))dη [sε(η,t ) ε ρ (η, t )sinh(s(x η))dη. (71) By p ξ (x, s) = s p(x, s) ε(x, t ) and (A6), a simple computation gives p ξ (, s) [ ( = sε(η,t ) ερ (η, t ) ) sinh(s(1 η))dη cosh(s) ( sε(η,t ) ερ (η, t ) ) 1 cosh(s(1 η))dη sinh(s) ε(, t ) cosh(s) sinh(s(1 η))cosh(s) = ε ρ (η, t ) dη cosh(s) ε ρ (η, t ) ε x (η, t ) ε x (η, t ) cosh(s(1 η))sinh(s) dη cosh(s) sinh(s(1 η))cosh(s) dη cosh(s) sinh(s(1 η))sinh(s) dη. cosh(s) (7) UsingtheinverseLaplacetransformformulae(see Oberhettinger & Baddi, 1973), we have that when k is a non-negative integer, there are two cases as follows:

13 INTERNATIONAL JOURNAL OF CONTROL 45 Downloaded by [Beijing Institute of Technology at 17:9 19 September 17 if 16k 4m 1 m <16k 4m 1 m 1, p ξ (,) = () m 1 ε s (16k 4m 1 m,t ) () m 1 ε s ( 16k 4m 1 m, t ) () m11 ε x (16k 4m 1 m,t ) () m 1m ε x ( 16k 4m 1 m, t ), (73) and if 16k 4m 1 m 1 < 16k 4m 1 m, p ξ (,) = () m 1m ε s (16k 4m 1 m,t ) () m11 ε s ( 16k 4m 1 m, t ) () m 1m 1 ε x (16k4m 1 m,t ) () m 1 ε x ( 16k 4m 1 m, t ), (74) for all of m 1 =, 1,, 3 and m =, 1. Weonlyconsiderthecasethat16k <16k 1since other cases can be treated similarly. Let 16k <16k 1, then 16k < 1and < 16k. For t > and s [, t, let ρ(s) = 6k 16k xε x (x, s)ε s (x, s)dx xε x (x, s)ε s (x, s)dx. (75) Then we have ρ(s) 1 (ε(, s), ε s(, s)) T and ρ(s) = 16k [ ε s (16k,s) εx (16k,s) εs ( 16k, s) ε x ( 16k, s) 1 1 6k 16k [εx (x, s) ε s (x, s)dx [εx (x, s) ε s (x, s)dx. Integrate above over [, TforanyT > togive 16k [ ε s (16k,s) εx (16k,s) εs ( 16k, s) ε x ( 16k, s) ds = ρ(t ) ρ() 1 ( 6k [εx (x, s) εs (x, s)dx [εx (x, s)ε s )ds. (x, s)dx 16k Using (A11)and(5), we can easily get that as follows: [ ε s (16k,s) εx (16k,s) ε s ( 16k, s) ε x ( 16k, s) ds ω M ( 16k)ω (ε(, ), ε s(, )) T = ωm ( 16k)ω (ŵ w, ŵ 1 w 1 ) T (76) (77) which together with (A3), (A8) and(a9) yields(a1).