Output feedback stabilisation for a cascaded wave PDE-ODE system subject to boundary control matched disturbance

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1 INTERNATIONAL JOURNAL OF CONTROL, 216 VOL.89,NO.12, Output feedback stabilisation for a cascaded wave PDE-ODE system subject to boundary control matched disturbance ua-cheng Zhou a, Bao-Zhu Guo a,b and Ze-ao Wu a a The Key Laboratory of System and Control, Academy of Mathematics and Systems Science Academia Sinica Beijing, P.R. China; b School of Computer Science and Applied Mathematics University of the Witwatersrand Johannesburg, South Africa ABSTRACT In this paper, we consider output feedback stabilisation for a wave PDE-ODE system with Dirichlet boundary interconnection and external disturbance flowing the control end. We first design a variable structure unknown input type state observer which is shown to be exponentially convergent. Then, we estimate the disturbance in terms of the estimated state, an idea from active disturbance rejection control. These enable us to design an observer-based output feedback stabilising control to this uncertain PDE-ODE system. ARTICLE ISTORY Received 2 October 215 Accepted 22 February 216 KEYWORDS Output feedback; stabilisation; wave PDE-ODE system; backstepping; active disturbance rejection control; disturbance AMS SUBJECT CLASSIFICATIONS 35L5, 93B52, 37L15, 93D15, 93B51 1. Introduction In recent years, stabilisation for systems described by partial differential equations PDEs) subject to external disturbance has attracted increasing attention. This is because in many situations, the control is used not only to guarantee the system to be normally operated in an ideal operation environment but also to be normally operated in the environment with uncertainties coming from either as internal or external disturbance. The stability issue of system with disturbance is therefore significant both theoretically and practically. To deal with disturbances, many different approaches have been developed. The backstepping method which was originally applied to stabilise the PDEs without disturbance in Krstic and Smyshlyaev 28), SmyshlyaevandKrstic 24), together with the adaptive control method, is designed in Guo and Guo 213), Krstic21) to stabilise one-dimensional 1D) wave equations where the uncertainties are supposed to be unknown parameters in disturbance. The internal model principle for output regulationhasbeenappliedinbyrnes,lauko,gilliam,and Shubov 2),RebarberandWeiss23),amongmany others, to infinite dimensional systems to reject disturbance generated from exogenous system. The Lyapunov functional approach is presented Ge, Zhang, and e, 211; e, Zhang, and Ge, 212) for disturbance rejection, where boundary feedbacks are designed for 1D Euler- Bernoulli beam with spatial and boundary disturbance. Another powerful method in dealing with uncertainties is based on active disturbance rejection control ADRC) which was first proposed by an in 29.InGuoand Zhou 214, 215), the boundary controls are designed by ADRC for multi-dimensional wave equation and Kirchhoff plate equation with control-matched disturbance that depends on both time and spatial variables. Recently, the boundary sliding mode control SMC) is designed for 1D heat, wave, Euler-Bernoulli, and Schrödinger equations with boundary input disturbance in Cheng, Radisavljevic, and Su 211),GuoandJin213a, 213b), Guo and Liu 214), Pisano, Orlov, and Usai 211). owever, there are a few works on coupled systems. In Krstic 29, Chapter 16), stabilisation for a cascade system of wave equation with linear time invariant finite-dimensional system is considered, where the backstepping approach is applied but no disturbance is concerned. In Wang, Liu, Ren, and Chen 215), boundary feedback stabilisation for a cascade system of heat PDE-ODE system with Dirichlet/Neumann interconnection and external disturbanceflowingthecontrolendisdiscussedbythesmc and the backstepping method. Motivated mainly by Krstic 29, Chapter 16) and Wang et al. 215) where only state feedback is concerned, weareconcernedwith,inthispaper,stabilisationfor the following wave PDE-ODE cascade system through CONTACT ua-cheng Zhou hczhou@amss.ac.cn 216 Informa UK Limited, trading as Taylor & Francis Group

2 INTERNATIONAL JOURNAL OF CONTROL 2397 Ut) ) PDE u t,t),u1,t),u t 1,t)) u tt x, t) =u xx x, t) ux, ) u x,t)= u x 1,t)=Ut)) u t x, ) Bu,t) ODE Ẋt) =AXt)Bu,t) X) CXt) Figure 1. Block diagram of the coupled wave PDE-ODE system 1.1). Dirichlet interconnection: Ẋt) = AXt) Bu, t), t >, u tt x, t) = u xx x, t), x, 1), t >, u x, t) =, t, 1.1) u x 1, t) = U t) ), t, y out t) = CXt), u t, t), u1, t), u t 1, t)), where X R n 1 and u, u t ) 1, 1) L 2, 1) are the states of ODE and PDE, respectively, U L 2 loc, ) is the control input of the entire system, A is an n n matrix, B R n 1,and) is the external disturbance. It is supposed that A, B) is stabilisable, C, A)isobservable, and d loc 1, ) is uniformly bounded: ) M for some M > andallt. The objective of the present paper is to design an output feedback control that can stabilise system 1.1) depicted in Figure 1 in the state space = R n 1, 1) L 2, 1) by rejecting the disturbance ). The inner product of is given by X 1, f 1, g 1 ), X 2, f 2, g 2 ) = X 1 X 2 kf 1 1) f 2 1) f 1 x) f 2 x)dx g 1 x)g 2 x)dx, 1.2) for all X 1, f 1, g 1 ), X 2, f 2, g 2 ), andk > isapositive constant. We proceed as follows. In Section 2, we design a variable structure unknown input observer for system 1.1) and establish convergence of the observer. The exponentialstabilityoftheclosed-loopsystemispresented.insection 3, we transform system 1.1) into an equivalent target system for which a state feedback is easily designed. By ADRC approach, we design a disturbance estimator to estimate the disturbance. The observer and estimator based output feedback is then designed by compensating the disturbance in feedback. The closed-loop system is shown to be asymptotically stable, followed by concluding remarks in Section State observer In this section, we first design an unknown type state observer to recover the state of system 1.1) viatheoutput. By the estimated state, we can estimate the disturbance by an idea of extended state observer for ODEs. The disturbance is then compensated in the feedback loop. We design a variable structure unknown input type state observer for system 1.1)as follows: Xt) = A Xt) LC X CX) Bû, t), t >, û tt x, t) = û xx x, t), x, 1), t >, û x, t) = c 1 û t, t) u t, t), t, û x 1, t) U t) c 2 û t 1, t) u t 1, t) c 3 û1, t) u1, t) M 1 signû t 1, t) u t 1, t)) M 2 signû1, t) u1, t)), t, ûx, ) = û x), û t x, ) = û 1 x), 2.1) where M 1 > M, M 2 > M M 1,andc 1, c 2, c 3 are positive design parameters. The symbolic function is a multivalued function defined by 1, x >, signx) = 1, 1, x =, 1, x <. Toensurethefunctionontheright-handsideofthe fourth equation of 2.1) to be measurable, for any T > and f L 2, T), we restrict the set-valued composition of the symbolic function as follows: { sign f t)) = gt) gt) { } sign f t)), f t), = f t) L 2 ), f t), 1, f t) =, where = {τ fτ) = }. Introduce the variable Xt), ũx, t), ũ t x, t)) = Xt) Xt), ûx, t) ux, t), û t x, t) u t x, t))) 2.2)

3 2398 UA-CENG ZOU ET AL. to be the error. Then Xt), ũx, t), ũ t x, t)) satisfies Xt) = A LC) Xt) Bũ, t), ũ tt x, t) = ũ xx x, t), ũ x, t) = c 1 ũ t, t), 2.3) ũ x 1, t) c 2 ũ t 1, t) c 3 ũ1, t) M 1 signũ t 1, t)) M 2 signũ1, t)) ), ũx, ) = ũ x), ũ t x, ) = ũ 1 x). For the error system 2.2), we have Theorem 2.1 which shows that the observer 2.1) is convergent. Theorem 2.1: For any initial value X), ũ, ũ 1 ) R n 2, 1) 1, 1) satisfying compatible conditions: ũ ) = c 1 ũ 1 ), ũ 1) c 2ũ 1 1) c 3 ũ 1) M 1 signũ 1 1)) 2.4) M 2 signũ 1)) d), and d loc 1, ), 2.1) admits at least one partial differential inclusion solution Xt), ũ, t), ũ t, t)) C, ; R n 2, 1) 1, 1)). Moreover, any partial differential inclusion solution Xt), ũ, t), ũ t, t)) C, ; R n 2, 1) 1, 1)) is exponentially stable in the sense that Et) C 1 E)e ω 1t, 2.5) forsomepositiveconstantsc 1 and ω 1,whereEt) is given by Et) := Xt) 2 R n ũ2 1, t) ũt 2 x, t)dx. ũ 2 x x, t)dx Proof: We first prove the ũ part in 2.3). Actually, by Guo and Jin 215, Theorem1),the ũ part in 2.3) has at least one solution ũ, t), ũ t, t)) C, ; 2, 1) 1, 1)),whichsatisfies ũ 2 1, t) ũ 2 x x, t) ũ2 t x, t)dx C 2 e C 3t ũ 2 1) ũ 2 x) ũ2 1 x)dx. 2.6) Next we show the X part in 2.3). By the Sobolev embedding 1, 1) C, 1), ũ 2, t) C 4 ũ 2 1, t) ũ 2 x x, t)dx for some constant C 4 >, 2.7) and ũ, t) C, ). Using the constant variational formula, the solution of X part is given by Xt) = e ALC)t X) e ALC)ts) Bũ, s)ds. 2.8) Since A LC is urwitz, there are positive constants C 5 and C 6 > suchthat e ALC)t C 5 e C 6t. 2.9) This together with 2.6), 2.7), and 2.8)yields Xt) R n e ALC)t X) R n C 5 e C6t X) R n C 2 C 4 C 5 B ũ 2 1) ũ 2 x) ũ2 1 x)dx Since e ALC)ts) Bũ, s)ds R n e C 6ts) e C 3s ds. 2.1) e C3t e C 6t e C6ts) e C3s, C 6 C 3, ds = C 6 C ) e C6t t, C 6 = C 3, we obtain 2.5)from2.6)and2.1). 3. Disturbance estimator and output feedback control In this section, we design an observer-based output stabilising feedback for system 1.1). To this end, we first introduce a transformation X, u, u t ) X, v, v t )asfollows Krstic, 29, Chapter 16): Xt) = Xt), vx, t) = ux, t) v t x, t) = u t x, t) KBux, t) μx y)uy, t)dy mx y)u t y, t)dy γx)xt), μx y)u t y, t)dy m x y)uy, t)dy γx)axt), 3.1)

4 INTERNATIONAL JOURNAL OF CONTROL 2399 where μs) = s γξ)abdξ, ms) = γx) = KMx), Mx) = I s e γξ)bdξ, A 2 I x I. 3.2) with I being an n n identify matrix. This transformation brings system 1.1) into the following system: Ẋt) = A BK)Xt) Bv, t), v tt x, t) = v xx x, t), v x, t) =, t, v x 1, t) = U t) ) μ 1 y)uy, t)dy m 1 y)u t y, t)dyγ 1)Xt), 3.3) where c 1 is positive. We then obtain the target system following: Ẋt) = A BK)Xt) Bw, t), t >, w tt x, t) = w xx x, t), x, 1), t >, w x, t) = cw t, t), t, w x 1, t) = U t) ) μ 1 y)uy, t)dy m 1 y)u t y, t)dy 3.7) γ 1)Xt) cu t 1, t) ckbu1, t) c μ1 y)u t y, t)dy 1 c m 1y)uy, t)dycγ1)axt), t. The inverse of transformation 3.6) canbefoundas follows where K is chosen so that A BK is urwitz. The transformation 3.1)is invertible,that is Xt) = Xt), ux, t) = vx, t) σx y)vy, t)dy nx y)v t y, t)dy ρx)xt), u t x, t) = v t x, t) KBvx, t) where σs) = n x y)vy, t)dy σx y)v t y, t)dy ρx)axt), 3.4) s ρξ)abdξ,ns) = ρx) =KNx), Mx) = I I s ρξ)bdξ, A BK) 2 I x e. 3.5) We further introduce X, v, v t ) X, w, w t )by Xt) = Xt), wx, t) = vx, t) c v t y, t)dy, w t x, t) = v t x, t) cv x x, t), 3.6) Xt) = Xt), w x y, t) cw t y, t) vx, t) = w, t) dy, 1 c 2 v t x, t) = w t x, t) cw x x, t). 3.8) 1 c 2 For the target system 3.7), we can easily design a stabilising state feedback. Since we have state estimation through the observer claimed by Theorem 2.1, we design naturally an output feedback control as follows: U t) = U t) μ 1 y)cm 1 y)) ) ûy, t)dy m 1 y) cμ1 y) ) û t y, t)dy γ 1) cγ1)a ) Xt) ckbu1, t) { cu t 1, t)k u1, t) μ1 y)ûy, t)dy c c m1 y)û t y, t)dy γ1) Xt) û t x, t) KBûx, t) μx y)û t y, t)dy m x y)ûy, t)dy dx } γx)dxa Xt), t, 3.9)

5 24 UA-CENG ZOU ET AL. where U t) is an auxiliary control. Under feedback control 3.9), the closed-loop of target system 3.7)is Ẋt) = A BK)Xt) Bw, t), t >, w tt x, t) = w xx x, t), x, 1), t >, 3.1) w x, t) = cw t, t), t, w x 1, t) =kw1, t) U t) f t) ), t, where f t) = μ 1 y) cm 1 y)) ) ũy, t)dy m 1 y) cμ1 y) ) ũ t y, t)dy γ 1) cγ1)a ) Xt) { k μ1 y)ũy, t)dy m1 y)ũ t y, t)dy c γ1) Xt) c γx)dxa Xt) ũ t x, t) KBûx, t) μx y)ũ t y, t)dy } m x y)ũy, t)dy dx, 3.11) satisfies, by Theorem 2.1,that f t) C f e ω 1t, t, 3.12) forsomepositiveconstantsc f and ω 1. The remaining is to design a continuous control U t) to stabilise 3.1) in the presence of disturbances ) and ft). We write 3.1)as Xt) d w, t) = A w t, t) Xt) w, t) BU t) w t, t) f t) )), 3.13) where B =,,δx 1)),andA is a linear operator defined in R n 1, 1) L 2, 1) by X A BK)X Bf) X A f = g, f DA), g f g DA) ={X, f, g) R n 2, 1) 1, 1) : f ) = cg), f 1) =kf1)}. 3.14) Lemma 3.1: The operator A defined in 3.14) generates an exponential stable C -semigroup. Proof: It suffices to show that the following system Ẋt) = A BK)Xt) Bw, t), t >, w tt x, t) = w xx x, t), x, 1), t >, 3.15) w x, t) = cw t, t), t, w x 1, t) =kw1, t), t, isexponentiallystable.actually,itiswellknownthatthe w-part is exponentially stable in 1, 1) L 2, 1), thatis,thereareconstantsc 1, C 2 > suchthat w 2 1, t) C 1 e C 2t w 2 x x, t) w 2 t x, t) dx w 2 1, ) wx 2 x, ) w2 t x, )dx. By embedding inclusion, 1, 1) C, 1), w 2, t) C 3 w 2 1, t) w 2 x x, t)dx 3.16) for some constant C 3 >. 3.17) ence w, ) C, ). Apply the constant variational formula to obtain the solution of ODE part as Xt) = e ABK)t X) e ABK)ts) Bw, s)ds. 3.18) Since A BK is urwitz, there are positive constants C 4 and C 5 > suchthat e ABK)t C 4 e C 5t. 3.19) Since e C2t e C 5t e C5ts) e C2s, C 5 C 2, ds = C 5 C 2 3.2) e C5t t, C 5 = C 2, it follows from 3.17), 3.18), and 3.19) that X part in 3.15) is exponentially stable. This completes the proof of the lemma. The following result is straightforward Weiss, Staffans, & Tucsnak, 21) whereforadmissibilityof control operator, we refer to Weiss 1989). Proposition 3.1: The operator A defined in 3.14) generates a C -semigroup of contractions e At on and B is admissible to e At. Therefore, for any initial value X), w, ), ẇ, )) and control input U

6 INTERNATIONAL JOURNAL OF CONTROL 241 L 2 loc, ) and f, d) L2 loc, ))2, 3.13) admits a unique solution Xt), w, t), ẇ, t)). By Proposition 3.1, the weak) solution of 3.1) satisfies X Y X Y d w, f = w, A f w t g w t g Y 3.21) U t) f t) )B f, g Y, f, g) DA ), where Y, f, g) DA ) is called a test function. A simple computation shows that A, the adjoint operator of A, is given by A X f = g A BK) X g f X, f DA ), g 3.22) DA ) = { X, f, g) R n 2, 1) 1, 1) : f ) =cg), f 1) =kf1), B X = }, By 3.21), d X Y kw1, t) f 1) w x x, t) f x) w t x, t)gx)dx = X A BK) Y w x x, t)g x) w t x, t) f x)dx kw1, t)g1) g1)u f t) ). 3.23) Choose X, f x), gx)) =,, x) DA ) and substitute into 3.23)toobtain d w t x, t)xdx =k 1)w1, t) w, t) U f t) ). 3.24) Same as X, u, u t ) X, w, w t )by3.1) and3.6), we have X, û, û t ) X, ŵ, ŵ t ). Replace w, w t )in3.24) by ŵ, ŵ t ) to have d ŵ t x, t)xdx = d w t x, t)xdx d w t x, t)xdx =k 1)w1, t) w, t) U f t) ) Let Then d w t x, t)xdx = U f t) ) k 1)ŵ1, t) ŵ, t) k 1) w1, t) w, t) d w t x, t)xdx. zt) = ŵ t x, t)xdx, z 1 t) =k 1)ŵ1, t) ŵ, t), z 2 t) = k 1) w1, t) w, t) f t), z 3 t) = w t x, t)xdx. żt) = U ) z 1 t) z 2 t) ż 3 t), 3.25) and by Theorem 2.1 and 3.12), there exists a constant C z > suchthat z 2 t) z 3 t) C z e ω 1t, t. 3.26) Now we are in a position to estimate the disturbance ) by the active disturbance rejection approach developed in Guo and Zhao 211) where the constant high gain is used. The extended state observer with time varying high gain for system 3.25) is designed as { ẑt) = U ) z 1 t) rt)ẑt) zt), ) =r 2 t)ẑt) zt), 3.27) where r C R, R ) is a time varying gain that is required to satisfy ṙt) >, lim rt) =, ṙt) rt) M, t for some M >, 3.28) ) lim =, lim rt) rt)eω 1t = for ω 1 > in3.26). Lemma 3.2: Let ẑt), )) be the solution of 3.27). Then Proof: Let lim ẑt) zt) ) ) =. 3.29) zt) = rt)ẑt) zt), ) = ) ). 3.3)

7 242 UA-CENG ZOU ET AL. Then zt), )) satisfies zt) =rt) zt) rt) ) rt) zt) ṙt) rt)z 2 t) ż 3 t), ) =rt) zt) ). 3.31) Set ηt) = zt) rt)z 3 t).then3.31) is equivalent to ηt) =rt)ηt) rt) ) ṙt) rt) ηt) rt)z 2 t) r 2 t)z 3 t), ) =rt)ηt) r 2 t)z 3 t) ). 3.32) By the local Lipschitz condition of the right-hand side of 3.32), we know that 3.32) has a local classical solution. Now, we show that 3.32)hasaglobalsolutionbytheLyapunovfunctionpresentedbelow.Actually,wecandefine a Lyapunov function for system 3.32)as follows: V x 1, x 2 ) = x x2 2 x 1x ) Then Vx 1, x 2 ) is positive definite: 1 2 V x 1, x 2 ) x 2 1 x2 2 2V x 1, x 2 ), x 1, x 2 R. Differentiate V ηt), )) along the solution of 3.32)to obtain V ηt), )) = 2ηt)rt)ηt) rt) ) ṙt) rt) ηt) rt)z 2t) r 2 t)z 3 t) rt)ηt) rt) ) ṙt) rt) ηt) rt)z 2 t) r 2 t)z 3 t) ) ηt)rt)ηt) r 2 t)z 3 t) ) 3 )rt)ηt) r 2 t)z 3 t) ) = rt) 2ṙt) η 2 t) rt) d 2 t) )ηt) rt) 3 ) ηt) )ṙt) rt) rt)z 2t)2ηt) ) r 2 t)z 3 t)ηt) 2 ) 1 2 φt)v ηt), )) ηt), )) 3rt) z 2 t) 3r 2 t) z 3 t) 4 ) 1 2 φt)v ηt), )) 23rt) z 2 t) 3r 2 t) z 3 t) 4 ) V ηt), )), 3.34) where φt) = rt) 3sup ṙt) t rt). By assumption 3.28), lim t φt) =. Re-organising 3.34)gives d V ηt), )) 1 V 4 φt) ηt), )) 3rt) z 2 t) 3r 2 t) z 3 t) 4 ), 3.35) which shows that V ηt), )) V η), d))e 1 4 φs)ds s φτ)dτ ds. φs)ds 3rs) z 2s) 3r 2 s) z 3 s) 4 ds) e 1 4 e ) Equation 3.36) implies that the local solution never blows up, so the global solution of 3.32) exists. Since e 1 4 φs)ds as t, wecanapplythel ospital ruletotheseconermoftheright-handsideof3.36), to obtain lim 3rs) z 2s) 3r 2 s) z 3 s) 4 ds) e 4 1 e 4 1 φs)ds 3rt) z 2 t) 3r 2 t) z 3 t) 4 ) = 4 lim φt) = 4 lim 3 z 2 t) rt) φt) 3rt) z 3t) rt) φt) 4 ) s φτ)dτ ds rt) rt) φt) 3.37) =, where in the last equality of 3.37), we used 3.26) and 3.28). It then follows from 3.36)and3.37)that lim V ηt), )) =, which implies lim ηt) ) =. 3.38) Furthermore, since zt) = ηt) rt)z 3 t), it follows from 3.28)thatlim zt) =. By 3.28)andẑt) zt) = zt)/rt),wefinallyobtain lim = ẑt) zt) =. 3.39) Equation 3.29)thenfollowsfrom3.38)and3.39). Sincewehaveanestimationofdisturbanceanhesystem 3.1) withu t), )) = isstableft) isasmall perturbation by 3.12)), we compensate the disturbance

8 INTERNATIONAL JOURNAL OF CONTROL 243 by designing U t) = ). 3.4) It is seen that the term in the right-hand side of 3.4) is useocanceltheeffectofthedisturbance.thisisjustthe estimation/cancellation nature of ADRC. Under feedback control 3.4), the closed-loop of system 3.1)becomes Ẋt) = A BK)Xt) Bw, t), t >, w tt x, t) = w xx x, t), x, 1), t >, 3.41) w x, t) = cw t, t), t, w x 1, t) =kw1, t) f t) ), t. Lemma 3.3: For any initial value X), w, ), w t, )). If f t) ) as t, then 3.41) admits a unique solution X, w, ẇ) satisfying lim Xt) 2 R n w2 1, t) wx 2 x, t)dx w 2 t x, t)dx =. 3.42) Proof: The existence and uniqueness of solution of system 3.41) has been proved by Proposition 3.1. Since f t) ) ast,foranygivenκ>, we may suppose that f t) ) κ for all t > t for some t >. Now, we write the solution of 3.41)as X w, t) w t, t) X) = e At w, ) w t, ) X) = e At w, ) e Att ) e Ats) B f s) w t, ) ds))ds e Ats) B f s) ds))ds. t e Ats) B f s) ds))ds 3.43) The admissibility of B claimed by Proposition 3.1 implies that e Ats) B f s) ds))ds 2 C t f d) 2 L 2,t) C tt 2 f d) 2 L, ), t >, for some constant C t that is independent of f s) ds). Since by Lemma 3.1 e At is exponentially stable, it follows from Remark 2.6 of Weiss 1989) that e Ats) B f s) ds))ds t e Ats) B f s) ds))ds t L f d) L t, ) Lκ, where L is a constant which is independent of ),and { ut), t τ, u v)t) = τ vt), t >τ. Suppose that e At L e ω2t for some L, ω 2 >. We have Xt) X) w, t) L e ω 2t w, ) w t, t) w t, ) L C t t 2 eω 2tt ) d L,t,L 2 Ɣ)) Lκ. 3.44) Passing to the limit as t for 3.44), we finally obtain Xt) lim w, t) Lκ. 3.45) w t, t) This proves 3.42). Combining 3.9) and3.4), an output feedback control for the system 1.1) is designed as follows: U t) = ) μ 1 y) cm 1 y)) ) ûy, t)dy m 1 y) cμ1 y) ) û t y, t)dy γ 1) cγ1)a ) Xt) ckbu1, t) { cu t 1, t) k u1, t) μ1 y)ûy, t)dy c c m1 y)û t y, t)dy γ1) Xt) û t x, t) KBûx, t) μx y)û t y, t)dy m x y)ûy, t)dy dx } γx)dxa Xt) t. 3.46)

9 244 UA-CENG ZOU ET AL. The closed-loop system of 1.1)then becomes Ẋt) = AXt) Bu, t), t >, u tt x, t) = u xx x, t), x, 1), t >, u x, t) =, t, u x 1, t) = U t) ), t, Xt) = A Xt) LC X CX) Bû, t), t >, û tt x, t) = û xx x, t), x, 1), t >, û x, t) = c 1 û t, t) u t, t), t, 3.47) û x 1, t) U t) c 2 û t 1, t) u t 1, t) c 3 û1, t) u1, t) M 1 signû t 1, t) u t 1, t)) M 2 signû1, t) u1, t)), t, ẑt) = z 1 t) rt)ẑt) zt), ) =r 2 t)ẑt) zt), ûx, ) = û x), û t x, ) = û 1 x), x 1, where Ut) is given by 3.46). Theorem 3.1: With the output feedback control 3.46), for any initial value X), u, ), u t, )) R n 2, 1) 1, 1), X), û, ), û t, )) R n 2, 1) 1, 1), ẑ), d) R, any partial differential inclusion solution of closed-loop system 3.47) satisfies where Ft) = Xt) 2 R n u2 1, t) lim Ft) =, 3.48) u 2 x x, t)dx ut 2 x, t)dx Xt) 2 Rn û1, t) 2 û 2 x x, t)dx ẑt) ) ). ût 2 x, t)dx Proof: By the error variables ũx, t) = ûx, t) ux, t) in 2.2) and zt), )) = rt)ẑt) zt), ) )) in 3.3), the system 3.47) is equivalent to Ẋt) = AXt) Bu, t), t >, u tt x, t) = u xx x, t), x, 1), t >, u x, t) =, t, u x 1, t) = U t) ), t, 3.49) Xt) = A LC) Xt) Bũ, t), t >, ũ tt x, t) = ũ xx x, t), x, 1), t >, ũ x, t) = c 1 ũ t, t), t, ũ x 1, t) c 2 ũ t 1, t) c 3 ũ1, t) M 1 signũ t 1, t)) M 2 signũ1, t)) ), t, zt) =rt) zt) rt) ) rt) zt) ṙt) rt)z 2 t) ż 3 t), ) =rt) zt) ). The X, ũ) part of 3.49)isjustthe2.3)anhe z, d) part of 3.49)is3.31). By Theorem 2.1 and Lemma 3.2, ũ 2 x x, t) ũ2 t x, t)dx ũ2 1, t) ẑt) zt) ) ) ast. 3.5) Now, we show convergence of the X, u) part of3.49). To this end, we rewrite X, u)part asfollows: Ẋt) = AXt) Bu, t), u tt x, t) = u xx x, t), x, 1), t >, u x, t) =, u x 1, t) = f t) ) μ 1 y) cm 1 y)))uy, t)dy m 1 y) cμ1 y))u t y, t)dy γ 1) cγ1)a)xt) { ckbu1, t) cu t 1, t) k u1, t) μ1 y)uy, t)dy 3.51) 1 m1 y)u t y, t)dy γ1) Xt) 1 c u t x, t) KBux, t) c μx y)u t y, t)dy m x y)uy, t)dy dx } γx)dxaxt), where ft) is given by 3.11). owever, 3.51) isexactly thesameasthatintheclosed-loopsystem3.41) under the transformations 3.1) and3.6). The convergence of 3.51) followsfrom3.5), 3.11), and lemma 3.3. This ends the proof of the theorem. 4. Concluding remarks In this paper, we consider stabilisation for a wave PDE- ODE cascade system with boundary control and control matcheddisturbance.wefirstdesignanunknowninput

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