Distributed delayed stabilization of Korteweg-de Vries-Burgers equation under sampled in space measurements

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1 3rd Interntionl Symposium on Mthemticl Theory of Networks nd Systems Hong Kong University of Science nd Technology, Hong Kong, July 16-, 18 Distributed delyed stbiliztion of Korteweg-de Vries-Burgers eqution under smpled in spce mesurements Wen Kng 1 nd Emili Fridmn Abstrct This rticle dels with distributed delyed stbiliztion of 1-D Korteweg-de Vries-Burgers (KdVB) eqution under smpled in spce mesurements. The dely my be uncertin, but bounded by known upper bound. On the bsis of sptilly distributed point mesurements, we construct regionlly stbilizing controller pplied through distributed in spce shpe functions. The existing Lypunov-Krsovskii functionls for het eqution tht depend on the stte derivtive re not pplicble to KdVB eqution becuse of the third sptil derivtive. We suggest new Lypunov-Krsovskii functionl tht leds to regionl stbility conditions of the closed-loop system in terms of liner mtrix inequlities (LMIs). By solving these LMIs, n upper bound on dely tht preserves regionl stbility cn be found, together with n estimte on the set of initil conditions strting from which the stte trjectories of the system re exponentilly converging to zero. This estimte includes priori Lypunov-bsed bounds on the solutions of the open-loop system on the initil time intervl of the length of dely. I. INTRODUCTION In recent decdes, Korteweg-de Vries-Burgers (KdVB) eqution hs drwn lot of ttention s nonliner model of long wves in shllow wter in rectngulr chnnel in which the effects of dispersion, dissiption nd nonlinerity re present (see e.g. [5]). Without the diffusion term, KdVB eqution becomes KdV eqution, which hs been proposed s model of wves on shllow wter surfces. In recent decdes, control of KdV nd KdVB equtions hs been recently extensively studied by reserchers. Regionl boundry stbiliztion of KdV eqution vi the bckstepping method by the stte feedbck [4] or output feedbck controllers [33] hs been considered. Globl distributed stbiliztion [16] or boundry stbiliztion [31] of KdVB eqution ws suggested. For prcticl implementtion of controllers for KdVB eqution, it is importnt to ensure their robustness with respect to smll input dely. In this work, we study distributed stbilizing controllers for KdVB eqution suffering from uncertin input delys. Stbiliztion of systems described by PDEs or cscde of ODE-PDE subject to stte/input / output time-dely or disturbnce hs been studied (see e.g. [17]- [7]). In [8] input dely compenstion ws suggested. In [14] nd [4], dely-independent stbiliztion of het equtions with stte *This work ws supported by Isrel Science Foundtion (Grnt No 118/14). 1 Wen Kng is with School of Automtion nd Electricl Engineering, University of Science nd Technology Beijing, P. R. Chin, nd lso with the Deprtment of Electricl Engineering, Tel Aviv University, Isrel. Emil: kngwen@mss.c.cn Emili Fridmn is with the Deprtment of Electricl Engineering, Tel Aviv University, Isrel. Emil: emili@eng.tu.c.il dely ws considered. Robustness with respect to smll input/output time-vrying dely of semiliner het eqution with globlly Lipschitz nonlinerity ws studied in [9], [1], where distributed control under point nd verged mesurements ws introduced. In [9], [1] globl stbility conditions were derived by using Lypunov-Krsovskii functionls tht depended on the stte derivtive vi the descriptor method [7], [8]. In [1] similr control lws for rection-diffusion eqution were suggested, however, effects of input dely were not studied. The present pper ims t introducing distributed control of KdVB eqution under point mesurements with respect to constnt input dely. Without dely, sufficient LMI conditions for globl stbiliztion of KdVB eqution re the sme s for diffusion-rection equtions derived in [9] nd [1]. However, nlysis of stbility of KdVB eqution with dely is not trivil tsk since the existing Lypunov- Krsovskii functionls from [9] [11] tht depend on the stte derivtive re not pplicble. We introduce novel Lypunov functionl tht depends on the stte only nd leds to LMI-bsed conditions of the closed-loop system. By solving these LMIs, n upper bound on dely tht preserves regionl stbility cn be found together with n estimte on the domin of ttrction. As suggested for the cse of ODEs in [3], our estimte on the domin of ttrction is bsed on the Lypunov-bsed bounds of the solutions on the initil time intervl (of the length of dely), where the system is open-loop. Nottion. The Sobolev spce H k (, 1) with k Z is defined s H k (, 1) = {z : D α z L (, 1), α k} with norm z H k = { D α z L } 1. For squre α k mtrix P the nottion P > indictes tht P is symmetric nd positive-definite, the symbol denotes its symmetric elements. Lemm 1. (Wirtinger inequlity [38]): For < b, let g H 1 (, b) be sclr function with g() = or g(b) =. Then b g 4(b b [ ] ) dg(x) (x)dx π dx. dx Lemm. (Poincré inequlity [34]): For < b, let g H 1 (, b) be sclr function with b g(x)dx =. Then b g (b b [ ] ) dg(x) (x)dx π dx. dx The following extension of Sobolev inequlity will be useful: 66

2 MTNS 18, July 16-, 18 Lemm 3. Let z(x) H 1 (, 1) be sclr function. Then mx x [,1] z (x) z (x)dx + z x(x)dx. Proof: Since z( ) H 1 (, 1) implies z( ) C[, 1] (cf. [3], [35]), by men vlue theorem, there exists c (, 1) such tht z(c) = z(x)dx. Then, by integrtion by prts nd further ppliction of Jensen s nd Young s inequlities, for ll x 1 [, 1] we hve z (x 1 ) = z (c) + x 1 z(x)z c x (x)dx [ ] 1 = z(x)dx x1 + z(x)z c x (x)dx z (x)dx + z x(x)dx. II. SYSTEM DESCRIPTION Inspired by [6], we consider the following KdVB eqution involving both instbility nd dissiption under the periodic boundry conditions: z t (x, t) + z(x, t)z x (x, t) βz xx (x, t) λz(x, t) +z xxx (x, t) = N b j (x)u j (t h), < x < 1, t, z(, t) = z(1, t), z x (, t) = z x (1, t), z xx (, t) = z xx (1, t), z(x, ) = z (x), (1) where x (, 1), β >, λ >, z(x, t) is the stte of KdVB eqution, u j (t) R, (j = 1,,, N) re the control inputs, u j (t) =, t <, nd h > is constnt dely. The dely my be unknown, but bounded by known bound h >. For sufficiently lrge λ >, the open-loop system (with u j (t) ) my be unstble. As in [1], [9], [1], [3], we ssume tht the points = x < x 1 < < x N = 1 divide [, 1] into N intervls Ω j = [, x j ) tht re upper bounded by : < x j = j. The control inputs u j (t) enter (1) through the shpe functions b j (x) such tht { bj (x) =, x / Ω j, j = 1,, N. () b j (x) = 1, otherwise, We ssume further tht sensors provide point mesurements of the stte y j (t) = z( x j, t), x j = + x j, t > h. (3) We design regionlly stbilizing distributed controller { µyj (t), j = 1,, N, t > h, u j (t) = (4), t h, where µ is positive constnt to be determined lter, nd y j (t) is given by (3). Denote the chrcteristic function of the time intervl [, h] by χ [,h] (t). Under the control lw (4), the closed-loop system becomes z t (x, t)+z(x, t)z x (x, t) βz xx (x, t) λz(x, t) +z xxx (x, t) = µ N b j (x)(1 χ [,h] (t))[z(x, t h) f j (x, t h)], < x < 1, t, z(, t) = z(1, t), z x (, t) = z x (1, t), z xx (, t) = z xx (1, t), z(x, ) = z (x), (5) where f j (x, t h) = x x j z ζ (ζ, t h)dξ. (6) A. Well-posedness of (5) subject to (6) Define Hper(, 1 1) = {g H 1 (, 1) : g() = g(1)}, nd g 1 H = P per 1 1 g (x)dx + P [g (x)] dx. Here P 1 nd P re positive constnts tht re relted to the Lypunov- Krsovskii functionl (see (8) below) tht will be used for stbility nlysis. It is obvious tht Hper(, 1 1) is subspce of Sobolev spce H 1 (, 1). Moreover, the norm H 1 per (,1) is equivlent to H1 (,1). By the rguments of [9], the well-posedness of the system (5) subject to (6) cn be obtined: Lemm 4. Assume tht the initil vlue z H 3 (, 1) H 1 per(, 1) stisfies the comptible conditions: z () = z (1), z () = z (1). (7) Then, for ll T >, there exists unique solution to the system (5) subject to (6) from the clss where z C(, T ; H 1 per(, 1)), z t L (, T ; L (, 1)) L (, T ; H 1 per(, 1)). III. REGIONAL STABILITY OF THE DELAYED KDVB EQUATION Consider the following Lypunov-Krsovskii functionl: V ug = V P = P V R = R V Q = Q V S = S V W = W V (t) = V ug + V P + V R + V Q + V S + V W, (8) [ θ T P1 P P P 3 1 t z x(x, t)dx, 1 t 1 t 1 t ] [ θdx, θ = t z(x, t) z(x, s)ds e δ(t s) (s + h t)z (x, s)dsdx, e δ(t s) z (x, s)dsdx, e δ(t s) (s + h t)z x(x, s)dsdx e δ(t s) z x(x, s)dsdx. Here P >, R >, Q >, S >, nd W >. Moreover, we ssume tht for some constnt γ > the following LMI holds: P 1 γ P P 3 + Q e δh >. (9) h ], 67

3 MTNS 18, July 16-, 18 Due to Jensen s inequlity [1] V Q Qe δh Q e δh h t z (x, s)dsdx [ ] t z(x, s)ds dx. LMI (9) gurntees the positivity of the Lypunov-Krsovskii functionl: V (t) γ z (x, t)dx + P z x(x, t)dx. (1) Theorem 1. Consider the system (5) subject to (6). Given positive sclrs, µ > λ, h, δ, nd positive tuning prmeters δ 1 > λ, C > nd C 1 >. Let there exist sclrs γ, P 1 >, P <, P > 1, Q >, R >, S >, W >, α >, P 3 R nd ν R such tht (9), hold, where Φ = Λ = Ξ = W e δh + α, (11) Ξ z=c1, Ξ z= C1, (1) Λ z=c1, Λ z= C1, (13) Φ z=c1, Φ z= C1, (14) P 1 (δ 1 λ) ν Ξ P z P β Ξ = P 1 β P (δ 1 λ) + ν,, (15) Φ 11 Φ 1 ν Φ 15 P 1 µ Φ P µ Φ 5 Φ 33 P z P z P β Φ 44 P Φ 55 P µ Φ 66 P µ Φ 77 Λ 11 P ν Λ 15 Λ P 3 Λ 33 P z P z P β Λ 44 P Λ 55 Λ 66 Φ 11 = P + P 1 λ + Rh + Q + δp 1, Λ 11 = Φ 11 δ 1 P 1, Φ 1 = P 1 µ P, Φ 15 = Λ 15 = P 3 + δp + P λ, Φ = Λ = Qe δh, Φ 5 = P µ P 3, Φ 33 = P 1 β + P λ + ν + Sh + δp + W, Λ 33 = Φ 33 δ 1 P, Φ 44 = Λ 44 = P β, Φ 55 = Λ 55 = Re δh 1 h + δp 3, Φ 66 = Λ 66 = Se δh 1 h, Φ 77 = α π., (16), (17) (18) Denote If M = mx ) {(P 1 + P h + P 3 h + ( Rh + Qh) } 1 + W h)p + (e δ1h 1). (P + Sh P 1 1, MC < C 1, (19) then for ny initil stte z H 3 (, 1) Hper(, 1 1) stisfying the comptible conditions (7) nd z H 1 per < C, system (5) subject to (6) possesses unique solution. Moreover, the solution of (5) subject to (6) stisfies [ ] V (t) Me δ() P 1 z(x)dx + P [z (x)] dx for ll t h. () Proof: We divide the proof into three prts. Step 1: By rguments of [8], [3], we first derive simple bound on V (h) in terms of z such tht V (h) < C 1. Since the solution of the system (5) subject to (6) does not depend on the vlues of z(x, t) for t <, we redefine the initil condition to be constnt: Due to (1), we hve Then z(x, t) = z (x), t. (1) V ug () = [P 1 + P h + P 3 h ] z (x)dx. V () [P 1 + P h + P 3 h + Rh + Qh] z (x)dx +(P + Sh + W h) [z (x)] dx. () We consider V (t) = P 1 z (x, t)dx + P z x(x, t)dx. (3) Given δ >. Assume tht there exists δ 1 > such tht then V (t) δ 1 V (t), t [, h], (4) V (t) + δv (t) δ 1 V (t), t [, h], (5) V (t) e δ1t V (), t [, h], (6) V (t) e δt V () + (e δ1t 1)V (), t [, h]. (7) Substituting () into the inequlity (7), together with the condition (19), we hve [ V (h) M P 1 z (x)dx + P ] 1 [z (x)] dx < C1, (8) if z 1 H = P per 1 1 z (x)dx + P [z (x)] dx < C. (9) Step : We show next the LMIs (9), (1) nd (13) gurntee tht (4) nd (5) hold. 68

4 MTNS 18, July 16-, 18 Differentiting V ug long (5), for t [, h] we obtin We hve V ug = P 1 β z x(x, t)dx + P 1 λ P z(x, t)z x(x, t) t P β z x(x, t) t z x(x, s)dsdx +P z xx(x, t) t z x(x, s)dsdx +P λ z(x, t) t 1 +P +P 3 z (x, t)dx z(x, t)[z(x, t) z(x, t h)]dx t z(x, s)ds[z(x, t) z(x, t h)]dx. (3) V P = P β z xx(x, t)dx + P λ z x(x, t)dx +P z xx(x, t)z x (x, t)z(x, t)dx, (31) V Q +δv Q =Q z (x, t)dx Q e δh z (x, )dx, (3) nd V W +δv W =W z x(x, t)dx W e δh z x(x, )dx. (33) Further by pplying Jensen s inequlity we obtin V R + δv R Rh z (x, t)dx Re δh 1 h nd V S + δv S Sh z x(x, t)dx Se δh 1 h Additionlly, ν [ z(x, t)z xx(x, t)dx + ] 1 z x(x, t)dx [ t z(x, s)ds ] dx, (34) [ t z x(x, s)ds] dx. (35) = ν R. (36) We dd to V (t) + δv (t) the left-hnd side of (36). Then, by tking into ccount (3)-(35), for t [, h] we rrive t V (t) + δv (t) δ 1 V (t) ψ (x, t)λψ(x, t)dx W e δh z x(x, t h)dx, (37) where ψ(x, t) = col{z(x, t), z(x, t h), z x (x, t), z xx (x, t), t z(x, s)ds, t z x(x, s)ds}, (38) nd Λ is given by (17). Similrly, differentiting V (t) long (5) nd dding (36), we hve V (t) δ 1 V (t) = ξ (x, t)ξξ(x, t)dx, t [, h], (39) where ξ(x, t) = col{z(x, t), z x (x, t), z xx (x, t)} nd Ξ is given by (15). As in [36], first we ssume tht z(x, t) ( C 1, C 1 ) x [, 1], t [, h]. (4) From (37) nd (39), it follows tht if Ξ nd Λ for ll z ( C 1, C 1 ), then (4) nd (5) hold. Mtrices Ξ nd Λ given by (15) nd (17) re ffine in z. Thus, Ξ nd Λ for ll z ( C 1, C 1 ) if LMIs (1) nd (13) re fesible. Therefore, (1) nd (13) gurntee tht (4) nd (5) hold. We prove next (4). If the LMI (9) is fesible, then Lemm 3 leds to mx z(x, x 1 t) z (x, t)dx + [z x(x, t)] dx γ z (x, t)dx + P [z x(x, t)] dx. P 1 z (x, t)dx + P [z x(x, t)] dx = V (t) t [, h]. (41) Moreover, from (1), we hve mx z(x, x 1 t) γ z (x, t)dx + P [z x(x, t)] dx V (t) t [, h]. (4) Therefore, in mnner similr to the proof of [36], one cn show tht V (t) < C 1, V (t) < C 1 for ll t [, h]. Thus, (4) nd consequently, (4) nd (5) re true on [, h]. Step 3: We continue to find sufficient conditions in terms of LMIs to gurntee V (t) + δv (t) for ll t > h. Differentiting V nd integrting by prts, we obtin (3)-(35). For t > h, (3)-(31) become V ug = P 1 β z x(x, t)dx + P 1 λ z (x, t)dx P 1 µ N xj z(x, t)[z(x, ) f j (x, )]dx P z(x, t)z x(x, t) t P β z x(x, t) t z x(x, s)dsdx +P z xx(x, t) t z x(x, s)dsdx +P λ z(x, t) t P µ N xj [z(x, ) f j (x, )] t +P z(x, t)[z(x, t) z(x, t h)]dx 1 t +P 3 z(x, s)ds[z(x, t) z(x, t h)]dx, (43) nd V P = P β z xx(x, t)dx + P λ z x(x, t)dx +P z xx(x, t)z x (x, t)z(x, t)dx +P µ N xj z xx (x, t)[z(x, ) f j (x, )]dx. Lemm 1 yields xj Hence, π (44) fj (x, t h)dx = [ x j x x j z ζ (ζ, t h)dξ] dx xj zx(x, t h)dx t > h. α N xj [ ] zx(x, t h) π f j (x, t h) dx (45) 69

5 MTNS 18, July 16-, 18 holds for some constnt α >. By dding to V (t) + δv (t) the equlity (36), nd using (3)-(35), (43)-(45), we obtin V + δv V + δv + α N N xj [ zx(x, ) π f j ]dx (x, ) xj η Φηdx (W e δh α) z x(x, t h)dx, (46) where η = col{z(x, t), z(x, t h), z x (x, t), z xx (x, t), t z(x, s)ds, t z x(x, s)ds, f j (x, t h)}. Mtrix Φ given by (16) is ffine in z. Hence, Φ for ll z ( C 1, C 1 ) if (14) is stisfied. Therefore, (11) nd (14) gurntee V (t) + δv (t), which implies V (t) e δ() V (h) t h. (47) Using (8) nd (47), we obtin (). Fig. 1. Stte z(x, t) with h =.189 nd x j =.1 under the point mesurements IV. EXAMPLE Consider the system (1) with prmeters β =.5 nd λ = 15. Here for the control lw (4) with the point mesurements, by using Ylmip we verify LMI conditions of Theorem 1 with µ =, =.1, δ = 1, δ 1 =, C =.44, C 1 =.5. We find tht the closed-loop system (5) subject to (6) preserves the exponentil stbility for h.189 for ny initil vlues stisfying z H 1 per <.44. Next finite difference method is pplied to compute the stte of the closed-loop system (5) subject to (6) to illustrte the effect of the proposed feedbck control lw (4) with the point mesurements. We choose the sme vlues of prmeters nd the initil condition z (x) =.5 sin(πx), x 1. Hence, z H 1 per = z z <.44. The steps of spce nd time re tken s.5 nd 1 7, respectively. Simultion of solutions under the controller { z( xj, t), t >.189, u j (t) =, t.189 with x j = j = =.1, x j = + x j, j = 1,, 1, where the sptil domin is divided into ten subdomins, shows tht the closed-loop system is exponentilly stble (see Fig. 1). Enlrging the vlue of h until.1, we find tht the solution strting from the sme initil condition is unbounded (see Fig. ). The simultions of the solutions confirm the theoreticl results. V. CONCLUSIONS In this work, we studied distributed control of the KdVB eqution in the presence of uncertin nd bounded constnt dely under the sptilly distributed point mesurements. By constructing novel ugmented Lypunov function, we derived sufficient conditions ensuring tht the closed-loop system is regionlly stble. The future work will be devoted to the extension of the obtined results to the observer-bsed boundry control of nonliner PDEs. Fig.. Stte z(x, t) with h =.1 nd x j =.1 under the point mesurements REFERENCES [1] Azouni, A., Titi, E.S. (14) Feedbck control of nonliner dissiptive systems by finite determining prmeters - A rection-diffusion Prdigm. Evolution Equtions nd Control Theory, 3, pp [] Blogh, A., Krstic, M. () Boundry control of the Kortewegde Vries-Burgers eqution: further results on stbiliztion nd wellposedness, with numericl demonstrtion. IEEE Trnsctions on Automtic Control, 45, pp [3] Brezis, H. (11) Functionl nlysis, Sobolev spces nd prtil differentil equtions. New York: Springer-Verlg. [4] Cerp, E., Coron, J.-M. (13) Rpid stbiliztion for Kortewegde Vries eqution from the left Dirichlet boundry condition. IEEE Trnsctions on Automtic Control. 58, pp [5] Demiry, H. (4) A trvelling wve solution to the KdV-Burgers eqution. Applied Mthemtics nd Computtion, 154, pp [6] Feng, B.F., Kwhr, T. () Sttionry trvelling-wve solutions of n unstble KdV-Burgers eqution. Physic D: Nonliner Phenomen,137, pp [7] Fridmn, E. (1) New Lypunov-Krsovskii functionls for stbility of liner retrded nd neutrl type systems, Systems & Control Letters, 43, pp [8] Fridmn, E. (14) Introduction to Time-Dely Systems: Anlysis nd Control, Bsel: Birkhäuser. 7

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