Robustness analysis for the boundary control of the string equation
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1 Routne analyi for the oundary control of the tring equation Martin GUGAT Mario SIGALOTTI and Mariu TUCSNAK I INTRODUCTION AND MAIN RESULTS In thi paper we conider the infinite dimenional ytem determined whoe tate i determined y the equation z t x t z x t = wt x 0 t > 0 x zx 0 = z 0 x z t x 0 = z x x 0 z0 t = 0 t > 0 3 z t = a z t z t t > 0 x t The aove ytem i a model for the viration of a tring which i fixed at the end x = 0 and which i ujected to a oundary damping depending on the velocity and on the poition at the end x = The term w in the right hand ide of tand for a contant with repect to x perturating force In the cae w 0 the initial and oundary value prolem - ha een tudied in particular in [] and in [] In [] the author tudied the Riez ai generation and the optimal decay propertie for a = and = 0 wherea in [] it ha een hown that for any a > 0 and 0 the ytem determined y - i exponentially tale in the tate pace X = V L 0 where V = { ϕ H 0 φ0 = 0 } and H 0 tand for the Soolev pace of function in L 0 with generalized derivative in L 0 One of the main reult in [] aert that at given a the et decay rate of the olution of - i achieved for = 0 Thi mean that the preence of the poition term z t in the feedack law doe not improve the decay rate of the ytem The aim of thi work i to how that the poition term z t can improve the routne propertie of the conidered ytem with repect to time dependent perturation acting in the right hand ide of In order to give a precie tatement of our reult we conider an output law given y yt = z 0 t 5 x M Gugat i with Lehrtuhl II für Angewandte Mathematik Friedrich- Alexander Univerität Nürnerg-Erlangen Martentr 3 Erlangen Germany gugat@amuni-erlangende M Sigalotti and M Tucnak are with Intitut Élie Cartan UMR750 Univerité Henri Poincaré Nancy BP Vandœuvrelè-Nancy Cedex France marioigalotti@inriafr tucnak@iecnu-nancyfr Definition : For γ > 0 we ay that the ytem - 5 ha the γ-routne property if it olution with z 0 = z = 0 atify the etimate y L 0 γ w L 0 for all w L 0 6 For given a > 0 and 0 the infinum of the et of thoe γ atifying 6 denoted y γ 0 a i called the routne coefficient aociated to the pair a One of our main reult how that the introduction of the poition term in the feedack law improve the routne propertie of our ytem More preciely the following reult hold 3 Theorem : For a the routne coefficient defined aove atifie = γ 0 a 0 > inf γ 0a > 7 0 The aove reult give y analytical method ufficient condition for a and in order to have γ 0 a < We conjecture that thi condition i atified for a larger cla of pair a and we give numerical imulation upporting thi aertion An important quetion tackled in thi work conit in determining a pair a 0 0 minimizing γ 0 We how elow that uch a pair exit Moreover we give y numerical imulation approximate value for a 0 0 and γ 0 a 0 0 II COMPUTATION OF THE TRANSFER FUNCTION By applying the Laplace tranform with repect to the time to the aove ytem we get that ẑx ẑ x x = ŵ x 0 C 0 ẑ0 = 0 C 0 ẑ x = a ẑ C 0 3 From it follow that ẑx = A inhx B cohx ŵ for x 0 and C 0 From the aove relation and it follow that B = w o that ẑx = A inhx ŵ cohx ŵ for x 0 and C 0 A a conequence ẑ ŵ x = A cohx inhx 5 x
2 for x 0 and C 0 From 3 and 5 it follow that for every C 0 we have A coh ŵ inh = = a which implie that A inh ŵ coh ŵ A coh a inh = [ a = coh inh ] ŵ It follow that A = [ a ] coh inh ŵ coh a inh ŵ C 0 The aove relation comined to 5 and to 5 yield that the output y i related to the perturation w y the relation ŷ = Hŵ C 0 with the tranfer function G given y G = a coh inh coh a inh III ROBUSTNESS ANALYSIS C 0 6 We want to chooe among thoe a and for which the ytem - i exponentially tale thoe for which the norm of G defined in 6 in the Hardy pace H C 0 i the mallet poile By the maximum principle the maximum of G on C 0 will e attained on the imaginary axi By uing thi fact and the Paley-Wiener theorem it follow that: Propoition 3: For every a > 0 we have γ 0 a = up Giω ω R By uing the aove fact it follow that the relevant function we have to invetigate i f : R R defined y fω = Giω ω R After ome imple calculation we otain that fω = a co ω [ in ω ω co ω] a ω in ω ω co ω in ω It can e eaily checked that the aove formula implie that fω = inc ω a in ω co ω ω in ω a in ω co ω inc ω where for x R we have ued the notation inc x = inx x Notice that for a = and = 0 we have fω = inc ω o that γ0 0 = In the equel we dicu condition for the inequality up fω < 3 ω R and we prove Theorem Notice firt that f0 = In particular for = 0 we have f0 = Thu the inequality > 0 i a neceary condition for 3 Moreover we ee that f0 > / f = a So another neceary condition for 3 i the inequality < a 3 Thi implie in particular a < / and < / For all < / we have f0 > f/3 = 3 9 a 3 a Since thi value i trictly decreaing a a function of a and a < / thi implie f/ = It i eay to check that Theorem follow from the following equence of lemma Lemma 3: Aume that a [ / 0 / and < a 33 up fω < ω Proof Define P ω = a in ω ω co ω ω in = a a co ω ω ω co inc inc ω Qω = a in ω coω inc ω = a in ω ω ω coω co inc co ω inc ω
3 Hence for all ω and a we have P ω a ω inc inc ω ω Qω inc Let S = inc ω and S = inc = fω = S P ω Qω S a S /S S a S a S /S = S a < i equivalent to 33 and the aertion follow Remark 33: Note that for < / we have f0 > /8/ / = 078 Lemma 3: Let S = /3 Aume that a [ /S 0 / and [S ] S S < a S 3 up ω [3/ ] fω < Proof For all ω [3/ ] we have P ω a ω inc inc ω Qω ω inc Let S = inc ω and S = inc 3 8 fω = S P ω Qω S S = 3 a S S S a S S < S i equivalent to 3 and the aertion follow f = a a So another neceary condition for 3 i the inequality [ ] < a 35 Lemma 35: Let S = / Aume that a [ / 0 / and [S S S 8 ] < 9 [ < S ] a [S ] 36 up fω < ω [/ 3/] Proof For all ω [/ 3/] we have P ω a a inc ω inc ω Qω a inc ω inc ω Let S = inc ω and S = inc = S 6 /9 fω = S P ω Qω S S a a a S a a S S S S S S a S 8 9 a a S S a S 8 9 i equivalent to 36 and the aertion follow < Lemma 36: Let S = inc Aume that a [ / 0 / and [ S 3 co S ] [ ] < S in up fω < ω [ /] Proof For all ω [ /] we have P ω a co inc ω [ in S inc ω Qω a in inc ω ] a 37
4 Let S = inc ω and S = inc = in fω = S P ω S Qω S S a co S S a in S a co S S a in a co S S a in < i equivalent to 37 and the aertion follow Lemma 37: Aume that a [ / 0 / and < a and 5 53 < a 38 up fω < ω [0 ] Fig In lack the region {a γ 0 a < } 0 a Proof For all ω [0 ] we have P ω a ω Qω a 3 3 ω 3 a 3 36 ω where fω = S P ω Qω p q p = a ω q = a 3 3 ω 3 a 3 36 ω p q < i equivalent to 0 < 3 3 a 3 3 ω 3 a 3 36 ω which i valid for ω = 0 For ω = the inequality i equivalent to 38 and the aertion follow a Fig Detail of the region decried in Figure IV NUMERICAL SIMULATIONS AND CONCLUDING REMARKS In Theorem we gave a ufficient condition on a in order to have inf γ 0a < 0 In order to check if thi condition i harp we numerically evaluated the et of thoe a 0 0 for which γ 0 a < The reult hown in Figure ugget that the range of a for which hold i larger then the range otained in Theorem y analytical method Another concluion which can e drawn from our numerical imulation i that at given a > 0 the range of thoe > 0 atifying the condition may e ounded away from zero ee Figure
5 The main concluion of thi work i that a poition damping term in the feedack can improve the routne propertie of the oundary control of the wave equation The optimal choice of the damping coefficient ret in general an open prolem An intereting open prolem i the generalization of our approach to more general feedack or output law REFERENCES [] M CHERKAOUI F CONRAD AND N YEBARI Optimal decay rate of energy for a wave equation with oundary feedack Adv Math Sci Appl 00 pp [] S COX AND E ZUAZUA The rate at which energy decay in a tring damped at one end Indiana Univ Math J 995 pp
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