Boundary Feedback Stabilization of Periodic Fluid Flows in a Magnetohydrodynamic

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1 Bounary Feebac tabilization of Perioic Flui Flows in a Magnetohyroynamic Channel Ionuţ Munteanu Abstract-In this technical note, an electrically conucting -D channel flui flow, in the presence of a transverse magnetic fiel, is investigate. The governing equations are the magnetohyroynamics equations, which are a coupling between the Navier-toes an Maxwell equations. The stability of the Hartmann-Poiseuille profile is achieve by finiteimensional feebac controllers acting on both normal components of the velocity fiel an of the magnetic fiel, on the upper wall only. Inex Terms-Control esign, magnetohyroynamics, stability. I. Introuction In this note, we consier a two-imensional channel flow of an incompressible electrically conucting flui riven by a pressure graient an affecte by a constant transverse magnetic fiel B 0. This in of flow was first investigate both experimentally an theoretically by Hartmann [4]. The governing equations are the magnetohyroynamic equations MHD, for short, which are a combination between the Navier-toes equations an the Maxwell equations we refer to [8] for etails. They are given by ρu t ν u + uu x + vu y + CC x CB y = p x, ρv t ν v + uv x + vv y + BB y BC x = p y, B t µσ B + ub x + vb y Bu x Cu y = 0, C t µσ C + uc x + vc y Bv x Cv y = 0, u x + v y = 0, B x + C y = 0, t 0, x, y R,, where u, v is the velocity vector fiel, p is the scalar pressure, an B, C is the magnetic fiel. The positive constants ρ, ν, µ an σ represent the flui mass ensity, the inematic viscosity, the magnetic permeability an the electrical conuctivity, respectively; is the istance between the walls. We shall assume that the channel is with perfectly insulating walls. In [8], a fully evelope equilibrium flow, associate to the system, is establishe in the following way: the velocity is consiere with zero wallnormal component, while the magnetic fiel is assume with the wall-normal component equal to the external applie magnetic fiel, B 0. More precisely, U e y = Ha tanhha B e y = y Ha + Ha [ coshhay coshha sinhhay sinhha, Ce B 0, where y := y, Ha := B 0 σρν. Notice that U e + B e y = Ha + ], V e 0 an e Hay sinhha, y [, ]. 3 Estimate 3 will be use latter, in the proof of the ey result in emma. below. Defining the imensionless variables x := x, u, v := v 0 u, v, t := v 0 t, B, C := b 0 B, C with v 0 = ρν p e x an b0 = µ σ ρν p e x p e is the pressure corresponing to the steay-state solution, we can write Manuscript receive June, 0; revise December 5, 0 an January 5, 03; accepte... Date of publication... ; ate of current version... This wor was supporte by a grant of the Romanian National Authority for cientific Research, CNC-UEFICDI Project PN-II-ID-PCE I. Munteanu is with the Octav Mayer Institute of Mathematics Romanian Acaemy an Al. I. Cuza University of Iaşi, Department of Mathematics, Iaşi , Romania ionutmunteanu3@yahoo.com. Digital Object Ientifier the linearization of aroun the equilibrium profile, as see the linear part of [8, eqs. 3-8] u t R u + Ue u x + Uyv e + R N m B 0 C x R N m B 0 B y R N m B e yc = p x, v t R v + Ue v x + R N m B e yb + R N m B e B y R N m B e C x = p y, B t R m B + U e B x + B e yv B e u x B 0 u y U e y C = 0, 4 C t R m C + U e C x B e v x B 0 v y = 0, u x + v y = 0, B x + C y = 0, t 0, x R, y,, where R = v 0 ν, N = σb 0 ρv 0 an R m = µσv 0. The star notation has been roppe for simplicity. In orer to reuce the complexity of the presentation, in what follows, we shall assume that the velocity fiel, the magnetic fiel an the pressure are π perioic with respect to the x coorinate. Moreover, the magnetic Prantl number of the flui, i.e., Pr m := νµσ, is assume to be equal to one. uch a perioic MHD channel flow oes not irectly correspon to a specific laboratory flow. However, it is often stuie as an approximation to torus evices of plasma controlle fusion, such as the Toama an the reverse fiel pinch. Numerical simulations have shown that turbulence may appear in the movement of this in of flow, that is, the flow may become unstable see, for example, [3]. The same stabilizing algorithm, we evelop here, may be also applie to perioic channel MHD flows with ifferent magnetic Prantl numbers, see Remar. below. But, the presentation an the calculus may get to har to follow. It is easy to chec that, Pr m = implies N = R = R m =, after some eventual rescales. o, system 4, supplemente with bounary conitions, has the following form u t u + U e u x + U e yv + B 0 C x B 0 B y B e yc = p x, v t v + U e v x + B e yb + B e B y B e C x = p y, B t B + U e B x + B e yv B e u x B 0 u y U e y C = 0, C t C + U e C x B e v x B 0 v y = 0, u x + v y = 0, B x + C y = 0, ut, x + π, y = ut, x, y, vt, x + π, y = vt, x, y, Bt, x + π, y = Bt, x, y, Ct, x + π, y = Ct, x, y, pt, x + π, y = pt, x, y, ut, x, = ut, x, = vt, x, = vt, x, = 0, vt, x, = Ψt, x, Bt, x, = Bt, x, = 0, Ct, x, = 0, Ct, x, = Ξt, x, t 0, x R, y, an initial ata u 0, v 0, B 0, C 0. Here, Ψ an Ξ are the bounary controllers. Concerning the well-poseness of 5, see [8]. tabilization of this in of flows is also the subject of the wor [], that provies stabilizing feebac controllers, by using the bacstepping metho. This metho allows to obtain controllers that can be easily numerically compute, unlie to the present note, where, the Riccati approach provies controllers that are not easily manageable from computational point of view. But, the result in [] hols true only for flows with low values of the magnetic Reynols number R m, while, in the present paper, the algorithm wors for any value of R m. Other results, on this subject, are [7], [5], [9] an the references therein. The stabilizing approach, applie here, is base on the ieas evelope in [6]. II. The Fourier framewor an main properties of the operators involve iewise in [6], the main iea is to ecompose the linear system 5 in Fourier series, obtaining so an infinite parabolic system. et Q, Q = 0, π,, be the space of all functions u loc R,, that are π perioic in x. These functions are characterize by their Fourier series ux, y = u y expix, u = u, Z, Z such that u := π u y y Z <. We shall consier H the complexifie space of,. We enote also 5

2 by the norm in H an by <, > the scalar prouct. The prouct space H H is efine as a complex Hilbert space, in the stanar way. Finally, we shall enote by H m,, H0 m,, m =,,..., the stanar obolev spaces on,. From [0], we have the following result, nown as the Poincaré inequality, v π v, v H, H0,. 6 Here an in what follows, stans for the partial erivative y. Decomposing system 5 in Fourier series, we get u t u + u + iue u + U e v + ib 0 c B 0 b Be c = ip, v t v + v + iue v + B e b + B e b ibe c = p, b t b + b + iue b + B e v ib e u B 0 u Ue c = 0, c t c + c + iue c ib e v B 0 v = 0, iu + v = 0, ib + c = 0, t 0, y,, b = b = c = 0, u = u = v = 0, v =, c = ξ, 7 with initial ata u 0, v0, b0, 0. Where { {u t, y} Z, {v t, y} Z, {b t, y} Z, {c t, y} Z, {p t, y} Z, u 0 y }, { v 0 Z y }, { b 0 Z y } {, Z c 0 y }, { Z t} Z an {ξ t} Z are the Fourier coefficients of u, v, B, C, p, u 0, v 0, B 0, C 0, Ψ an Ξ, respectively. We stuy, first, the case where = 0. We tae 0 0 an ξ 0 0. Thus, system 7 gets the form u 0 t u 0 + Ue v 0 B 0 b 0 Be c 0 = 0, v 0 t v 0 + Be b 0 + B e b 0 = p 0, b 0 t b 0 + Be v 0 B 0 u 0 Ue c 0 = 0, c 0 t c 0 B 0v 0 = 0, v 0 = 0, c 0 = 0, t 0, y,, u 0 = u 0 = v 0 = v 0 = 0, b 0 = b 0 = c 0 = c = 0. The fifth an the sixth equation of 8 imply that v 0 0 an c 0 0. Now, let us sum the first with the thir equation of 8, multiply scalarly by u 0 +b 0 an tae the real part of the result, then, subtract the thir equation from the first one of 8, multiply scalarly by u 0 b 0 an tae the real part of the result. It yiels, after some computations, that for some C, γ > 0. u 0 + b 0 C exp γ t u b 0 0, t 0, 9 From now on, we consier only 0. We set := u + b, := v +c, D := u b, D := v c an 0 := u0 +b0, 0 := v0 +c0, D0 := u 0 b0, D0 := v0 c0. Next, we sum the first equation with the thir one of 7, the secon equation with the fourth one of 7. This way, 7 reuces to a two equations system, from which we eliminate the pressure by aing to the secon equation, multiplie by, the first one, ifferentiate with respect to y an multiplie by i using as-well the ivergence free conitions; for more etails, see [6]. o, we en with just one equation expresse in terms of the unnowns an D. Then, we subtract the thir equation from the first one of 7, the fourth equation from the secon one of 7, an reuce the pressure as before. Consequently, we obtain the following bounary controlle system with just two unnowns, an D, + t + + B 0 [ + id e ] [id e + B 0 ] + [4 + i 3 D e ] + i[ e D ] = 0, D + D t + D B 0D [ + i e ]D [i e 0 B 0 ]D + [4 + i 3 e ]D + i[d e ] = 0, with = = = 0, = := + ξ, D = D = D = 0, D = D := ξ, an initial ata 0, D0, Z \ {0}. Here e := U e + B e, D e := U e B e. Remar.: Notice that for flows with Pr m, it is still possible to reuce the number of the unnowns form the system 7, by using the Elsasser variables. Inee, first we write the MHD equations in terms of the Elsasser variables, i.e., z ± := u, v ± B, C, as z ± t a + z ± a z + z z ± = 8 p + B +C, z ± = 0, where a + = R + R m an a = R see [, ec..]. Then, ecomposing this system in Fourier series, reucing R m an using the the pressure from it in fact, reucing the term p + B +C ivergence free conition, one may obtain, for each Z \ {0}, a system with only two unnowns, similar to 0, but, of highly complexity than it. In the spirit of [6], we introuce the next linear operators : D H H H H an F : DF H H H H, efine as D T := + D + D T, F D T := + B 0 [ + id e ] [id e + B 0 ] + [ 4 + i 3 D e ] + i[ e D] D B 0 D [ + i e ]D [i e B 0 ]D + [ 4 + i 3 e ]D + i[d e ] here T means the transpose matrix. With D = H, H 0, an DF = H 4, H 0,, respectively. Moreover, we efine the operator A := F, DA = { D T H H : DT DF }. We have emma.: The operator A generates a C 0 analytic semigroup on H H an for each λ ρ A the resolvent set of A, λi +A is compact. Moreover, there exists M > 0 such that σ A {λ C : Reλ < 0}, > M. Here σ A is the spectrum of A. Proof The proof is similar to the proof of [, emma ]. As in [6, Remar 3.], emma. implies that for all > M, taing D 0, the solution to 0 satisfies the following exponential ecay t + D t C exp γ t 0 + D 0, t 0, 3 for some C, γ > 0. Hence, it remains to control system 0 for 0 < M only. Next, the main effort is to show an unique continuation type result for the eigenvalues of the ual operator A := F, of A, similar to that one in [6, emma 4.]. The ual operator F, of F, is given by F DT = B 0 [ id e ] + [id e + B 0 ] + [ 4 i 3 D e ] + id e D D + B 0 D [ i e ]D + [i e B 0 ]D + [ 4 i 3 e ]D + i e D 4 with DF = H 4, H 0,. By emma., the operator A has a countable set of eigenvalues, enote by { λ j} we enote by j= m j the corresponing multiplicities. Besies, there is only a finite number N of eigenvalues for which Reλ j 0, j =,..., N, the unstable eigenvalues. We enote by M := m m N, the sum of the multiplicities. Finally, let { φ j := } φ j φ j T an { φ j= j := φ } j φ j T the corresponing eigenfunctions of the operator A an its ual A, respectively. Now, we can j= put forwar the ey result. emma.: et any 0 < M. Then, there exists µ C such that, for any λ j, an unstable eigenvalue of A, we have Re φ j + µ φ j > 0. Proof Fix Z, such that 0 < M. et us enote by λ := λ j, an the corresponing eigenfunction φ := φ j. Consier, first, that we have A φ = λφ. Thus, λ + F φ = 0, where F is the ual of the operator F, efine in 4. For a function f : [, ] C, we enote by f ˇ : [, ] C the following function: f ˇy := f y, y [, ]. It is easy to see that Š e = D e, an, moreover, λ + F φ φ T = λ + F φˇ ˇ φ T = 0. et us enote by T := φ + φˇ φ + φˇ T, henceforth λ + F T = 0. We have two cases: 0 or 0, or both an equal to zero. Assume that = 0 an = 0. ince ˇ,

3 3 we also have = 0. Denoting by Ψ := +, via equation λ + F T = 0 an the form of the operators an F, we euce that Ψ + B 0 Ψ + id e + λψ + id e + ˇ = 0, Ψ = Ψ 5 = 0. calar multiplying 5 by Ψ an taing the real part of the result, we obtain Ψ + + Reλ Ψ + Re i D e + ˇ Ψy = 0. 6 ince Ψ = + + 4, using both the Poincaré inequality 6 an relation 3, we get, by 6, that π + + Reλ Ψ i D e + ˇ Ψy = i e + ˇ ˇΨy 4 + Ψ. 7 III. Main results The aim of this paper is to prove the following result. Theorem 3.: There exist finite-imensional feebac controllers Ψ an Ξ, of the form Ψt, x = [ R v + c v c T t ] T expix, 0< M 3 an Ξt, x = [ R v + c v c T t ] T expix, 0< M 4 such that, once inserte into 5, the corresponing solution of the close-loop system 5 satisfies: ut, vt, bt, ct C exp αt u 0, v 0, b 0, c 0, t 0, ince λ is an unstable eigenvalue, we have Reλ 0. Moreover, 4 +π 4 > 0 for all > 0. o, relation 7 implies that 0. Therefore, we have φ = φˇ. We claim that φ 0 or φ 0. Inee, assume by contraiction that φ = φ = 0. ince φ = φˇ, we see that φ = φ = 0. Now, arguing as above, that is, taing into account that λ + F φ φ T = 0 an enoting by Φ := φ + φ, we have Φ + B 0 Φ + id e + λφ + id e φ + φˇ = 0, Φ = Φ 8 = 0, that yiels φ 0 φ. But, this is in contraiction with the fact that φ φ T is an eigenfunction. Therefore, when = = 0 we have that φ 0 or φ 0. imilar results we get if we tae χ χ T := φ φˇ φ φˇ T an argue as before. More precisely, χ = χ = 0 implies φ 0 or φ 0. o, we have: either 0 or 0 an χ 0 or χ 0, or φ 0 or φ 0. The first case implies the existence of a θ C such that [ + θ ] + [χ + θ χ ] 0. That is φ + θ φ 0. We conclue that, in any case, there exists some µ C such that φ + µ φ 0. Then, replacing, eventually, φ by φ + µ φ φ, we get that φ + µ φ > 0. Now, let us treat the case of generalize eigenfunctions. et us assume that there exists φ φ T,..., φ J φ J T, for some J N, such that λ + A φ φ T = 0 an λ + A j φ j φ j T = 0, j =,..., J. Replacing, eventually, φ j φ j T by φ j φ j T + q j φ φ T, with q j > 0 big enough, j =,..., J, an taing into account the above results, we complete the proof. Next, in a similar way as in [6], we introuce the Dirichlet operator D, associate to F, as follows: let θ > 0, large enough, for each, ξ C we enote by D ξ T := w H 4, H 4,, the solution to the equation θw + F w = 0, y,, w = w = 0, w = 0, w = ξ T 9. Then, liewise in [6], we introuce, F, Ã the extensions to H H of the operators, F an A, respectively, an arguing as in there, we conclue that the system 0 can be equivalently rewritten as t D T t + Ã D T t = θ + F D D, t > 0 0 Equation 0 is unerstoo in the following wea sense t D T, φ + D T, A φ = T [ ] D, θ + F D φ, φ DA, where the ual [ ] θ + F D is given by [ ] θ + F D φ φ T = φ φ T, φ φ T H 4, H0,. for some C, α > 0. Here R : H H H H are linear self-ajoint operators such that they satisfy Riccati algebraic equations R z 0, A z 0 + H H R z 0 C = z 0 H H, z 0 H H, for all 0 < M, M given in emma.. A = F, where F is given by an is given by. Finally, stans for the scalar prouct in C. Proof It is easy to see that, once we fin feebac stabilizing controllers, D for 0, for all 0 < M, taing then Ψt, x := t + D t expix 0< M an Ξt, x := t D t expix, 0< M in system 5, the theorem follows immeiately. Inee, since, D exponentially stabilize system 0, for 0 < M, together with relation 3 yiels: once inserte, D D 0, > M into system 0, the corresponing solution of the close-loop system 0 satisfies t + D t Ce δt 0 D0 T, t 0, Z \ {0}, for some C, δ > 0. Remember that = v + c an D = v c. Thus, the above relation implies that, once inserte := + D an ξ := D, 0 < M, an = ξ = 0, for > M, into the system 7, the corresponing solutions v an c of the close-loop system 7 satisfy an v c t + D t C e δt v 0 c0 T, t 0 t + D t C e δt v 0 c0 T, t 0, for all Z \ {0}. Then, arguing as in [6, Remar 8.] an [6, Th. 5.], we get the esire result. Therefore, from now on, we focus on the feebac stabilization of the system 0, 0 < M. For simplicity, we are going to omit the symbol, also we are going to reefine some symbols, i.e., z := D T, B := θ + F D. With these notations, 0 becomes t z + A z = B D z 0 = z 0 := 0 D 0 } M T, t > 0, T 5. We enote by ZN u := span { φ j, an j= Zs N := span { } φ j. Then we j=m+ introuce the projection P N : H H ZN u, an its ajoint P N, efine by P N := λi + A λ; P N πi := λi + A Γ πi λ, Γ

4 4 where Γ its conjugate Γ, respectively separates the unstable spectrum from the stable one of A A, respectively. We set Au N := P N A, AN s := I P N A, for the restrictions of A to ZN u an ZN s, respectively. These projections commute with A. We then have that the spectra of A on ZN u an Z s N coincie with { } λ M j an { } λ j= j, j=m+ respectively. Moreover, since A generates a C 0 analytic semigroup on H H, its restriction AN s, to Z s N, generates liewise a C 0 analytic semigroup on ZN s. This implies that A s N satisfies the spectrum etermine growth conition on ZN s, i.e., e As N t H H C α0 exp α 0 t, t 0, 6 for some 0 < α 0 < Reλ N +. The system 5 can accoringly be ecompose into its stable an unstable parts, as z = z N + ζ N where z N := P N z, ζ N := I P N z, where applying P N an I P N on 5, we obtain on Z u N : on Z s N : t z N + N z N = P N B t ζ N +A s N ζ N = I P N B D T, zn 0 = P N z 0, 7 D T, ζn 0 = I P N z 0, 8 respectively. Here, I stans for the ientity map. Note that, by 6, system 8 is stable, therefore it remains to stabilize the finite-imensional unstable part 7. First, let us esign a stabilizing controller, in open-loop form. To this en, via the Gramm-chmit proceure, we can rearrange the basis { } φ M i, i= of the space ZN u, in such a way to have φ i, φ j = δi j, i, j =,..., M. 9 M In this basis, we ecompose z N, as follows z N t, y = z i tφ i y, where i= z i t C, t 0, i =,..., M. Introucing this ecomposition in 7, we get M t z i + z i Au N φ i = P N B D T. 30 i= Using relation 9, we obtain, after scalar multiplying 30 by φ j, that M t z j + z i i= A u N φ i, φ j = PN B D T, φ j, j =,..., M. 3 By the iempotency of the projector P N, we can assume without any loss of generality that P N φ j = φ j, j =,..., M. Thus, taing into account an the above relation, we see that PN B D T, φ j = D T, B P N φ j φ j T = φ j + D φ j = φ j + µ φ j 3, j =,..., M, where an D were chosen such that D = µ, with µ given by emma.. We set an Z := z... z M T, B := φ + µ φ... φ M + µ φ M T N φ, φ N φ, φ... N φ M, φ Λ := N φ, φ N φ, φ... N φ M, φ N φ, φ M N φ, φ M... N φ M, φ M Therefore, by 3 an the above notations, 3 can be rewritten as t Z + Λ Z = B. 33 et β > 0, large enough. We claim that the controller := β z z M exponentially stabilizes the system 33. Inee, introucing this into the system 33, we get where t Z + Λ + βg Z = 0, 34 G := φ + µ φ... φ + µ φ φ M + µ φ M... φ M + µ φ M. By emma. we now that Re φ j + µ φ j > 0, j =,..., M. Hence, for β > 0 large enough, the matrix Λ + βg is a Hurwitz matrix. This guarantees the exponential stability of the system 30, once inserte the controllers, D, efine above, into it. Thus, returning form CM to ZN u, the open-loop controllers, D, actually, stabilize system 7, as wante. Then, arguing as in [6, Th. 5.] an using relation 6, one can show that these controllers stabilize system 5, in fact. Finally, we solve the following minimization problem with quaratic cost functional φz 0 := min z t + t + D t t, 0 subject to, D 0, ; H an t z + A z = B D T, t > 0; z 0 = z The functional φ is well efine an, using it, we introuce the operator R : H H H H, as φz 0 = R z 0, z 0, that satisfy the following Riccati algebraic equation R z 0, A z 0 + R z 0 = z 0, z 0 H H. In conclusion, applying the maximum principle, we obtain that the feebac DT = R z exponentially stabilizes system 5 for more etails, see [6, Th. 8.]. Thereby, completing the proof of the theorem. Acnowlegments. The author woul lie to express their gratitue to the referees for their careful assessment as well as for their fruitful suggestions regaring the initial version of the paper. References [] V. Barbu, tabilization of a plane channel flow by noise wall normal controllers, ystems & Control etters, vol. 59, no. 0, pp , 00. [] E. Falgarone an T. Passot, Turbulence an Magnetic Fiels in Astrophysics. New Yor: pringer-verlag, 003. [3] F. Hamba an M. Tsuchiya, Cross-helicity ynamo effect in magnetohyroynamic turbulent channel flow, Physics of Plasma, vol. 7, no., pp. -4, 00. [4] J. Hartmann, Theory of the laminar flow of an electrically conuctive liqui in a homogeneous magnetic fiel, Det Kgl. Danse Viensabernes elsab Mathematis-fysise Meelelser, vol. XV, no. 6, pp. -7, 937. [5] D. ee an H. Choi, Magnetohyroynamic turbulent flow in a channel at low magnetic Reynols number, Journal of Flui Mechanics, vol. 439, pp , 00. [6] I. Munteanu, Normal feebac stabilization of perioic flows in a two imensional channel, Journal of Optimization Theory an Applications, vol. 5, no., pp , 0. [7] I. Munteanu, Normal feebac stabilization for linearize perioic

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