Electronic Journal of Differential Equations, Vol. 06 06), No. 7, pp. 0. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STABILIZATION OF THE WAVE EQUATION WITH VARIABLE COEFFICIENTS AND A DYNAMICAL BOUNDARY CONTROL ZHIFEI ZHANG Abstract. In this article we consider the boundary stabilization of the wave equation with variable coefficients and a dynamical Neumann boundary control. The dynamics on the boundary comes from the acceleration terms which can not be ignored in some physical applications. It has been known that addition of dynamics to the boundary may change drastically the stability properties of the underlying system. In this paper by applying a boundary feedback control we obtain the exponential decay for the solutions. Our proof relies on the Geometric multiplier skills and the energy perturbed approach.. Introduction Let be a bounded domain in R N N ) with smooth boundary Γ. We assume Γ = Γ 0 with Γ 0 =. We consider the following wave equation with dynamical Neumann boundary condition u tt div Ax) u = 0, ux, t) = 0, in 0, ), on Γ 0 0, ), mx)u tt x, t) + νa ux, t) = Ct), on 0, ), ux, 0) = u 0 x), u t x, 0) = u x), in,.) where div X denotes the divergence of the vector field X in the Euclidean metric, Ax) = a ij x)) are symmetric and positive definite matrices for all x R N and a ij x) are smooth functions on R N. Let νa u = n i,j= a ijx) xj uν i, where ν = ν, ν,, ν n ) T denotes the outward unit normal vector of the boundary and ν A = Aν. Here Ct) is the boundary feedback control. We say that equation.) is with dynamical boundary conditions when mx) 0. In some physical applications one has to take the acceleration terms into account on the boundary. Usually in this case to describe what really happened we need the models with dynamical boundary conditions. They are not only important theoretically but also have strong backgrounds for physical applications. There are numerous of these applications in the bio-medical domain [7, 0]) as well as in 00 Mathematics Subject Classification. 34H05, 35L05, 49K5, 93C0, 93D5. Key words and phrases. Exponential decay; wave equation with variable coefficients; dynamical boundary control. c 06 Texas State University. Submitted December 9, 05. Published January 5, 06.
Z. ZHANG EJDE-06/7 applications related to noise suppression and control of elastic structures [, 3, 8, ]). Moreover, in this article we assume that mx) L ); mx) m 0 > 0, x. Our motivating example is the hybrid wave equation u tt u = 0, ux, t) = 0, mu tt x, t) + ν ux, t) = ν u t, in 0, ), on Γ 0 0, ), on 0, ), ux, 0) = u 0 x), u t x, 0) = u x), in..) It is shown that the above system with m > 0 is not uniformly stable, while the case when m = 0 the system is exponentially stable under suitable geometrical conditions on. That is to say, the dynamical terms on the boundary may change the stability property of the system. For details, see [0,, 7]. The purpose of this paper is to study how the dynamic boundary conditions on affect the decay of the system. In this article, we shall design a collocated boundary feedback to obtain the exponential stabilization of the system.). We set Ct) = βu t x, t) γ νa u t,.3) where the constants β and γ are positive numbers such that βγ < m. Before we state and prove our results, let us first recall some works related to the problem we address. Wave equations with acceleration terms in the dynamical boundary conditions have been studied within the framework of the model u tt u + gu t ) = f, Ku)u tt x, t) + ν ux, t) = Ct), in 0, ), on Γ 0, ), ux, 0) = u 0 x), u t x, 0) = u x), in..4) In the one dimensional case, problem.4) with Ct) = hu t ) has been studied in [5, 8], and with Ct) = αu + fu xt αu t ) has been considered in [6]. For N-dimensional case with N [4, 9] discussed the asymptotic stabilization and the existence of the solutions, respectively. In [5], the case when Ct) involves an unknown disturbance was considered, again in the one dimensional case. For equations with other type of dynamical boundary conditions, see [,, 6] for wave equations with acoustic boundary conditions and [0] for wave equations with viscoelastic damping and dynamic boundary conditions acting on a surface of local reaction, and the references therein. In equation.) we adopt the feedback law given in.3) to obtain the exponential stabilization of the following closed loop system: We define u tt div Ax) u = 0, ux, t) = 0, in 0, ), on Γ 0 0, ), mx)u tt x, t) + νa ux, t) = βu t x, t) γ νa u t, ux, 0) = u 0 x), u t x, 0) = u x), in. g = A x), x on 0, ),.5) as a Riemannian metric on and consider the couple, g) as a Riemannian manifold. Let D denote the Levi-Civita connection of the metric g. For each
EJDE-06/7 STABILIZATION OF THE WAVE EQUATION 3 x, the metric g induces an inner product on R n x by X, Y g = A x)x, Y, X g = X, X g, X, Y R n x, where, denotes the standard metric of the Euclidean space R n. To obtain the stabilization of the problem.5), the following geometrical hypotheses are assumed. Geometrical assumptions. There exists a vector field H on Riemannian manifold, g) such that the following properties hold: A) DH, ) is strictly positive definite on : there exists a constant ρ > 0 such that for all x, for all X M x the tangent space at x): A) DHX, X) D X H, X g ρ X..6) H ν 0 on Γ 0..7) Remark.. For any Riemannian manifold M, the existence of such a vector field H in A) has been proved in [], where some examples are given. See also [4]. For the Euclidean metric, taking the vector field H = x x 0 and we have DHX, X) = X, which means assumption A) always holds true with ρ = for the Euclidean case. Energy of the system.). Before we go to the stabilization of the system, we should first define an energy connected with the natural energy of the hybrid system. We set ηx, t) = mu t x, t) + γ νa u, x. Let u be a regular solution of system.5). Then we associate to system.5) the energy functional Et) as Et) = u t + g u ) g dx + m βγ Γ η dγ. The main result of this paper reads as follows. Theorem.. Let the geometrical assumptions A) and A) hold. Then there exist constants C > 0 and ω > 0 such that Et) Ce ωt E0), t 0..8) This article is organized as follows. In the next section, we discuss the wellposedness of the nonlinear close-loop system by semigroup theory. Section 3 devotes to the proof of the exponential stability. We construct two auxiliary functions to estimate the energy, thus to obtain the main result finally.. Well-posedness of the closed loop system In this section, we study well-posedness results for system.5) using semigroup theory. Denote H Γ 0 = {u H ) u Γ0 = 0}. We consider the unknown in the state space, denoted by U = u, v = u t, η ) T, Υ := H Γ 0 ) L ) L ),.)
4 Z. ZHANG EJDE-06/7 with the norm defined by U = u, v, η) T = g u g + v ) dx + m βγ η dγ..) The closed-loop system.5) can be rewritten in the abstract form U = AU, U 0 = u 0, u, η 0 ) T,.3) where the operator A is defined by u v A v = div g u η γ η + m γ β)vx, t) with domain DA) := { u, v, η) T H ) L ) L ) : div g u L ), η = mv Γ + γ νa u }..4) We will show that A generates a C 0 semigroup on Υ. Now we state the wellposedness result. Theorem.. For any initial datum U 0 Υ, there exists a unique solution U C[0, ), Υ) of system.3). Moreover, if U 0 DA), then U C[0, ), DA)) C [0, ), DA)). Proof. Step. We prove that A is dissipative. We know Υ is a Hilbert space equipped with the adequate scalar product, Υ and norm U defined by.). For U DA), a simple computation leads to AU, U Υ = d dt U = u t νa udγ + ηη t dγ Γ m βγ Γ = u t νa udγ η dγ + m βγ)γ γ ηu tx, t)dγ = u t νa udγ η dγ mγ m βγ)γ ) η dγ + mγ γ ηu tx, t)dγ m = γ u t + γ m ν A u ) dγ m βγ)γ ) η dγ + mγ γ ηu tx, t)dγ..5) Now we handle the items in.5) by applying Hölder inequality γ ηu tx, t)dγ η dγ + k m u t dγ,.6) k mγ 4γ
EJDE-06/7 STABILIZATION OF THE WAVE EQUATION 5 where k > 0 is to be chosen later. Substituting the inequality.6) to.5) yields γ m AU, U Υ m ν A udγ γ k m) u 4γ t dγ m βγ)γ mγ η dγ γ m ν A udγ βγ m + βγ ) k mγ Γ u t dγ βm + βγ) m η dγ, m βγ).7) by choosing k = m m+βγ, which is negative noticing that βγ < m. Step. We will show that λi A is surjective for a fixed λ > 0. Given ā, b, c) T Υ, we seek a solution U = u, v, η) T DA) of that is, satisfying u λi A) v = η ā b c ; λu v = ā, quadin, λv div Ax) u = b, in, λη + γ η + β m γ )vx, t) = c, on, η = mv Γ + γ νa u, on,.8) where we take t as a parameter for granted. Suppose that we have found u with the appropriate regularity, then from the equation.8)-,.8)-3 and.8)-4 we have v := λu ā, η = λ + γ ) c β m γ )λu ā)). Now we state the process on how to get u. Eliminating v and noticing η = mv Γ + γ νa ux, t), we find that the function u satisfies λ u div Ax) u = λā + b, in, u = 0, on Γ 0, νa u = γ η m γ λu ā), on..9) We obtain a weak formulation of system.9) by multiplying the equation by ψ and using Green s formula λ λmλ + β) uψ + Ax) u, ψ )dx + uψ λγ +.0) mλ + β = b + λā)ψdx + c + λγ + λγ + ā)ψ, for any ψ H 0 ) = {ψ H ) ψ Γ0 = 0}. As the left hand side of.0) is coercive on H ), Lax-Milgram Theorem guarantees the existence and uniqueness of a solution u H ) of.9). Step 3. Finally, the well-posedness result follows from Lummer-Phillips Theorem.
6 Z. ZHANG EJDE-06/7 3. Exponential Stability In this section, we show the exponential stability of the system by energy perturbed approach. Here we define two auxiliary functions V t) = Hu)u t dx, V t) = div 0 H ρ)uu t. To estimate the functions, we need some lemmas from [4] and [3]. Lemma 3. [4, Theorem.]). Suppose that ux, t) is a solution of the equation u tt + div A u = 0. Then V t) = B + I, where we denote the boundary term B Γ) = νa uhu)dγ + u t g u Γ g)h νdγ, 3.) Γ and the internal term I = D g H g u, g u)dx u t g u g) div 0 H dx, here div 0 H is the divergence of H in the Euclidean metric of R n. Lemma 3. [4, Theorem.]). Suppose that ux, t) is a solution of the equation u tt + div A u = 0. Then V t) = B + I, where we denote the boundary term by B Γ) = u νa div 0 H)dΓ + div 0 H ρ)u νa udγ, 4 Γ Γ and the internal term I = 0 H ρ)u div t g u g)dx + 4 u div νa div 0 H)dx. Lemma 3.3 [3, Lemma 7.]). Let > 0 be given small. Let u solves the problem.). Then T g u { T gdγdt C T, νa u) + u ) } t dγdt + u H. + 0,T )) 0 3.) According to Lemmas 3., 3. and 3.3 we obtain: Lemma 3.4. Suppose that the geometrical assumptions A) and A) hold. Let u solve.). Then there exist constants c, c, c 3 > 0 such that ρ Et) + V t) + V t) ρ m βγ η dγ + c u t + g u g)dγ + c u dx, 3.3) V t) c Et), V t) c 3 Et). 3.4)
EJDE-06/7 STABILIZATION OF THE WAVE EQUATION 7 Proof. Obviously the estimate 3.4) is true. Now we prove the inequality 3.3). First we estimate the boundary terms B, B given in Lemma 3. and Lemma 3.. Since u Γ0 = 0, we have g u = νa u ν A ν A, 3.5) g which induces g u g = νa u g ν A. 3.6) g Similarly we have Hu) = H, g u g = νa u ν A H ν. 3.7) g Substituting the equalities 3.6) and 3.7) to 3.) yields B Γ 0 ) = u t g u g)h νdγ 0, Γ where we notice the geometrical assumption A) is true. It is obvious that B Γ) = B Γ 0 ) + B ) c u t + g u g)dγ. 3.8) Since u Γ0 = 0, we have B Γ 0 ) = 0. And we have B Γ) = B Γ 0 ) + B ) c g u gdγ. 3.9) Next, we estimate the internal terms I and I. By the geometrical assumption A), we have I ρ g u gdx u t g u g)operatornamediv 0 H dx. 3.0) It is obvious that I div 0 H ρ)u t g u g)dx + c u dx. 3.) Combining the above inequalities 3.8), 3.9), 3.0) and 3.) we complete the proof. The following is the observability inequality for the system.). Lemma 3.5. Suppose that the geometrical assumptions A) and A) hold. Let u solve problem.). Then for any given > 0, there exists a time T 0 > 0 and a positive constant C T,,ρ such that { T E0) C T,,ρ u t + νa u) + η ) } dγdt + u H /+ 0,T )), 3.) 0 for all T > T 0. Proof. For any small enough, integrating the inequality 3.3) on the interval, T ) yields ρ T ρ T Et)dt + V T ) V ) + V T ) V ) m βγ η dγ + c T u t + g u g)dγ + c T u dx.
8 Z. ZHANG EJDE-06/7 Then we use inequality 3.) in Lemma 3.3 and inequality 3.4) in Lemma 3.4 to obtain T { T Et)dt C T,,ρ η + u t + νa u g)dγ 0 3.3) ) + u H /+ 0,T )) }dx + c 0 ET ) + E), where the constant C T,,ρ depends on c, ρ, m βγ, meas) and the constant c 0 = 4 ρ max{c, c 3 }. Here c, c 3 are the constants given in inequality 3.4). We notice that ) E0) + c 0 ET ) + E) = = c0++ Et)dt + c0++ E0) Et) ) dt + c 0 E) E0) ) + c0 ET ) E0) ) c0++ Et)dt T + c 0 Ėτ)dτ 0 c0++ c0++ max{t,c0++} Et)dt + 3c 4 0 t Ėτ)dτ ) dt + c 0 Ėτ)dτ where we used the following inequality known from.7), 0 0 u t + νa u) + η ) dγdt, Ėt) = AU, U Υ c 4 u t + νa u) + η ) dγdt. 3.4) Now we shall take T 0 = c 0 + + to guarantee that T > c 0 + +, for all T > T 0. Substituting 3.4) in 3.3) completes the proof. In what follows we use the compactness-uniqueness argument to absorb the lower order term in 3.). We list the lemma and omit the proof, which could be found in [4, 5, 9, 3, 6] and many others. Lemma 3.6. Suppose that the geometrical assumptions A) and A) hold. Let u solve problem.). Then for any T > T 0, there exists a positive constant C depending on T,, ρ, meas) such that T E0) C u t + νa u) + η ) dγdt. 0 Proof of Theorem.. From.7) we know that Ėt) = AU, U Υ γ m ν A udγ + βγ u t dγ + m + βγ c 5 u t + νa u) + η ) dγdt, βm + βγ) m η dγ m βγ) 3.5) where c 5 = min{ γ m, βγ m + βγ, βm + βγ) m m βγ) }.
EJDE-06/7 STABILIZATION OF THE WAVE EQUATION 9 By Lemma 3.6 and the above inequality 3.5), we have, that for all T > T 0, T E0) C u t + νa u) + η ) dγdt C T Ėt) = C ET ) E0)), 0 c 5 0 c 5 which yields ET ) C c 5 C E0). The exponential decay result.8) follows from the above inequality. Acknowledgments. This research was supported by the National Natural Science Foundation of China under grant nos. 64736, 657334 and 578. References [] K. T. Andrews, K. L. Kuttler, M. Shillor; Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 97 3) 996), 78-795. [] J. T. Beale, S. I. Rosencrans; Acoustic boundary conditions, Bull. Amer. Math. Soc., 80 974), 76-78. [3] B. M. Budak, A. A. Samarskii, A. N. Tikhonov; A Collection of Problems on Mathematical Physics A.R.M. Robson, Trans.), The Macmillan Co, New York, 964. [4] B. Chentouf, A. Guesmia; Neumann-boundary stabilization of the wave equation with damping control and applications, Communications in Applied Analysis, 4 00), no. 4, 54-566. [5] F. Conrad, G. O Dowd, F. Saouri; Asymptotic behaviour for a model of flexible cable with tip masses, Asymptotic Analysis, 30 00), 33-330. [6] F. Conrad, A. Mifdal; Uniform stabilization of a hybrid system with a class of nonliner feedback laws, Advances in Mathematical Sciences and Application, Tokyo, Vol., No. 00), 549-569. [7] S. Čanić and A. Mikelić; Effective equations modeling the ow of a viscous incompressible fluid through a long elastic tube arising in the study of blood ow through small arteries, SIAM J. Appl. Dyn. Syst., 003), 43-463. [8] G. G. Doronin, N. A. Larkin; Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions, Nonlinear Anal., 8 00), 9-34. [9] G. G. Doronin, N. A. Larkin, A. J. Souza; A hyperbolic problem with nonlinear second-order boundary damping, Electron. J. Differ. Equ. 998) -0. Paper 8. [0] N. Fourrer, I. Lasiecka; Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Eqns. and Control Th. EECT), pp. 63-667, 03. [] P. J. Graber; Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, Journal of Evolution Equations, 0), 4-64. [] J. Lagnese; Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 983), 63-8. [3] I. Lasiecka, R. Triggiani; Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 5 99) 89-4. [4] I. Lasiecka, J. L. Lions, R. Triggiani; Nonhomogeneous boundary value problems for secondorder hyperbolic operators, J. Math. Pures Appl., 65986), 49-9. [5] I. Lasiecka, R. Triggiani; Exact controllability of the wave equation with Neumann boundary control, Appl. Math. and Optimiz., 9 989), 43-90. [6] J. Li, S. G. Chai; Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback. Nonlinear Anal., 05), 05-7. [7] W. Littman, B. Liu; On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 999), 7-34. [8] P. M. Morse, K. U. Ingard; Theoretical Acoustics, Princeton University Press, 987. [9] Z. H. Ning, Q. X. Yan; Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback, J. Math. Anal. Appl., 367 00), 67-73.
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