Internal exact controllability and uniform decay rates for a model of dynamical elasticity equations for incompressible materials with a pressure term

This paper is concerned with the internal exact controllability of the following model of dynamical elasticity equations for incompressible materials with a pressure term, φ′′ − ∆φ = −∇p, and it is also devoted to the study of the uniform decay rates of the energy associated with the same model subject to a locally distributed nonlinear damping, φ′′ − ∆φ + a(x)g(φ′) = −∇p, where Ω is a bounded connected open set of Rn (n ≥ 2) with regular boundary Γ, φ = (φ1(x, t), . . . , φn(x, t)), x = (x1, . . . , xn) are n-dimensional vectors and p denotes a pressure term. The function a(x) is assumed to be nonnegative and essentially bounded and, in addition, a(x) ≥ a0 > 0 a.e. in ω ⊂ Ω, where ω satisfies the geometric control condition. The first result is obtained by applying HUM (Hilbert Uniqueness Method) due to J. L. Lions while the second one is obtained by employing ideas first introduced in the literature by Lasiecka and Tataru.


Description of the problem.
Let Ω be a bounded connected open set of R n (n ≥ 2) with regular boundary Γ.Let Q = Ω×]0, T[ be a cylinder whose lateral boundary is given by Σ = Γ×]0, T[.Corresponding author.Email: mastudillo86@gmail.comConsider the following problem (1.1) System (1.1) was studied by J. L. Lions [26], motivated by dynamical elasticity equations for incompressible materials.Assuming that Ω is strictly star-sharped with respect to the origin, that is, there exists γ > 0 such that (where m(x) = x = (x 1 , . . ., x n ) and ν is the exterior unitary normal) J.L. Lions [26] proved that the normal derivative of the solution φ of the (1.1) belongs to (L 2 (Σ)) n while A. R. Santos [35] established the boundary exact controllability for problem (1.1).In this direction it is worth mentioning the work due to Cavalcanti et al. [10], in which boundary exact controllability of the viscoelastic equation φ − ∆φ − t 0 g(t − s)∆φ(s) ds = −∇p, has been studied.System (1.1) may be obtained from Newton's second law considering small deflections of Ω, where Ω is a solid body composed of elastic, isotropic, incompressible materials (like some rubber types).For more information on the physical interpretation of this model see A. R. Santos [34] and Cavalcanti et al. [10].
Inspired by the above mentioned works we study, in natural way, in Section 2, the internal exact controllability of the system where ∆φ = (∆φ 1 , . . ., ∆φ n ), φ = (φ 1 , . . ., φ n ), div φ = ∑ n i=1 ∂φ i ∂x i and p = p(x, t) is the pressure term.In addition, ω ⊂ Ω and χ ω is the characteristic function of ω where ω is a neighbourhood of the boundary Γ satisfying the well-known geometric control conditions.
The exact controllability problem for system (1.3) is formulated as follows: given T > 0 large enough, for every initial date {φ 0 , φ 1 } in a suitable space, it is possible to find a control h such that the solution of (1.3) satisfies φ(x, T) = φ (x, T) = 0.
Next, in Section 3, we are going to investigate the uniform decay rates of the energy associated with problem (1.1) subject to a locally distributed nonlinear damping as follows (1.4) where a ∈ L ∞ (Ω) is a nonnegative function such that a(x) ≥ a 0 > 0 in ω ⊂ Ω (1.5) and g : R n −→ R n s −→ g(s) = [g i (s i )] i=1,...,n (1.6) where, for all i = 1, . . ., n, g i : R −→ R is a function such that g i is continuous monotonic increasing and g i (0) = 0, for some positive constants k and K. (1.7)

Main goal, methodology and previous results.
The main goal of this paper is twofold: First, to obtain the exact internal controllability of the system (1.3) and then use this result to prove that the solutions of problem (1.4) decay exponentially to zero.System (1.1) was first introduced by J. L. Lions [26].In his work, J. L. Lions proved the hidden regularity property holds for this linear system, under the condition that the domain is star-shaped.Taking advantage of this property and an inverse inequality due to Cavalcanti et al. [10], we are able to obtain the direct and inverse inequalities needed to obtain the internal controllability.
In order to obtain the stabilization result, we use an approach first introduced by Lasiecka and Tataru in [20] which allows us not to impose any growth condition of the function g near the origin and show that the energy decays as fast as the solution of an associated differential equation.Indeed, we are able to establish general decay rates of the energy given by and driven by the solution S(t) of the nonlinear ODE where q is a strictly increasing function in connection with the damping term g(u t ).Moreover, under some extra conditions on the class of nonlinear dissipation and assuming that the pressure is constant, we give examples of the explicit decay rates.A different but related approach is provided by Alabau-Boussouira and Ammari in [3], where the authors obtained sharp, simple and quasi-optimal decay rates for nonlinearly damped abstract infinite-dimensional systems.The method employed by the authors relies on an observability inequality for the conservative system and some comparison properties, combining optimal geometric conditions as provided by Bardos et al. [6] with an optimal-weight convexity method of Alabau-Boussouira (see [1] and [2]).
On the other hand, although the bibliography concerning the wave equation subject to a locally distributed damping is truly long, see, for instance, [1,2,6,[12][13][14][15][16][17][18]21,22,[27][28][29]31,38,40], and a long list of references therein, it seems that there exist just few papers in connection with control or stabilization for the model of dynamical elasticity equations for incompressible materials introduced by J. L. Lions [26].In [33] Oliveira and Charão also studied the decay properties of the solutions of an incompressible vector wave equation with a locally distributed nonlinear damping.However, in order to obtain the algebraic decay rate to zero of the solution, the authors imposed some extra technical conditions on the function g like the fact that the partial derivative of g is positive definite which will not be necessary in our present study.In order to get this result, the authors used multiplier methods and a lemma due to Nakao.The exponential decay rate of the energy for the case of an incompressible vector wave equation with localized linear dissipation was obtained by Araruna et al. [4].In a more general framework, a problem that is similar to (1.1), is studied in [19], where Ammari, Feireisl and Nicaise proved exponential and polynomial decay rates for an acoustic system with spatially distributed damping.

Internal exact controllability
In what follows, we consider the Hilbert spaces and equipped with their respective inner products We also consider and We have that V is dense in V with topology induced by V and We observe that throughout this paper repeated indexes indicate summation from 1 to n.

Direct and inverse inequalities
Let us consider the following problem (2.8) Firstly, observe that following the arguments [37] we can show that for regular initial data the problem (2.8) is equivalent to the problem where A is Stokes operator defined as follows: A : (H 2 (Ω)) n ∩ V → H given by Au := P (−∆u) where P : (L 2 (Ω)) n → H is the orthogonal projector in (L 2 (Ω)) n onto H and ∆ : (H 2 (Ω)) n ∩ V → (L 2 (Ω)) n is the Laplace operator with Dirichlet boundary conditions.
In this section we are going to obtain the direct and inverse inequalities to problem (2.8) which is enough to apply HUM (Hilbert Uniqueness Method) in order to obtain the above mentioned exact controllability.For this end we will employ the multiplier technique.The main results of this section are Theorem 2.1 and Theorem 2.4 below.Theorem 2.1.Let {φ 0 , φ 1 } ∈ H × V and φ the ultra weak solution of the problem (2.8).Then, there exists a constant C 1 > 0 such that (2.10) Proof.Since φ is the ultra weak solution to problem (2.8) then making use of standard properties of ultra weak solutions for linear problems (see Cavalcanti [10,Section 5] or Lions [25, Chap. 1, Section 4]) there exists C 0 > 0 such that where We assume that ω ⊂ Ω is a neighborhood of Γ(x 0 ) where (2.17) Proof.Suppose that the following estimate holds where θ is solution of the problem (2.8) with initial data {θ 0 , θ 1 } ∈ V × H. Then we have the desired result.Indeed, take {φ 0 , Let us define where η satisfies (2.19) and φ is the solution to problem (2.21) Thus we have that ψ is the solution to problem From (2.18) and (2.22) we have We will prove (2.18) in several steps.By Theorem 3.3 due to Cavalcanti et al. [10], for T > T 0 = 2R 0 the following inequality holds where θ is the weak solution of the problem (2.8) with the initial data {θ 0 , Then, for all regular solutions of (2.8), the following identity holds Proof.Let us consider Integrating by parts with respect to x k and using the fact that φ = 0 on Σ, we get (2.24) Integrating by parts the first integral, with respect to x i , we obtain Since div φ = 0 on Q, we conclude that Therefore, and the proof is finished.
Lemma 2.6.Let T > T 0 and ε > 0 be such that T − 2ε > T 0 .Let θ be the solution of problem (2.8) with initial data {θ 0 , θ 1 } ∈ V × H.Then, there exists C > 0 such that (2.26) From (2.23) and (2.26) we have analogously to what we have done in (2.27), we obtain, and making the change of variable s = t + ε, we infer Lemma 2.7.Let T > T 0 and ε > 0 be such that T − 2ε > T 0 .Let θ be the solution of problem (2.8) with initial data {θ 0 , θ 1 } ∈ V × H.Then, there exists C > 0 such that where ∇θ means  (2.32) Proof.Initially considering the regular initial data, one obtains the general result using density arguments.In equation (2.8) 1 , taking the inner product in (L 2 (Ω)) n of θ − ∆θ + ∇p and r • ∇θ, and integrating in [0, T], we obtain (2.33) Introduce the notations Next, we are going to estimate these terms.Using integration by parts and the properties of the function r(x, t) defined in (2.30), we deduce J 1 , satisfies Then Gauss's Formula yields and since θ i (x, t) = 0 on Σ, we deduce that Thus (2.35) Furthermore, we have for the term J 2 From Gauss's formula and (2.13), we have Notice that (2.37) From Gauss's formula and (2.13), we deduce that Thus from (2.36), (2.37), (2.38) and (2.13), we have For the term J 3 we have (2.41) Note that by Lemma 2.5 we have that Using the properties of r(x, t) and estimating others term on the right side of equality (2.41), we obtain that Therefore, by Lemma 2.6 and (2.43) we have (2.44) , by (2.44) and choosing δ small enough we have Since the inequality above is valid for all T > T 0 , in particular for T − 2ε, proceeding as in the demonstration of Lemma 2.6 we have the desired result.
Remark 2.8.According to the proof of Lemma 2.3 in J. L. Lions [25] we can construct a neighbourhood ω of Γ 0 such that Ω ∩ ω ⊂ ω and we can to build a vector field r for ŵ.Then, we get analogously, that (2.45) The function r can be chosen as follows r(x, t) = ρ 2 (x)η(t) where η ∈ C 1 (0, T) and it satisfies Proposition 2.9.Let us consider T > T 0 and ε > 0 such that T − 2ε > T 0 and θ the solution of problem (2.8) with initial data {θ 0 , θ 1 } ∈ V × H.Then, there exists a constant C > 0 such that Proof.Initially we consider regular initial data and one obtains the general result using density arguments.In equation (2.8) 1 , taking the inner product in (L 2 (Ω)) n of θ − ∆θ + ∇p and rθ and integrating in [0, T], we obtain (2.47) Next, we are going to estimate the terms in (2.47).Denote by Using the properties of the function r(x, t) defined in (2.45) and making use of the equality (θ i , rθ i ) = (θ i , rθ i ) + (θ i , r θ i ) + (θ i , rθ i ), we have that I 1 can be estimated as follows Notice that using Gauss's formula and taking θ = 0 on Σ into account, we infer that I 2 verifies (2.50) By Young's inequality we obtain that Making use of the inequality ab ≤ 1 2 a 2 + 1 2 b 2 , we can write (2.52) By the Cauchy-Schwarz inequality and by Young's inequality we have that for any δ > 0.

.69)
Since V → H is compact, then from Theorem 5.1 due to J. L. Lions [24], we have (2.70) From (2.61) and (2.69) we have that and ψ is independent of t in ω.
3 Uniform decay rate

Wellposedness
Let us consider the following problem ) n , u 1 ∈ V and assume that hypotheses (1.5), (1.6) and (1.7) hold.Then, there exists a unique function u : Proof.Let P be the orthogonal projector in (L 2 (Ω)) n onto H and consider the problem where the operator A is the Stokes operator, that is, is defined by We have by [7, Proposition IV.5.9] that Observe that since v ∈ V ⊂ H then by integration by parts Moreover the operator A mentioned above is linear and maximal monotone in Thus b is a form bilinear, continuous and coercive in V × V, so we can define an operator where I is the identity operator.From the Lax-Milgram theorem this operator is an isomorphism from V onto V .Therefore A is maximal monotone.
Since I is continuous and monotone then by [5, Corollary 1.3], we conclude that The operator B is defined by Also note that since the projection is self-adjoint and v ∈ V ⊂ H, we have The operator B is monotone.Indeed, since P is linear and self-adjoint we have because a is nonnegative and g i is monotonic by hypothesis.Then B is monotone.We claim that B is hemicontinuous.In fact, Take Let In the same way we obtain We observe that, since g i is continuous we have that lim Thus, from Lebesgue's dominated convergence theorem we conclude that that = (a(g(u)), P (w)) Thus by (3.6) and (3.7) we obtain and therefore B is hemicontinuous.Moreover, we have that I + A + B is coercive.In fact, using condition (1.7), then ( We can reformulate the problem (3.2) to obtain Thus we have a matrix operator A : H → H, where H = V × H, defined by In order to prove the maximality of A, it is sufficient to prove that R(I + A) = H, that is, given (v 0 , h 0 ) ∈ H, we have to show that there exists (v, h) ∈ D(A) such that Combining the above identities we deduce Therefore, it is sufficient to prove that I + A + B is maximal monotone in V × V , that is, we have to show that R(I + A + B) = V .Since B is monotone and hemicontinuous, I + A is maximal monotone em V × V and I + A + B is coercive, we conclude by [5,Corollary 1.3] that (I + A) + B is maximal monotone.Then, (3.11) possesses a unique solution h ∈ V. Since v = v 0 + h and Av + Bh = h 0 − h, we conclude that v ∈ V and Av + Bh ∈ H. Consequently, the system (3.10) has a unique solution (v, h) ∈ D(A), and, therefore, A is maximal monotone in H.
Finally, from the above and making use of Theorem 3.1 due to Brezis [8] and given {u 0 , u 1 } ∈ D(A) × V there exists a unique u(t) regular solution of problem (3.2) in the class Now we are going to recover the pressure term of the regular solution.Let v ∈ V be any time-independent test function.For any t > 0 we thus have Since v ∈ V ⊂ H, we obtain as in (3.3) and by (3.4) we have This being true for any v ∈ V, by [7, Theorem IV.2.3] we obtain that there exists a unique p(t) ∈ L 2 0 (Ω) such that u (t) − ∆u(t) + ag(u (t)) = −∇p, that is, e. in (0, T).
Theorem 3.2.Let u 0 ∈ V, u 1 ∈ H and assume that hypothesis (1.5), (1.6) and (1.7) hold.Then, there exists a unique weak solution u of problem (3.1) Proof.Proceeding analogously to Theorem 3.1, we prove that A is maximal monotone in H = V × H. Thus given {u 0 , u 1 } ∈ V × H, by [8,Theorem 3.4] there exists an unique u(t) weak solution of problem (3.2) in the class To determine the pressure term of the weak solution, we note that u ∈ L 1 (0, T; V ).Then Let us consider {ϕ m } m∈N a sequence of functions in V such that ϕ m → 0 in (D(Ω)) n .Then we have | u (t), ϕ m | → 0.
Thus, u (t) is a linear and continuous form in V with the norm of (D(Ω)) n .Then by the Hahn-Banach theorem u (t) can be continuously extended to (D(Ω)) n , which will still be denoted by u (t).In this sense, L = u (t) − ∆u(t) + a(x)g(u (t)) is a linear and continuous form in D(Ω) a.e. in [0, T).Similar to the statement previously made, we have L ∈ (D (Q)) n and L(ϕ) = 0 in (D (0, T)) n ∀ϕ ∈ V. From above and Rham's theorem (cf.J. L. Lions [24] and R. Teman [39]), we have

Stability result
The energy related to problem (3.1) is given by In addition, multiplying the regular solution of problem (3.1) by u , performing integrations by parts having in mind that div u = 0 in Ω × (0, T) we deduce the following identity of the energy: By standard density arguments the above identity (3.15) remains valid for weak solutions of (3.1) as well.
Before presenting our stability result, we will define some needed functions.For this purpose, we are following ideas first introduced in the literature by Lasiecka and Tataru [20].For the reader's comprehension, we will repeat them briefly.Let h : R → R be a concave, strictly increasing function with h(0) = 0, and such that from which we deduce having in mind that h(s Note that such a function can be straightforwardly constructed, given the hypotheses of g given in (1.6) and (1.7).With this function, we define

.18)
As r is monotone increasing, cI + r is invertible for all c ≥ 0. For L a positive constant, we then set, respectively, where C ia a positive constant that will be determined later.The function z is easily seen to be positive, continuous, and strictly increasing with z(0) = 0. Finally, let q(x) = x − (I + z) −1 (x). (3.20) We can now proceed to state our stability result.with lim t→∞ S(t) = 0, where the contraction semigroup S(t) is the solution of the differential equation where q is given in (3.20).Here the constant c (from definition Proof.Writing u = v + w we have that (3.1) is equivalent to From (1.5), Remark 2.11 and from Lemma 2.2 due to Cavalcanti et al. [10], we have where c depends on T. Let Then using hypothesis (1.7), we obtain Moreover, from (3.17) and from the fact that h is concave and increasing, having in mind that a(x) ≤ a ∞ + 1 and a(x) 1+ a ∞ < a(x) we deduce that with jointly with the identity of the energy allows to conclude that E(T) ≤ c 1+c E(0) for all T > T 0 , and, therefore, since we are dealing with an autonomous system, E(2T) ≤ c 1+c E(T) ≤ c 1+c 2 E(0), and, by iteration, E(nT) ≤ c 1+c n E(0) for all n ∈ N and T > T 0 which allows to conclude the desired exponential decay.
If p is constant everywhere, we can give a wide assortment of examples which we borrowed from [11, Section 8, Corollary 8.1 and Corollary 8.2] by following ideas firstly introduced in [1,2].Indeed, in this specific case denoting As a consequence, we deduce, for the purely wave equation, as before that which implies that decay of each E i (T) is driven by the Corollary 8.1 and Corollary 8.2 due to [11] and the decay associate with the full energy is given by If g i (s i ) = s p i for all i = 1, . . ., n, the decay of E(t) would be the sum of decays of the same type.However, if g i (s i ) = s p i i the decay would be the worst one given by the largest p i .Example 3.6.We take g i (s i ) = s 3 i e Example 3.8.We take g i (s i ) = |s i | θ−1 s i , 0 < θ < 1.In this case the analysis is identical to the case of example 1 since g −1 (s i ) = s 1 θ i , s i > 0 and 1 θ > 1.Thus the decay rates in that case become Summarizing we can choose g i (s i ) as the above examples so that for each one we have a decay E i (t) but the total one E(t) := ∑ n i=1 E i (t) will be driven by the worst one.

. 34 )Example 3 . 5 . 2 i 2 i 2 − 2 +
Let us see some examples: We consider g i (s i ) = s p i , p > 1 at the origin.Since the function s p+1 is convex for p ≥ 1 we will be solving S i,t + S p+1 be integrated directly, of course.However, for sake of illustration of the general formula we findG(s, S 0 ) = p+1 .Thus E i (t) ≤ C(E i (0)) E i (0) −p+1 t(p − 1) 2 −p+1 .
(Ω) + u i (t) 2 L 2 (Ω) , i = 1, ..., neach portion of the full energy verifies the identity of the energyE i (t 2 ) − E i (t 1 ) = − (x)g i (u i )u i dxdt, i = 1, . .., n,and, furthermore, for each i = 1, . . ., n the wave equation is in place, namely a for s i at the origin.Since the function s 2 i e We consider g i (si ) = s i |s i |e − 1 |s i | for s i near zero .Since the function s 3/2i e is convex on [0, s 0 ] for some small s 0 we are led to differential equationS i,t + S 3/2 i eFunction G(S, S 0 ) is given by G(S, S 0 ) = − e i