Exact Controllability of a Second-Order Integro-Differential Equation with a Pressure Term

This paper is concerned with the boundary exact controllability of the equation u 00 u Z t 0 g(t ) u( )d = r p where Q is a nite cylinder )0; T ( , is a bounded domain of R n ; u = (u1(x; t); ; u2(x; t)), x = (x1; ; xn) are n dimensional vectors and p denotes a pressure term. The result is obtained by applying HUM (Hilbert Uniqueness Method) due to J.L.Lions. The above equation is a simple model of dynamical elasticity equations for incompressible materials with memory.


Introduction
Let Ω be a bounded domain of R n with regular boundary Γ.Let Q = Ω×]0, T [ be a cylinder whose lateral boundary is given by Σ = Γ×]0, T [.Let us consider the following system * corresponding author e-mail: marcelo@gauss.dma.uem.brEJQTDE, 1998 No. 9, p. 1 ∂x i and p = p(x, t) is the pressure term.
The exact controllability problem for system ( * ) is formulated as follows: given T > 0 large enough , for every initial data {u 0 , u 1 } in a suitable space, it is possible to find a control v such that the solution of ( * ) satisfies u(T ) = u (T ) = 0.
Let Ω be a three dimensional solid body, made of an elastic, isotropic and incompressible material (like some rubber types) under external forces f .From the Newton's Second Law and considering small deflections of Ω we get Mathematically, the incompressibility condition is represented by where I is the identity matrix.
Noting that under small deflections, the quadratic terms of the determinant are neglectible, we obtain that div u = 0 in Ω for all t which leads to the model The presence of the memory term in equation ( * ) is related to the viscoelastic properties of the material.EJQTDE, 1998 No. 9, p. 2 Assuming that Ω is strictly star-shaped with respect to the origin, that is, there exists γ > 0 such that and ν is the exterior unitary normal) and g = 0, J.L.Lions [9] proved that the normal derivative of the solution u of ( * ) belongs to (L 2 (Σ)) n while in A. Rocha [1] the exact controllability was establish.
Inspired by the above mentioned works we study, in a natural way, the exact boundary controllability of system ( * ) when the kernel g is small, which is the goal of this paper.For this end we employ the multiplier technique to obtain the direct and inverse inequalities.However, the convolution term g * ∆u brought up some technical difficulties which were bypassed by transforming the problem ( * ) into an equivalent one using the standard Volterra equations theory.
We can find in the Literature several works in connection with memory terms.The reader is referred to the works of J. U. Kim [3], G. Leugering [7], I. Lasiecka [6] and the classical book of J. Lagnese and J. L. Lions [5].
Our paper is organized as follows.In section 2 we give the notations, present standard and auxiliar results and state the main one.In section 3 we obtain the direct and inverse inequalities related with ( * ) when g = 0 while in section 4 we obtain the same inequalities to the general case.In section 5 we study the ultra weak solutions of ( * ) which is enough to apply HUM in order to obtain the above mentioned exact controllability.

Notations and Main Result
In what follows we consider the Hilbert spaces and equipped with their respective inner products EJQTDE, 1998 No. 9, p. 3 and We also consider We have that V is dense in V with the topology induced by V and Also, we observe that if f ∈ (D (Ω)) n and satisfies f (v) = 0 for all v ∈ V, then there exists p ∈ D (Ω) such that f = −∇p.In particular, if f ∈ (H −1 (Ω)) n and f (v) = 0 for all v ∈ V, we have the same conclusion with p ∈ L 2 (Ω)/R.
In addition, let us define Let g : [0, ∞[→]0, ∞[ be a real funtion satisfying the following hypotheses: ) where C 1 , C 2 and C 3 are positive constants and g(0) < ε, where ε > 0 is sufficiently small.(2.13) As an example of a function which satisfies the conditions (2.9)-(2.13)above one can cite EJQTDE, 1998 No. 9, p. 4 Let us consider the following problem We have the auxiliar results The proof of the above result is obtained by applying Galerkin's approximation with two estimates.The uniqueness is obtained by the usual energy method.
Remark 1.Using Galerkin's method with additional estimates, we can also prove that if In this case we have p ∈ H 2 (Ω).
EJQTDE, 1998 No. 9, p. 5 From Lemma 2.1 and using density arguments we can prove the following result: (ii) The linear mapping (iii) The solution u obtained in (i) satisfies (iv) The solution u is such that where p ∈ D (Q).
From the item (iii) of the above lemma and making use of Gronwall's lemma we obtain the following energy inequality where C(T ) is a positive constant which depends on T > 0 and EJQTDE, 1998 No. 9, p. 6 is the energy related with problem (2.14).
Now, we are in position to state our main result: Theorem 2.1.Provided that the hypotheses (1.2), (2.9)-(2.13)hold there exists for every initial data {u 0 , u 1 } ∈ H × V , a control v ∈ Z such that the ultra weak solution of the system

Direct and Inverse Inequalities with null kernel
In this section we are going to obtain the direct and inverse inequalities to problem (2.14) when the kernel of the memory g = 0.For this end we will employ the multiplier technique.Although the result below plays an essential role in our intent, we will omit its proof since it is exactly as if we were dealing with the wave equation whose proof can be found in J. L. Lions [8], lemma 3.7, pp.40-43.
n × W × L 1 (0, T ; W ).Then, for each strong solution u of (2.14) with g = 0, we have the following identity where ∇u means and (1.2) holds.Then, for each weak solution u of (2.14) there exists a positive constant C > 0 such that EJQTDE, 1998 No. 9, p. 8 Proof.Assuming the direct inequality proven for regular initial data, one obtains the general result using density arguments and extension by continuity.Let us consider then From the identity (3.
where C is a positive constant.
Next, we are going to prove that Indeed, from (2.17), using Gauss's formula and taking into account that div u = 0 in Q we have In order to give a sense to ∂u ∂ν when u is a weak solution of problem (2.14) let us consider Y the space of the weak solutions of the same problem when

Since problem (2.14) is a linear one and has uniqueness of solution the linear map
No. 9, p. 9 is injective.Therefore, the vector space Y is a Banach space with the norm Representing by X the space of the strong solutions of (2.14), that is, when we have X → Y with injection continuous and dense.
Considering the linear map Consequently γ : X → (L 2 (Σ)) n is continuous in X with the topology induced by Y and since X is dense in Y with respect to the same topology, the map γ has a linear and continuous extension γ : Y → L 2 (Σ) n defined as follows: Considering u ∈ Y , there exists u µ ∈ X such that Motivated by the above definition, the normal derivative ∂u ∂ν of u ∈ Y is given by ∂u ∂ν = lim µ→∞ ∂u µ ∂ν .
Theorem 3.3 ( Inverse Inequality) For all T ≥ T 0 and for all weak solution u of (2.14) with f = 0 we have the following inequality Proof.We proceed as in Theorem 3.2, that is, we consider initially regular initial data and then we obtain the desired result by a density argument.From (3.1) and (3.4) we can write In what follows we are going to proceed as in J. L. Lions [8] using arguments due to L. F. Ho [2] and V. Komornik [4]. Since and taking into account that (3.10)It follows from (3.9) and (3.10) that This concludes the proof. 2

Direct and Inverse Inequalities in the general case
In this section we will establish the direct and inverse inequalities related with the weak solutions of the homogeneous problem using the inequalities obtained in section 3 when g = 0. We begin making some considerations.In what follows X will represent the cylinder Q or its lateral boundary Σ.Let be the linear operator defined by We note that from Cauchy-Schwarz inequality and Fubini's theorem we have EJQTDE, 1998 No. 9, p. 12 Then, K is well defined and since ||g|| L 1 (0,∞) < 1 we conclude that ||K|| L(L 2 (X)) < 1 and consequently the operator (I − K) −1 exists and belongs to L(L 2 (X) ).Moreover, in this case we have where the above equality is understood in the space L(L 2 (X) ).
By standard Volterra equations theory for any θ ∈ L 2 (X) there exists a unique solution of the Volterra equation Furthermore, ϕ and θ are related by the equations where and Also, g and h are related by the formula From (4.1) and ( 4.3) we obtain the equivalent problem for θ From the direct and inverse inequalities obtained in section 3 there exists L 1 , L 2 and L 3 positive constants such that Provided that the asumptions (2.11)-(2.13)hold and from (4.12) we deduce T ];H) and therefore, using lemma (2.2) (ii), we infer Finally, from assumption (2.13) and since |v|≤ λ||v||, ∀v ∈ V from (4.19) we have Choosing ε small enough, we conclude from the last inequality that there exists a positive constant C such that The direct inequality follows imediately from (4.7) and (4.8).This concludes the desired result. 2