Interior Controllability of a Broad Class of Reaction Diffusion Equations

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z L2 Ω given by z′ −Az 1ωu t , t ∈ 0, τ , where Ω is a domain in R, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u ∈ L2 0, t1;L Ω and A : D A ⊂ Z → Z is an unbounded linear operator with the following spectral decomposition: Az ∑∞ j 1 λj ∑γj k 1〈z, φj,k〉φj,k . The eigenvalues 0 < λ1 < λ2 < · · · < · · ·λn → ∞ of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace, and {φj,k} is a complete orthonormal set of eigenvectors of A. The operator −A generates a strongly continuous semigroup {T t } given by T t z ∑∞ j 1 e −λj t ∑γj k 1〈z, φj,k〉φj,k . Our result can be applied to the nD heat equation, the OrnsteinUhlenbeck equation, the Laguerre equation, and the Jacobi equation.


Introduction
In this paper we prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert space Z L 2 Ω given by z −Az 1 ω u t , t ∈ 0, τ , where Ω is a domain in R n , ω is an open nonempty subset of Ω, 1 ω denotes the characteristic function of the set ω, and the distributed control u ∈ L 2 0, t 1 ; L 2 Ω and A : D A ⊂ Z → Z Mathematical Problems in Engineering is an unbounded linear operator.Here we assume the following spectral decomposition for A: λ j E j z, 1.2 with •, • denoting an inner product in Z, and The eigenvalues 0 < λ 1 < λ 2 < • • • < λ j < • • • λ n → ∞ of A have finite multiplicity γ j equal to the dimension of the corresponding eigenspace, and {φ j,k } is a complete orthonormal set of eigenvectors of A. So, {E j } is a complete family of orthogonal projections in Z and z ∞ j 1 E j z, z ∈ Z.The operator −A generates a strongly continuous semigroup {T t } given by 1.4 Systems of the form 1.1 are thoroughly studied in 1, 2 , but the interior controllability is not considered there.
Examples of this class of equations are the following well-known partial differential equations.
Example 1.1.The interior controllability of the heat equation, where Example 1.2 see 3, 4 . 1 The interior controllability of the Ornstein-Uhlenbeck equation is 2 The interior controllability of the Laguerre equation is 3 The interior controllability of the Jacobi equation is To complete the exposure of this introduction, we mention some works done by other authors showing the difference between our results and those of them: the interior approximate controllability is very well-known fascinate and important subject in systems theory; there are some works done by 5-9 .
Particularly, Zuazua in 9 proves the interior approximate controllability of the heat equation 1.5 in two different ways.In the first one, he uses the Hahn-Banach theorem, integrating by parts the adjoint equation, the Carleman estimates and the Holmgren Uniqueness theorem 10 .But, the Carleman estimates depend on the Laplacian operator Δ, so it may not be applied to those equations that do not involve the Laplacian operator, like the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.
The second method is constructive and uses a variational technique: let us fix the control time τ > 0, the initial and final state, z 0 0, z 1 ∈ L 2 Ω , respectively, and > 0. The control steering the initial state z 0 to a ball of radius > 0, and center z 1 is given by the point in which the following functional achieves its minimum value: where ϕ is the solution of the corresponding adjoint equation with initial data ϕ τ .The technique given here is motivated by the following results.

Main Theorem
In this section we will prove the main result of this paper on the controllability of the linear system 1.1 .But before that, we will give the definition of approximate controllability for this system.To this end, the system 1.1 can be written as follows: where the operator B ω : Z → Z is defined by B ω f 1 ω f.For all z 0 ∈ Z and u ∈ L 2 0, τ; Z the initial value problem 2.1 admits only one mild solution given by Definition 2.1 exact controllability .The system 2.1 is said to be exactly controllable on 0, τ if for every z 0 , z 1 ∈ Z there exists u ∈ L 2 0, τ; Z such that the solution z of 2.2 corresponding to u satisfies z τ z 1 .
Definition 2.2 approximate controllability .The system 2.1 is said to be approximately controllable on 0, τ if for every z 0 , z 1 ∈ Z, ε > 0 there exists u ∈ L 2 0, τ; Z such that the solution z of 2.2 corresponding to u satisfies Remark 2.3.The following result was proved in 12 .If the semigroup {T t } is compact, then the system z −Az B ω u t can never be exactly controllable on time τ > 0, which is the case of the heat equations, the Ornstein-Uhlenbeck equation, the Laguerre equation, the Jacobi equation, and many others partial differential equations.
The following theorem can be found in a general form for evolution equation in 2 .
Theorem 2.4.The system 2.1 is approximately controllable on 0, τ if, and only if, Now, we are ready to formulate and prove the main theorem of this paper.

Theorem 2.5.
If for an open nonempty set ω ⊂ Ω the restrictions φ ω j,k φ j,k | ω to ω are linearly independent functions on ω, then for all τ > 0 the system 2.1 is approximately controllable on 0, τ .
Proof.We will apply Theorem 2.4 to prove the approximate controllability of system 2.1 .To this end, we observe that B ω B * ω and T * t T t .Suppose that B * ω T * t z 0, ∀t ∈ 0, τ .Then, z, φ j,k φ j,k x 0, ∀x ∈ ω.

Applications
As an application of our result we will prove the controllability of the nD heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation and the Jacobi equation.

The Interior Controllability of the Heat Equation 1.5
In this subsection we will prove the controllability of system 1.5 , but before that, we will prove the following theorem.To this end, first, we will consider the following definition and results from 13 .Definition 3.2.A differential operator L is say to be hypoelliptic analytic if for each open subset Ω of R n and each distribution u ∈ D Ω , we have that: if L u is an analytic function in Ω, then u is an analytic function in Ω.

Corollary 3.3 see 13, page 15 . Every second-order elliptic operator with constant coefficients is hypoelliptic analytic.
Proof of Theorem 3.1.Let φ be an eigenfuction of −Δ with corresponding eigenvalue λ > 0.Then, the second-order differential operator L Δ λ is an elliptic operator according to 13, Definiton 7.2, page 97 .Therefore, applying the foregoing corollary we get that L Δ λ hypoelliptic analytic.
On the other hand, we know that Lφ Δφ λφ 0, which is trivially an analytic function, then φ is an analytic function in Ω.Now, we will make the abstract formulation of the problem, and to this end, let us consider Z L 2 Ω and the linear unbounded operator A : D A ⊂ Z → Z defined by Aφ −Δφ, where It is well-known that this operator A has spectral decomposition given by 1.2 and the system 1.5 can be written as an abstract equation in the space where the control function u belongs to L 2 0, τ; Z , and the operator B ω : Z → Z is defined by B ω f 1 ω f.Theorem 3.4.For all open nonempty set ω ⊂ Ω and τ > 0 the system 3.2 is approximately controllable on 0, τ .ii Laguerre operator:

The Interior
can be represented in the form of 1.2 .This was done in 3, 4 , where they prove that the eigenfunctions in these cases are polynomial functions in multiple variables, which are trivially analytic functions.

Final Remark
The result presented in this paper can be formulated in a more general setting.Indeed, we can consider the following evolution equation in a general Hilbert space Z: where A : D A ⊂ Z → Z is an unbounded linear operator in Z with the spectral decomposition given by 1.2 , the control u ∈ L 2 0, τ; Z and B : Z → Z is a linear and bounded operator linear and continuous .
In this case the characteristic function set is a particular operator B, and the following theorem is a generalization of Theorem 2.5.Hence, from Lemma 1.4, we obtain that γ j k 1 z, φ j,k B * φ j,k 0, j 1, 2, . . . .Since B * φ j,k are linearly independent on Z, we obtain that z, φ j,k 0, j 1, 2, . . . .Therefore, E j z 0, j 1, 2, 3, . .., which implies that z 0. So, the system 4.1 is approximately controllable on 0, τ .Remark 4.2.As future researches, we will try to use this technique to study the controllability of other partial differential equations such as the thermoelastic plate equation, the equation modelling the damped flexible beam, and the strongly damped wave equation.

Corollary 2 . 6 .
If φ j,k are analytic functions on Ω, then for all open nonempty set ω ⊂ Ω and τ > 0 the system 2.1 is approximately controllable on 0, τ .Proof.It is enough to prove that, for all open nonempty set ω ⊂ Ω the restrictions φ ω j,k φ j,k | ω to ω are linearly independent functions on ω, which follows directly from Theorem 1.3.

Theorem 3 . 1 .
The eigenfunctions of the operator −Δ with Dirichlet boundary conditions on Ω are real analytic functions in Ω.