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 given by , , where is a domain in , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control and is an unbounded linear operator with the following spectral decomposition: . The eigenvalues of have finite multiplicity equal to the dimension of the corresponding eigenspace, and is a complete orthonormal set of eigenvectors of . The operator generates a strongly continuous semigroup given by . Our result can be applied to the D heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.
In this paper we prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert space given by where is a domain in , is an open nonempty subset of , denotes the characteristic function of the set and the distributed control and is an unbounded linear operator. Here we assume the following spectral decomposition for : with denoting an inner product in , and The eigenvalues of have finite multiplicity equal to the dimension of the corresponding eigenspace, and is a complete orthonormal set of eigenvectors of . So, is a complete family of orthogonal projections in and The operator generates a strongly continuous semigroup given by 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 is a bounded domain in of class , is an open nonempty subset of , denotes the characteristic function of the set , and the distributed control .
Example 1.2 (see [3, 4]). (1) The interior controllability of the Ornstein-Uhlenbeck equation is
where is the Gaussian measure in and is an open nonempty subset of .
(2) The interior controllability of the Laguerre equation is where , is the Gamma measure in and is an open nonempty subset of .
(3) The interior controllability of the Jacobi equation is where , , is the Jacobi measure in and is an open nonempty subset of .
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  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 . But, the Carleman estimates depend on the Laplacian operator , 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 , the initial and final state, , , respectively, and . The control steering the initial state to a ball of radius and center is given by the point in which the following functional achieves its minimum value: where is the solution of the corresponding adjoint equation with initial data .
The technique given here is motivated by the following results.
Theorem 1.3 (see [11, Theorem 1.23, page 20]). Suppose is open, nonempty and connected set, and is real analytic function in with on a nonempty open subset of . Then, in .
Lemma 1.4 (see [1, Lemma 3.14, page 62]). Let and be two sequences of real numbers such that . Then if and only if
2. 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 is defined by . For all and the initial value problem (2.1) admits only one mild solution given by
Remark 2.3. The following result was proved in . If the semigroup is compact, then the system can never be exactly controllable on time , 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 .
Theorem 2.4. The system (2.1) is approximately controllable on 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 the restrictions to are linearly independent functions on , then for all the system (2.1) is approximately controllable on .
Proof. We will apply Theorem 2.4 to prove the approximate controllability of system (2.1). To this end, we observe that and . Suppose that . Then, Hence, from Lemma 1.4, we obtain that Since are linearly independent on , we obtain that Therefore, , which implies that . So, the system (2.1) is approximately controllable on .
Corollary 2.6. If are analytic functions on , then for all open nonempty set and the system (2.1) is approximately controllable on .
Proof. It is enough to prove that, for all open nonempty set the restrictions to are linearly independent functions on , which follows directly from Theorem 1.3.
As an application of our result we will prove the controllability of the D heat equation, the Ornstein-Uhlenbeck equation, the Laguerre equation and the Jacobi equation.
3.1. 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.
Theorem 3.1. The eigenfunctions of the operator with Dirichlet boundary conditions on are real analytic functions in .
To this end, first, we will consider the following definition and results from .
Definition 3.2. A differential operator is say to be hypoelliptic analytic if for each open subset of and each distribution , we have that: if is an analytic function in , then is an analytic function in .
Corollary 3.3 (see [13, page 15]). Every second-order elliptic operator with constant coefficients is hypoelliptic analytic.
Proof of Theorem 3.1. Let be an eigenfuction of with corresponding eigenvalue . Then, the second-order differential operator is an elliptic operator according to [13, Definiton 7.2, page 97]. Therefore, applying the foregoing corollary we get that hypoelliptic analytic.
On the other hand, we know that , which is trivially an analytic function, then is an analytic function in .
Now, we will make the abstract formulation of the problem, and to this end, let us consider and the linear unbounded operator defined by , where It is well-known that this operator 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 belongs to , and the operator is defined by .
Theorem 3.4. For all open nonempty set and the system (3.2) is approximately controllable on .
Proof. It is enough to prove that the operators (i)Ornstein-Uhlenbeck operator: , defined on ,with in the space (ii)Laguerre operator: , defined on , with , in the space ;(iii)Jacobi operator: , , with , in the space 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.
4. 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 : where is an unbounded linear operator in with the spectral decomposition given by (1.2), the control and is a linear and bounded operator (linear and continuous).
In this case the characteristic function set is a particular operator , and the following theorem is a generalization of Theorem 2.5.
Theorem 4.1. If the vectors are linearly independent in , then the system (4.1) is approximately controllable on .
Proof. From [2, Theorem 4.1.7, part (b)-(iii)], it is enough to prove that To this end, we observe that Hence, from Lemma 1.4, we obtain that Since are linearly independent on , we obtain that Therefore, , which implies that . So, the system (4.1) is approximately controllable on .
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.
This work was supported by the CDHT-ULA-Project: C-1667-09-05-AA and the DID-USB-Project: S1-IC-CB-011-07.
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