1. Introducción

The Timoshenko beam theory was developed by Ukrainian-born scientist Stephen Timoshenko in the beginning of the 20th century. The model takes into account shear deformation and rotational inertia effects, making it suitable for describing the behavior of short beams, sandwich composite beams or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order, but unlike ordinary beam theory - i.e. Bernoulli-Euler theory, there is also a second order spatial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter, and thus the distance between opposing shear forces decreases.

This paper has been motivated by the works in[2],[8],[9],[10],[12] and[13], where a new technique is used to prove the approximate controllability of some diffusion process.

Following[2],[9] and[13], in this paper we study the interior approximate controllability of the following Timoshenko Type Equation

(1)

Where Ω is a sufficiently regular bounded domain in and such that , ω is an open nonempty subset of , denotes the characteristic function of the set ω and the distributed control

Specifically, we prove the following statement: For all the system is approximately controllable on Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time

But, before proving this result, we study the approximate controllability of the following Timoshenkotype equation with the controls acting in the whole set Ω using some result from[8].

(2)

Where.

Of course, the interior approximate controllability of this equation is more interesting problem from the applications point of view since the control is acting only in a subset or part of Ω. Our technique is simple and rests on the shoulders of the following fundamental results:

Theorem 1.1.[10] The eigenfunctions of -Δ with

Dirichlet boundary condition on are real analytic functions.

Theorem 1.2.[1]Suppose is an open, nonempty and connected set, and f is areal analytic function in Ω with on a non-empty open subset ω of Ω. Then in Ω.

2. Abstract Formulation of the Problem

Let and consider the linear unbounded operator defined by , where

(3)

The operator A has the following very well known properties: the spectrum of A consists of only eigenvalues

,each one with multiplicity equal to the dimension of the corresponding eigenspace.

a) There exists a complete orthonormal set of eigenvectors of A.

b) For all we have

(4)

Whereis the inner product in X and

(5)

So, is a family of complete orthogonal projections in z and

(6)

c) generates an analytic semigroupgiven by

(7)

d) The fractional powered spaces are given by:

,

with the norm

,

And

(8)

Also, for we define, which is a Hilbert Space with norm givenby

Hence, the equations (1) and (2) can be written as abstract systems of ordinary differential equations in the Hilbert space

(9)

(10)

where

(11)

and

(12)

is a linear unbounded operator with domain

and

Now, using the following Lemma from[11] we can prove that the linear unbounded operator given by the linear equation (9) generates a strongly continuous semigroup which decays exponentially to zero.

Lemma2.1.Let Z be a separable Hilbert space and two families of bounded linear operators in Z with being a complete family of orthogonal projections such that

(13)

Define the following family of linear operators

(14)

Then

a) T(t) is a linear bounded operator if

(15)

for some continuous real-valued function g(t).

b) Under the condition (15) is a C₀ -semigroup in the Hilbert space Z whose infinitesimal generator is given by

(16)

With

(17)

c) The spectrum of is given by

(18)

Theorem 2.2.The operator given by (12) is the infinitesimal generator of a strongly continuo semigroup represented by

(19)

Whereis a complete family orthogonal projections in the Hilbert space given by

(20)

and

(21)

Where

Therefore, , the ingenvalues

are simple and

Where are the roots of the characteristic equation

and this semigroup decays exponentially to zero; that is to say,

where

and

The following gap condition plays an important role in this paper

(22)

3. Controllability of the System (10)

In this section we shall prove the approximate controllability of the system (10). But, before we shall give the definition of approximate controllability for this system. To this end, for all and the initial value problem

(23)

were , admits only one mild solution given by

(24)

Definition 3.1.(Approximate Controllability) The system (10) is said to be approximately controllable on if for every there exists such that the solution of (23) corresponding to verifies:

Definition 3.2. For the system (10) we define the following concepts:

a) The controllability mapping is defined by

(25)

b) The grammian mapping is given by

that is to say

Theorem 3.3.The system (10) is approximately controllable on if, and only if, one of the following statements holds:

Proposition 3.4.The following equality holds:

(26)

Prof.From (11) we know that Then,

And

Since

we get that

On the other hand,

Therefore,

REMARK 3.1.If ω is a nonempty open sub set of Ω such that , then

Now, we shall use the equality (26) in order to characterize the approximate control ability of the system (10) in terms of the following family of finite dimensional control problems,

(27)

where

Proposition 3.5. The operator

can be written as follows

where,

Proof. From condition (26) and the representation (19) of T(t) we obtain

where,

Theorem 3.6.a) The system (10) is approximately controllable on if, and only if, each of the following system

(28)

is approximately controllable.

b)The system (10) is approximately controllable on if, and only if

Proof.a) For the purpose of contradiction, let us assume that system (10) is approximately controllable on and there exists j such that the system

is not approximately controllable on. Then, there exists such that:

(29)

On the other hand, from part (iii) of Theorem 3.3 we have that:

Now, letting , we obtain:

This implies that, which contradicts the assumption. Therefore, (28) is approximately controllable for all j.

If for all j system (28) is approximately controllable, then by Theorem 3.3 part (ii),

Clearly that, for all , there exists such that. Then, using Proposition 3.5, we get for all z in Z that

Hence, (10) is approximately controllable and (a) is proved.

b) follows immediately from (a) and Theorem 3.3.

Next, we shall use the following result: Consider the following finite dimensional controlsystem

(30)

Where A and B are matrixes of dimensions and respectively.

Theorem 3.7. (see[Lee and Marcus(1967)]). (Kalman) The system (30) is controllably on if, and only if,

That is to say,

where is the vector space generated by .

Theorem 3.8. The system (10) is approximatelycontroll-able on.

Proof. It is enough to prove the controllability of the finite dimensional system (28) with

and

So,

where

Therefore, the controllability of the system (28) is equivalent to the controllability of each finite dimensional systems,

(31)

where, and the system (31) is controllable if, and only if,

which can be verified trivially. Therefore, system (31) is controllable, and consequently, system (10) is also approximately controllable applying Theorem 3.6.

4. Proof of the Main Theorem

In this section we shall prove the main result of this paper on the approximate controllability of the linear system (9). To this end, we observe that the definition of controllability for system (9) is similar to the one given to system (10). And, for all and the initial value problem

(32)

admits only one mild solution given by

(33)

Consider the following bounded linear operator:

(34)

Whose adjoint operator is given by

(35)

The following lemma is trivial:

Lemma 4.1.The equation (9) is approximately controllable on if, and only if,.

The following result is well known from linear operator theory:

Lemma 4.2.Let W and Z be Hilbert spaces and

the adjoint operator of the linear operator . Then,

As a consequence of the foregoing Lemma one can prove the following result:

Lemma 4.3.Let W and Z be Hilbert spaces and

the adjoint operator of the linear operator Then if, and only if, one of the following statements holds:

The following theorem follows directly from (35), lemma 4.1 and lemma 4.3.

Theorem 4.4.(9) is approximately controllable on if, and only if,

(36)

For the proof of the main theorem of this paper we shall use the following version ofLemma 3.14 from [3] and Lemma 4.4 from [2].

Lemma 4.5.Let be sequences of real numbers such that

(37)

Then, for any we have that

(38)

if, and only if,

(39)

Proof. (Lemma 4.5) By analytic extension we obtain

(40)

Now, dividing this expression by we get

From (37) we have that and for and, for, then passing to the limit when we obtain that.

Then, we have that

Now, dividing this expression by we get

From (37) we have that and for and, for , then passing to the limit when we obtain that.

Then, we have that

In general, if we continue with this process and divide this expression by, we get that

From (37) we have that

Then, passing to the limit when we obtain that So, continuing with this procedure we get that

and

Repeating this procedure from here, we would obtain that,and continuing this way we get

Now, we are ready to formulate and prove the main theorem of this work.

Theorem 4.6.(Main Result) For all nonempty open subset ωof Ω and the system(9) is approximately controllable on. Moreover, a sequence of controls steering thesystem (9) from initial state to an neighborhood of the final state at time is given by

and the error of this approximation is given by

Proof. We shall apply Theorem 4.4 to prove the controllability of system (9). To this end, we observe that

and, since the eigenvalues of the matrix are simple, there exists a family of complete complementary projections such that

Therefore,

where

Now, suppose that . Then,

The assumption (22) implies that the sequence

satisfies the conditions on Lemma 4.5. In fact, we have trivially that for and from (22) we obtain:

Therefore,

Then, from Lemma 4.5 we obtain for all that

Since

we obtain that,

Then,

On the other hand, from Theorem 1.1 we know that are analytic functions, which implies the analyticity of. Then, from Theorem 1.2 we get for that

From Theorem 3.8, the system (10) is approximately controllable. So, from part iii) of Theorem 3.3 we conclude that .

Therefore,

Then, from Theorem 4.4 we obtain that system (9) is approximately controllable.

Now, given the initial and the final states and, we consider the sequence of controls

Then,

From part c) of Lemma 4.3 we know that

Therefore,

This completes the proof of the Theorem.

Corollary 4.7.The approximate controllability of the system (9) is equivalent to the approximate controllability of the system (10).

5. Final Remark

The result presented in this paper can be formulated in a more general setting. Indeed, we can consider the following Timoshenko Type Equation in a general Hilbert space Z and

(41)

where, is an unbounded linear operator in Z with the spectral decomposition given by

with eigenvalues

each one with multiplicity equal to the dimension of the corresponding eigenspace.

a) There exists a complete orthonormal set of eigenvectors of A.

b) For all we have

c)

(42)

The controls and are linear and bounded.When , the operators and are particular cases of and.