Interior Controllability of a Timoshenko Type Equation

In this paper we prove the interior controllability of the following Timoshenko Type Equation �


Introduction
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.
Specifically, we prove the following statement: For all > 0 the system is approximately controllable on [0, ]. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time > 0. 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].
Where ∈ 2 �[0, ]; 2 (Ω)�, = 1,2. 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:

Abstract Formulation of the Problem
Let = 2 (Ω) and consider the linear unbounded operator The operator A has the following very well known properties: the spectrum of A consists of only eigenvalues 0 < 1 < 2 < ⋯ < → ∞, each one with multiplicity equal to the dimension of the corresponding eigenspace.
a) There exists a complete orthonormal set � , � of (4) Where〈⋅,⋅〉is the inner product in X and , ∈ (6) c) − generates an analytic semigroup{ ( )} ≥0 given by The fractional powered spaces are given by: 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 1 = 1 × × 1 × as follows: is a linear unbounded operator with domain ( ) = ( 2 ) × ( ) × ( 2 ) × ( )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.

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 0 ∈ and ℱ ∈ 2 ([0, ]; ) the initial value problem were = 2 (Ω) × 2 (Ω), admits only one mild solution given by The system (10) is said to be approximately controllable on [0, ] if for every 0 , 1 ∈ 1 , > 0there exists ℱ ∈ 2 (0, ; ) such that the solution ( ) of (23) corresponding to ℱ verifies: (0) = 0 and ‖ ( ) − 1 ‖ < . Definition 3.2. For the system (10) we define the following concepts: a) The controllability mapping (25) b) The grammian mapping : → is given by = * that is to say (10) is approximately controllable on [0, ]if, and only if, one of the following statements holds: i).
Prof.From (11) we know that = � 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, where,

Proof. From condition (26) and the representation (19) of T(t) we obtain
Theorem 3.6.a) The system (10)  This implies that = 0 , 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 ∈ ( ≠ 0), there exists ∈ ℕ such that ≠ 0. Then, using Proposition 3.5, we get for all 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 ′ = ( ) + ( ), > 0, ∈ ℝ , ∈ ℝ , (30) Where A and B are matrixes of dimensions × and × respectively. Proof. It is enough to prove the controllability of the finite dimensional system (28) with Therefore, the controllability of the system (28) is equivalent to the controllability of each finite dimensional systems, where,ℱ ∈ 2 (0, ; ℝ 2 )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.

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 0 ∈ 1 and ℱ ∈ 2 (0, ; ) the initial value problem admits only one mild solution given by The following theorem follows directly from (35) Then, for any > 0 we have that if, and only if, (39) Proof. (Lemma 4.5) By analytic extension we obtain ∀ ∈[0,∞). (40) Now, dividing this expression by 2 (1) we get From (37) we have that 1 ( ) − 2 (1) < 0 and 2 ( ) − 2 (1) < 0 for ≥ 1 and ( ) − 2 (1) < 0 , for ≥ 3, ≥ 1, then passing to the limit when → ∞ we obtain that 21 = 0. Then, we have that Now, dividing this expression by 3 (1) we get From (37) we have that 1 ( ) − 3 (1) < 0 and 2 ( ) − 3 (1) < 0for ≥ 1 and 3 ( ) − 3 (1) < 0, for ≥ 2, then passing to the limit when → ∞ we obtain that 31 = 0. Then, we have that =0, ∀ ∈[0,∞). In general, if we continue with this process and divide this expression by (1) , we get that Then, passing to the limit when → ∞ we obtain that 1 = 0. So, continuing with this procedure we get that 11 = 21 = 31 = ⋯ 1 = 0 Repeating this procedure from here, we would obtain that 12 = 22 = 32 = ⋯ = 2 = 0 ,and continuing this way we get Now, we are ready to formulate and prove the main theorem of this work.   From Theorem 3.8, the system (10) is approximately controllable. So, from part iii) of Theorem 3.3 we conclude that = 0.