Estimates Related to the Extended Spectral Control of the Wave Equation

In this work, we show through a numerical approach the extended spectral controllability of the wave equation over a portion of a regular domain, and that the spectrum obtained is quasi-symmetrical. Numerical simulations also verify the analytical estimates demonstrated in Niane and al. [4] and the graphic illustrations relating to this spectral controllability were done.


Introduce the Problem
For the wave equation in Niane and al. [4], the authors has been proved the existence of a C ∞ -control with a support in an open portion ω of the domain Ω as small as they wish control a finite number of uncontrolled eigenmodes.Furthermore the disruption on other modes is inversely proportional to the control duration.Thus, the estimates related to the spectral controllability have been analytically proven for the wave equation.
In the sequel, let Ω be a bounded smooth domain of R 2 , T > 0 the maximum observability time and ω an open subset of Ω.We consider the wave equation with state φ where φ 0 ∈ H 1 0 (Ω) and φ 1 ∈ L 2 (Ω).

Explore Importance of the Problem
The problem of exact internal controllability of the wave equation is to find a control v ∈ L 2 (0, T ; L 2 (ω)) such that if φ is the solution of (1), then φ(T ) = φ (T ) = 0.
Thus, we know that three major difficult achievements of these controls appear: • If ω is too widely open as support controls.
• If ω is too small on the usual functional spaces.
• If the time t of observability is too small.
To avoid these difficulties, we use the concept of extended spectral controllability of seeking control whose support is smaller and exactly control a finite number of specific modes of the system (1) and we study the structure of the associated spectrum.The effect of pertubation caused by such control is not estimated and its evaluation conditions the possibility to use such a method to control the state of distributed systems.
Thus, the objective of this work is to prove that at a certain time depending on the number of controlled and support modes of control, the perturbation is inversely proportional to the time control and the spectrum is quasi symetric.A consequence of this result is that for the initial data in L 2 (Ω) × H −1 (Ω) and an open non-empty ω of the domain Ω, there is an internal control class C ∞ which brings the system to as near state as we want a state considered in L 2 (Ω) × H −1 (Ω).

Reminders on the Spectral Theory
Consider the homogeneous wave equation and let the family (w j ) for { j ∈ N } the sequences of eigenfunctions of the operator −∆ϕ and (λ j ) j∈N the sequence of eigenvalues associated rows in ascending order.If N ∈ N , posing the vector space generated by the eigenfunctions w 1 , ..., w N .If initial conditions (ϕ 0 , ϕ 1 ) ∈ F 2 N , then we have: And the only solution of the homogeneous wave equation (2) ϕ ∈ C(O, T ; F N ).Moreover, it is known by Lions[6,7]: For j ∈ N , let ζ j = cos( λ j t)w j (x) and ξ j = sin( λ j t) Theorem 1 (Niane [3,4]) Let N ∈ N * and the initial couple (y 0 , y 1 ) ∈ F 2 N , it exist T 0 > 0 such that for all T > T 0 , there exist a C ∞ control v with support in ω × ]0, T [ such that the solution y of (1) verify: 1.The orthogonal projection of (y(T ), y (T )) on where C(N, ω) depend only of N and ω.
Remark 1 Thus allow the flow of time, we can bring the system to a condition as nearly as we want its state equilibrium.

Main Results: Differential Characterization and Decomposition Theorem
Theorem 2 Let the quadratic functional J defined over F N × F N by where ϕ is the solution of (1) for the initial data (ϕ 0 , ϕ 1 ), has a unique minimum attained at a point (ψ 0 , ψ 1 ) caracterised by for all ψ solution of (2) corresponding at initial data (ψ 0 , ψ 1 ) ∈ F N × F N . Proof.
Functional J is convex and differentiable, so if it admits a minimum, it is characterized by equation ( 8).So just to show that (8) has a unique solution.
To do this, we can divide the relation (2.7) by T (It may be noted that this division is not essential to show existence and uniqueness).
Therefore, A is a Gram matrix; Knowing that the system {ζ i } ∪ {ξ i } for i ∈ {1, ..., N} form a free family, we deduce that the matrix A is positive definite, so reversible.So, the system (14) admits a unique solution.
Hence the existence and uniqueness of the minimum of the functional J

Decomposition Theorem
Now, let's get down to give a representation of the solution of the linear system (14).
Let D be the diagonal square matrix of order 2N defined as follows Theorem 3 1.The matrix A ( 14) is writen in the form where B is a matrix of which the euclidean norm is majored by a constant R N independant of T and I the unit matrix.
2. There exist a constant T 0 > 0 such that for all T > 0, the solution of the linear system ( 14) is writen: where Proof.∀i, j ∈ {1, ..., N} we obtain by simple calculations the components of the stiffness matrix: • If i j, then we have: • Then, we have • Also, if i j, we have • Finally, • At least i j, we have Of the various components of the matrix A is deducted 1. of Theorem For the 2. of Theorem , it will suffice to prove that To do this, pose the constant where R N = ||D −1 ||||B||; And for T > T 0 , according to Von Neuman Lemma (see Brezis [1]), we deduce that: Hence the solution Φ of the linear system ( 14) is written as We deduce the 2. of Theorem 3. Numerical approaches to estimates for extended spectral controllability Consider the plan regular domain Ω following: Let us return to the linear system ( 14) of order 2N where A is the stiffness matrix, Φ the unknown vector, T time The finite element method we'll use allows us to achieve high levels of convergence with the Free-fem ++ software; Once the triangulation is made, the assembly of the stiffness matrix and the second member vector set, we can obtain estimates related to increases standards for extended internal spectral controllability of the wave equation.3.1 Variational Formulation Let (T h ) a regular triangulation of the domain Ω h into triangles of diameter less than h, the positive and constant step tending to 0.
We define the space W h approximate solutions by: And, thereafter we define the discrete problem: Find u h ∈ W h such that: ∀i, j ∈ {1, ..., N} i.
ii.If i j, then iii.Afterwards, iv.Also, vi.Also, By assembling the various components of (i.) relations to (vi.), one draws the stiffness matrix A.
Let Y h the second member of the discrete space W h defined by: And,

Mesh Viewing
Considering the regular domain Ω meshed below: And so: 1.After assembly of the stiffness matrix and the second member vector, Freefem++ allows us to determine the spectrum of the rigidity matrix.And also determining the values approximate coefficients (eigenvalues) and solutions of the associated linear system.
2. In the domain Ω set (figure 2 and 3), by restricting to a portion of ω of Ω and an internal control, we see that for values below the caused disturbance is inversely proportional to the duration of the control t on a fixed interval ]0, T ]; Hence the values obtained are quasi-symmetric numerically relative to the values of each sets of eigenvalues see fig. 4   -All eigenvalues obtained in all spectra are ranked in ascending order as demonstrated by the spectral theory.But the symmetry of the spectrum is not linked to the rank of a fixed eigenvalue.
-Numerical simulations amply illustrate two types of almost or quasi-symmetric spectra.
-And, this confirms the virtual symmetry of each spectrum recorded in Table 2 by an eigenvalue.
-Moreover, it can be seen that even by further refining the mesh and by increasing the number of series of eigenvalues obtained, the observations remain globally the same.

Conclusion
Thus, the aim of this work was to prove that from a certain time depending on the number of eigen modes controlled and the support of the control, the perturbation is inversely proportional to the control time.A consequence of this result is that for appropriately selected initial data, there is an internal control over a portion of the considered domain to a near state that we wants.And, the theoretical results obtained have been proved numerically.

Figure
Figure 8. Spectrum 5 The approximation errors in the calculation of the eigenvalues associated with this domain (Ω h ) range from −8.953.10 −14 and +4.13202.10−14 with Freefem ++ Software.