OBSERVABILITY AND UNIQUENESS THEOREM FOR A COUPLED HYPERBOLIC SYSTEM

We deal with the inverse inequality for a coupled hyperbolic system with dissipation. The inverse inequality is an indispensable inequality that appears in the Hilbert UniquenessMethod (HUM), to establish equivalence of normswhich guarantees uniqueness and boundary exact controllability results. The term observability is due to the mathematician Ho (1986) who used it in his works relating it to the inverse inequality. We obtain the inverse inequality by the Lagrange multiplier method under certain conditions.


Introduction.
Several approaches are known concerning the principle of unique continuation.One of them consists in the classical principle of identity for analytical function, that is, holomorphic (analytic) functions which are defined in some region can frequently be extended to holomorphic functions in some larger region.These extensions are uniquely determined by the given functions (cf.[6,7,10]).Another extension process, introduced by Holgren, is based on the resolution of the homogeneous boundary value problem for the wave equation, that is, given {f (x),f 1 (x)} in Ᏸ(Ω) × Ᏸ(Ω), let us consider the homogeneous boundary value problem: It is well known that in this case (1.1) has a strong solution and Then we define in Ᏸ(Ω) × Ᏸ(Ω) the quadratic form: where Σ 0 is a part of the lateral boundary Σ of the cylinder Q, with positive measure.The quadratic form (1.3) is a seminorm in Ᏸ(Ω)×Ᏸ(Ω).We have the following result: if u is a solution of (1.1) with {f ,f 1 } ∈ Ᏸ(Ω)×Ᏸ(Ω), and (∂u/∂ν) = 0 on Σ 0 , then this implies that u is zero in Q, i.e., the seminorm (1.3) is, indeed, a norm in Ᏸ(Ω)×Ᏸ(Ω).This is true due to the Holgren's theorem (cf.[3,8]).Our purpose in this paper is to establish a result of unique continuation in the direction of the Holgren's theorem based on a result of Ruiz [11].We present this approach as follows.
Let Ω be a bounded domain in R n with boundary Γ = ∂Ω.In the cylinder Q = Ω×]0,T [ we consider the initial boundary value problem for the wave equation (1.1).We are interested in the following result: there exists an open subset Ᏸ ⊂ Ω such that there exists a time T > 0 and a constant C > 0, so that (1.4) Lions (cf.[9]) proved that this estimate holds if Ω is of class C 2 , and Ᏸ is a neighborhood of the part of the boundary , where x 0 is some point of R n , and ν(x) is the unit outer normal.
Bardos, Lebeau and Rauch [2] showed that, when Ω is of class C ∞ , inequality (1.4) holds if Ᏸ satisfies some "geometric control property," that is: there exists some T > 0 such that every ray of geometric optics intersects the set Ᏸ × (0,T ).
2. Problem formulation.In the cylinder Ω×]0,T [ we consider the following initial boundary value problem for the coupled hyperbolic system with dissipation where u = (u 1 ,...,u m ), (v 1 ,...,v m ), A = A * and B(x) = B * (x) are square matrices of order m, and we are using the Einstein summation convention.
We assume that We denote by Ᏼ the real Hilbert space of quadruples {u, v, u 1 ,v 1 } of m-compound vector-functions such that 3) The inner product in Ᏼ is defined by the formula (2.4) We defined an unbounded operator Ꮽ in Ᏼ given by: In standard way, we can check that the domain Ᏸ(Ꮽ * ) of the adjoint operator is given by: (2.6) Thus, the operator Ꮽ is skew selfadjoint and from Stone's theorem it follows that it generates a 1-parameter group of unitary operators ᐁ(t) in Ᏼ.Moreover, ᐁ(t) is strongly continuous in t and ᐁ(t){f ,g,f 1 ,g 1 } is strongly differentiable with respect to t for and possesses the following regularity: Now, we introduce the following notation: (2.9) Since the operator ᐁ(t) is unitary, we have We also observe that, for Ᏺ = {f ,g,f 1 ,g 1 } ∈ Ᏼ, ᐁ(t)Ᏺ is a weak solution in Ᏼ to the abstract Cauchy problem in the following sense:

Observability and uniqueness theorem.
In this section, we obtain the inverse inequality by using the Lagrange multiplier method with certain conditions over an arbitrary neighborhood Ᏸ ⊂ Ω.
It is easy to verify that the unperturbed wave equation of (2.1a) and (2.1b) is variational and that the expression is its Lagrangian.The invariance of the elementary action ᏸdx 1 •••dx n dt under the 1-parameter group of dilations in all variables with infinitesimal operator leads to a conservation law (cf.[1, E. Noether's theorem, page 88]).Now let {f ,g,f 1 ,g 1 } ∈ D(Ꮽ) and let u(x, t), v(x, y) be a solution of problem (2.1).After integration by parts, over the cylinder Ω×]0,T [, that conservation law we then obtain the identity: where the pair (u, v) is a solution of (2.1); ϕ(x), C(x) are smooth functions in Ω, and We observe that identities of the kind in (3.3) are common in [4,5].We have that (3.5) From these inequalities we obtain From the identity it follows that So, using this inequality, from (3.6) we obtain (3.9) We then arrive at the following inequality: (3.10) Now for an arbitrary point x 0 ∈ R n we find: Let Ᏸ ⊂ Ω be an arbitrary neighborhood of Γ 0 .There exists a function ϕ(x) such that (i) (∂ϕ/∂ν) ≤ 0 on Γ , (ii) ϕ 0 (x) ≥ µ > 0, in Ω\Ᏸ, where ϕ 0 We assume that and similarly, From these inequalities we obtain (3.16) Inequalities (3.10) and (3.16) lead to (3.17) In the general case, we can choose function ϕ(x) in the following way: and extend it in Ᏸ satisfying the property (i).Then, Thus, from (3.17), we arrive at the inequality: ( Now we can assume that there exists a function ϕ(x) which satisfy (i), (ii), and (iii) ∆ 2 ϕ − 2∆ϕ 0 (x) ≤ 0, in Ω. (We can construct some examples of these functions under certain assumptions on Ω and Ᏸ.)In this case, from (3.17), it follows (3.21) We choose ε such that µ > εC 0 (we can set ε = (µ/2C 0 )), and Then (3.20) and (3.21) can be rewritten in the following way: where C = C(ϕ, b 0 ,a 0 , Ω).

Conclusion.
We arrive at the following assertion: in the cylinder Ω×]0,T [, we consider the initial boundary value problem for the following coupled hyperbolic system with dissipation: Here where Ᏸ is an arbitrary neighborhood of the boundary (or a part of the boundary).We have that: (1) If Ᏸ is an arbitrary neighborhood of Γ 0 = {x ∈ Γ ; (x − x 0 , ν(x)) ≥ 0}, then there exists T 0 > 0 such that, for T > T 0 , the following inequality holds: (2) If Ᏸ and Ω are such that there exists a function ϕ(t) with properties (i), (ii), and (iii), then, for T > T 0 , we have Remark 4.1.Let us consider the coupled system: where A, B, and These new terms considered add to the right-hand side of the main identity (3.3) (and the rest of the formulas) the following expression: (4.7) Thus, we obtain under the same assumption: (1) In the first case, (4.8) (2) In the second case: i.e., in this situation only the first case makes sense.Thus, we have the following results of unique continuation, i.e., if Ᏸ ⊂ Ω is an arbitrary neighborhood of Γ 0 satisfying (i) and (ii), with support (G) ⊆ Ᏸ, and the pair (u, v) is a strong solution of the initial boundary value problem for the coupled system (4.