Observability and uniqueness theorem for a coupled hyperbolic system

Boris V. Kapitonov,Joel Souza

doi:10.1155/s0161171200002477

Abstract

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 Uniqueness Method (HUM), to establish equivalence of norms which 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.

Highlights

Several approaches are known concerning the principle of unique continuation
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:
We have the following results of unique continuation, i.e., if Ᏸ ⊂ Ω is an arbitrary neighborhood of Γ0 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 (4.5) with homogeneous Dirichlet condition and u ≡ v ≡ 0, in Ᏸ×]0, T [, (4.10)

Summary

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. Observability, inverse inequality, uniqueness theorem, unique continuation. We have the following result: if u is a solution of (1.1) with {f , f1} ∈ Ᏸ(Ω)×Ᏸ(Ω), and (∂u/∂ν) = 0 on Σ0, this

Results

Conclusion