ABSTRACT
Following the finite-dimensional stabilization problems in the preceding chapter, we are mainly interested in control problems governed by partial differential equations. In this chapter, some preliminary results on elliptic problems necessary for the following chapters are discussed. In this monograph, a uniformly elliptic operator ℒ of order 2 equipped with a boundary operator τ is studied. Let Ω be a bounded domain in ℝm with the boundary Γ which consists of a finite number of smooth components of (m – 1)-dimension. The pair (ℒ, τ) is a standard one, and described as () ℒ u = − ∑ i , j = 1 m ∂ ∂ x i ( a i j ( x ) ∂ u ∂ x j ) + ∑ i = 1 m b i ( x ) ∂ u ∂ x i + c ( x ) u , τ u = α ( ξ ) u + ( 1 − α ( ξ ) ) ∂ u ∂ v , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315367798/bad9a720-4aeb-4c28-9c18-ba1aa5ac9d9d/content/eq203.tif"/>
