Null controllability properties of some degenerate parabolic equations.

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The problem in 2D : simplest assumptions on the degeneracy A(x) ∼ ( λ1 (x) 0 0 λ 2 (x) ) with 0 ≤ λ 1 (x) ≤ λ 2 (x). ε 1 (x) denotes the eigenvector associated to λ 1 (x), and ε 2 (x) the eigenvector associated to λ 2 (x). Simplest assumptions (H s (A)) : Ω is of class C 4 , and there exists some α ≥ 0, and some neighborhood of the boundary Γ such that : ◮ λ 1 (x) = d(x, Γ) α for all x ∈ V(Γ), ◮ ε 1 (x) = ν(p Γ (x)) for all x ∈ V(Γ), where p Γ (x) is the projection of x on the boundary Γ, ◮ 0 < m ≤ λ 2 (x) ≤ M for all x ∈ Ω. 8/26
The control problem We consider ⎧ ⎪⎨ u t − div (A(x)∇u) = h(x, t)χ ω , boundary conditions, ⎪⎩ initial condition, under (H s (A)) . ◮ Question 1 : well-posedness yes, in natural weighted Sobolev spaces. ◮ Question 2 : null controllability : given T > 0, u 0 ∈ L 2 (Ω), does there exist h ∈ such that u(T ) = 0 9/26
Page 1 and 2: Carleman estimates and applications
Page 3 and 4: Presentation of the problem Ω boun
Page 5 and 6: An example in N-D ◮ biology : the
Page 7: ◮ α ≥ 2 : well-posed with the
Page 11 and 12: Associated observability estimate T
Page 13 and 14: Associated observability estimate T
Page 15 and 16: Extensions ◮ more general degener
Page 17 and 18: Useful 1-D Hardy type inequalities
Page 19 and 20: A useful improved version (J. Vanco
Page 21 and 22: Useful 2-D (N-D) Hardy type inequal
Page 23 and 24: How to obtain the Carleman estimate
Page 25 and 26: But the Hardy type inequalities all

Null controllability properties of some degenerate parabolic equations.

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