Carleman Estimates for a Class of Degenerate Parabolic Operators

Abstract

Given α ∈ [0, 2) and f ∈ L 2 ((0, T) × (0, 1)), we derive new Carleman estimates for the degenerate parabolic problem wt + (x α wx)x = f , where (t, x) ∈ (0, T) × (0, 1), associated to the boundary conditions w(t, 1) = 0 and w(t, 0) = 0 if 0 ≤ α < 1 or (x α wx)(t, 0) = 0 if 1 ≤ α < 2. The proof is based on the choice of suitable weighted functions and Hardy-type inequalities. As a consequence, for all 0 ≤ α < 2 and ω ⊂⊂ (0, 1), we deduce null controllability results for the degenerate one-dimensional heat equation ut − (x α ux)x = hχω with the same boundary conditions as above.

Null controllability of Kolmogorov-type equations

“…Proof of Proposition 9: Let n ∈ Z and g n be dened by (14). Then g n ≡ 0 on (0, T ) × (a, b) and g n solves (15).…”

“…Then, null controllability holds if and only if γ ∈ (0, 1) (see [13,14]), while, for γ ≥ 1, the best result one can show is regional null controllability(see [12]), which consists in controlling the solution within the domain of inuence of the control. Several extensions of the above results are available in one space dimension, see [1,34] for equations in divergence form, [11,10] for nondivergence form operators, and [9,24] for cascade systems.…”

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

“…Then, null controllability holds if and only if γ ∈ (0, 1) (see [13,14]), while, for γ ≥ 1, the best result one can show is "regional null controllability"(see [12]), which consists in controlling the solution within …”

Controllability of Parabolic Equations by the Flatness Approach

“…(Actually any r ≤ −1 is also admissible, by picking p > 1 sufficiently close to 1 in (1.8).) Note that our result applies as well to a(x) = (1 − x) r with 0 < r < 1, yielding a positive null controllability result when the control is applied at the point (x = 1) where the diffusion coefficient degenerates (see [5,Section 2.7]). Note also that the coefficient a(x) is allowed to be degenerate/singular at a sequence of points: consider e.g.…”

“…After the pioneering work in [11,15,20], mainly concerned with the one-dimensional case, there has been significant progress in the general N-dimensional case [13,14,19] by using Carleman estimates. The more recent developments of the theory were concerned with discontinuous coefficients [1,2], degenerate coefficients [4,5,6,12], or singular coefficients [8,10].…”