Carleman estimates for one-dimensional degenerate heat equations

Abstract

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain.We derive new Carleman estimates for the degenerate parabolic equationwhere the function a mainly satisfies 1)), a > 0 on (0, 1) and 1 √ a ∈ L 1 (0, 1).We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0) = 0 and/or a(1) = 0. A typical example is a(x) = x α (1 − x) β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equationThe present paper completes and improves previous works [7,8] where this problem was solved in the case a(x) = x α with α ∈ [0, 2).

Null controllability of Kolmogorov-type equations

“…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

“…Several extensions of the above results are available in one space dimension, see [1,37] for equations in divergence form, [11,10] for nondivergence form operators, and [9,25] for cascade systems. Fewer results are available for multidimensional problems, mainly in the case of two dimensional parabolic operators which simply degenerate in the normal direction to the boundary of the space domain, see [15].…”

“…In the case of γ ∈ [1/2, 1], our weight β will be the classical one. On the other hand, for γ ∈ (0, 1/2) we follow the strategy of [1,11,37], adapting the weight β to the nonsmooth coefficient |x| 2γ .…”

Approximate controllability of the coupled degenerate system with two boundary controls

“…In this paper, we investigate the approximate controllability of the coupled degenerate system Recently, the controllability of the following degenerate parabolic equation has been investigated; see references [1][2][3][4][5]: u t -x p u x x + c(x, t)u = hχ ω , (x, t) ∈ (0, 1) × (0, T), (1.6) where c ∈ L ∞ ((0, 1) × (0, T)). The degenerate equation (1.6) can be obtained by suitable transformations of the Prandtl equations; see [6].…”

“…The authors prove that the problem (1.6), (1.7) or (1.8) and (1.9) is null controllable if 0 < p < 2, and the problem is not null controllable if p ≥ 2, see the references [1][2][3][4][5]. On the other hand, it is shown that, for every p > 0, the problem (1.6), (1.7) or (1.8) and (1.9) is approximate controllability; see [7,8].…”