Carleman estimates and null controllability of a class of singular parabolic equations

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

Here, (1.4) is degenerate at the boundary y = 0, and its reaction term is singular. Controllability theory, containing approximate controllability and exact controllability, has been widely investigated for semilinear uniformly parabolic equations over the last forty years and there have been a great number of results (see for instance [3,[18][19][20]31] and the references therein for a detailed account). The study of the controllability for semilinear degenerate or singular parabolic equations just began about ten years ago and it is far from being completely solved although many results have been known. Parabolic equations may be not exact controllable for a general terminal datum since there is a smoothing effect for their solutions. As usual, one considers null controllability. That is to say, the zero function is chosen as the terminal datum.
The null controllability of system (1.1)-(1.3) is based on the Carleman estimate for solutions to the conjugate problem x ∈ (0, 1), (1.14) where v T ∈ I α . A Carleman estimate is a weighted inequality that relates a global (weighted) energy with a weighted local norm of the solution. Since (1.12) is singular, the reaction term cannot be controlled by the diffusion term generally in establishing the Carleman estimate for solutions to problem (1.12)-(1.14). Therefore, there should be some further restrictions on c t or c xx . In the present paper, we prescribe the restriction on c t as Moreover, the Carleman estimate still holds if there are convection and reaction terms with suitable weights in (1.12). Precisely, we can establish the Carleman estimate of solutions to the problem of and estimate its solutions uniformly with respect to η, The key for the Carleman estimate is the local one near the singular point x = 0. To show the idea on the choice of the weights, we use the method of undetermined functions to determine the suitable weights. By complicated and detailed estimates, we establish the local Carleman estimate, uniformly with respect to η ∈ (0, 1), near the singular point x = 0. Combining this local Carleman estimate and the classical one for uniformly parabolic equations, we get the uniform Carleman estimate and thus the uniform observability inequality for solutions to problem (1.16)-(1.18), which imply the ones for solutions to problem (1.15), (1.13), (1.14) by a limit process as η → 0 + . Owing to the uniform observability inequality, we can prove that the linear system is null controllable and the control function is uniformly bounded by considering a family of functional minimum problems, where b, c, c t , γ ∈ L ∞ (Q T ) and u 0 ∈ L 2 (0, 1). It is noted that there is no other restriction on b and γ except for b, γ ∈ L ∞ (Q T ). By a fixed point argument, we can show that the semilinear system of subject to (1.20) and (1.21) is null controllable and the control function is uniformly bounded, wherep andq are two measurable functions in Q T × ℝ such that for (x, t) ∈ Q T and u, v ∈ ℝ, with some K > 0. Then, by the uniform estimates on the control functions and solutions of system (1.22), (1.20), (1.21) and a limit process as η → 0 + , we prove that the semilinear system of the singular parabolic equation (1. 27) In particular, in the case 0 < α ≤ 2, the system of The paper is organized as follows: In Section 2, we prove the well-posedness of the singular problems whereĝ is a measurable function in Q T × ℝ 2 such that for (x, t) ∈ Q T and u, v, z, η ∈ ℝ, Definition 2.1. A function u ∈ L 2 (0, T; H 1 0 (0, 1)) with x α/2 u ∈ L ∞ (0, T; L 2 (0, 1)) is said to be a solution to problem (1.25) ) is said to be a solution to problem (2.1), (1.13), (1.14) if ), respectively. That is to say, u and v have some continuity with respect to the time variable so that they make sense at a time.

Well-posedness for semilinear singular problems
Using the uniform estimates in Propositions 2.5 and 2.6, we can show the well-posedness of the problem of the linear equation and for each ε ∈ (0, T),

Estimates near the singular point
In this subsection, we consider the regularized problem (1.16)-(1.18) and we always assume that α > 0, 0 ≤ β < 2, 0 < x 0 < x 1 < 1, 0 < η < 1, v T ∈ H 1 0 (0, 1), and b, c, c t , γ ∈ L ∞ (Q T ) satisfying We first cut off v in the following way: Set It follows from (1.16) and (3.3) that w satisfies Remark 3.1. The convection term and the second reaction term of (1.16) are regarded as a known function in (3.5) since they can be controlled by the diffusion term for the Carleman estimate. However, the first reaction term of (1.16) has to be treated as a reaction term in (3.5).
Reformulate (3.5) in a similar way as for the classical Carleman estimate. For s > 0, set Here, θ takes the form of 4 , t ∈ (0, T), (3.8) as usual, which satisfies with C 0 > 0 depending only on T, while ψ ∈ C 2 ([0, 1]) is a negative function and will be determined below (see (3.15)). It follows from (3.5) and (3.7) that z satisfies Decompose L s z into It follows from (3.10) that The following lemma gives the formula of the left-hand side of (3.11).

Lemma 3.2.
For each s > 0, Proof. Note that We compute the four integrals on the right-hand side of (3.13), respectively. Integrating by parts and using Then (3.12) follows by substituting the above four identities into (3.13).

Remark 3.3. The penultimate term in (3.12) is 2s ∬
Note that the assumption on c is only c, c t ∈ L ∞ (Q T ). So this term is treated as an integral not of z 2 but of zz x .
Owing to (3.12), in order to get a global weighted energy, one should choose ψ such that (3.14) A choice of ψ ∈ C 2 ([0, 1]) being a negative function and satisfying (3.14) is which satisfies For such ψ, one gets the following lemma.
Proof. Substituting the definition of φ into (3.12) and using (3.16), we get after a direct calculation that We estimate the last five terms on the right-hand side of (3.18). On one hand, one obtains from (3.9), the Hölder inequality and Lemma 2.4 that On the other hand, a direct calculation and Lemma 2.4 show for each δ ∈ (0, 1], where C 1 > 0 depends only on N, T, α and β. Choose δ ∈ (0, 1) so small that Then it follows from (3.21) that with C 2 > 0 depending only on N, T, α and β. By substituting (3.19) and (3.22) into (3.18) and choosing s 0 = 2C 0 + C 2 0 + 4C 1 + C 2 + 1 4 , one gets (3.17).

Remark 3.5.
In Lemma 3.4, M 0 depends on c t , which is caused because we distribute cz into L + s z. If cz is distributed into L − s z, then M 0 will depend on c x and c xx .

Remark 3.6. Lemma 3.4 still holds if
But s 0 and M 0 depend also on c 0 .
Below we prove the following Caccioppoli inequality.

Remark 3.9.
The assumption on c t , the factors (x + η) α/2 and (x + η) α/2−1 in the convection term and the second reaction term of (1.16) are necessary when one establishes the Carleman estimate in such a way.
As a corollary of Theorems 3.10 and 3.11, one can get the following Carleman estimate and observability inequality for the singular problem (1.15), (1.13), (1.14).

Null controllability
In this section, we prove the null controllability of system (1. 25