Correction: Gayte Delgado, I.; Marín-Gayte, I. A New Method for the Exact Controllability of Linear Parabolic Equations. Mathematics 2025, 13, 344

Text Correction

There was an error in the original publication [1]. In Section 3, after the Equation (10), it is written:

$${\beta }_1^{\prime }\le {\displaystyle \frac{\left(‖y\left(T\right)‖-‖{\mathsf{\Psi }}_{v_1}\left(T\right)‖\right)c}{{\mathrm{s}\mathrm{u}\mathrm{p}}_{\mathsf{\Omega }\times \left(0,T\right)}{w}_1}} \mathrm{i}\mathrm{n} [0,T],$$

and it must be written as follows.

A correction has been made to Section 3. The Exact Controllability to Zero, Paragraphs 6 and 7:

We choose β₁ ∈ C¹([0, T]), satisfying (6), and w₁ ∈ W(0, T), verifying (7), the hypothesis of Theorem 3, and besides,

w₁ ∈ L^∞(Ω × (0, T)),

$${\beta }_1^{\prime }\le {\displaystyle \frac{\left(‖y\left(T\right)‖-‖{\mathsf{\Psi }}_{v_1}\left(T\right)‖\right)c}{\underset{\mathsf{\Omega }\times \left(0,T\right)\backslash \tilde{I}}{\mathrm{s}\mathrm{u}\mathrm{p}}{w}_1}} \mathrm{i}\mathrm{n} [0,T]\backslash \tilde{I},$$

being $\tilde{I}$ an interval in (0, T) such that I ⊊ $\tilde{I}$ ⊊ (0, T),

$${\beta }_1^{\prime } \text{≤} 0 \mathrm{in} \tilde{I}$$

and

$${\beta }_1^{\prime }<-{\displaystyle \frac{‖{\mathsf{\Psi }}_{v_1}\left(T\right)‖ \underset{B\times I}{\mathrm{s}\mathrm{u}\mathrm{p}} u }{\underset{B\times I}{\inf w_1}}} \mathrm{i}\mathrm{n} I.$$

Note that $\underset{B\times I}{\inf w_1}$ > 0. Then, it is verified that

$${\beta }_1^{\prime }{w}_1\le \left(‖y\left(T\right)‖-‖{\mathsf{\Psi }}_{v_1}\left(T\right)‖\right)u \mathrm{i}\mathrm{n} \mathsf{\Omega }\times \left[0,T\right].$$

(11)

Effectively, since

$${\beta }_1^{\prime }\underset{\mathsf{\Omega }\times \left(0,T\right)\backslash \tilde{I}}{\mathrm{s}\mathrm{u}\mathrm{p}}{w}_1\le \left(‖y\left(T\right)‖-‖{\mathsf{\Psi }}_{v_1}\left(T\right)‖\right)c \mathrm{i}\mathrm{n} \left[0,T\right]\backslash \tilde{I},$$

if ${\beta }_1^{\prime }\left(t\right)$ > 0, we have that

$${\beta }_1^{\prime }\left(t\right) w_1\left(x,t\right)\le {\beta }_1^{\prime }\left(t\right)\underset{\mathsf{\Omega }\times \left(0,T\right)\backslash \tilde{I}}{\mathrm{s}\mathrm{u}\mathrm{p}}{w}_1,$$

in Ω × [0, T]\$\tilde{I}$ and, since ${\beta }_1^{\prime }\left(t\right)$ ≤ 0 in $\tilde{I}$, the inequality above is also true in Ω × $\tilde{I}$, then (11) is obvious because $\left(‖y\left(T\right)‖-‖{\mathsf{\Psi }}_{v_1}\left(T\right)‖\right)$c > 0. It is also verified.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.