Null controllability for a fourth order parabolic equation

Abstract

In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of eigenfunctions of the Laplacian.fourth order parabolic equations, null controllability, Lebeau-Rabbiano inequality

Null controllability of semi-linear fourth order parabolic equations

“…The author in [19], using the same ideas as in [20], proved the boundary null controllability of a linear fourth order parabolic equation in dimension 1 where the control acts on the whole boundary. Using the ideas of [27], the author in [32] proved the null controllability of a linear fourth order parabolic equation in higher dimension. Moreover, the author in [6] proved the null controllability of the linear Kuramoto-Sivashinsky equation by using a moments method and spectral analysis.…”

“…By using the definition of κ and l for p =p and combining this with the Carleman inequality presented in Lemma 2.7, we can estimate the first two terms in the left-hand side of (32). In other words, we have…”

“…for λ ≥ C 0 and s ≥ C 0 (T + T 1/2 ). Before we estimate the following terms in (32), let us present a Lemma which will be useful for the sequel : Lemma 2.10. There exists a constant C 0 (Ω) > 0 such that…”

Hierarchical exact controllability of the fourth order parabolic equations

“…Until now, there are some results about the controllability for fourth order parabolic equations in both one dimension (see [7,8,9,10,11,17,21,28]) and the higher dimensions (see [12,15,18,22,29,36]). In particular, the approximate controllability and non-approximate controllability of higher order parabolic equations were studied in [14].…”

Carleman estimate and null controllability of a fourth order parabolic equation in dimension N ≥ 2

“…Among them, the approximate controllability and non-approximate controllability of higher order parabolic equations were studied in [11]. In addition, in [27] the author proved the null controllability of system (1) by using the ideas of [23]. Consequently, as far as we now, a Carleman inequality for a fourth order parabolic equation was an open problem whenever N ≥ 2.…”