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Article Dans Une Revue Evolution Equations and Control Theory Année : 2024

Null-controllability for weakly dissipative heat-like equations

Paul Alphonse
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Armand Koenig

Résumé

We study the null-controllability properties of heat-like equations posed on the whole Euclidean space $\mathbb R^n$. These evolution equations are associated with Fourier multipliers of the form $\rho(\vert D_x\vert)$, where $\rho\colon[0,+\infty)\rightarrow\mathbb C$ is a measurable function such that $\Re\rho$ is bounded from below. We consider the ``weakly dissipative'' case, a typical example of which is given by the fractional heat equations associated with the multipliers $\rho(\xi) = \xi^s$ in the regime $s\in(0,1)$, for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier $\rho$, in particular assuming that $\rho(\xi) = o(\xi)$, we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier $\rho$. Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.
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Dates et versions

hal-03913881 , version 1 (27-12-2022)
hal-03913881 , version 2 (13-09-2023)
hal-03913881 , version 3 (23-04-2024)

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Citer

Paul Alphonse, Armand Koenig. Null-controllability for weakly dissipative heat-like equations. Evolution Equations and Control Theory, 2024, 13 (3), pp.973-988. ⟨10.3934/eect.2024013⟩. ⟨hal-03913881v3⟩
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