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Contrôlabilité de quelques équations aux dérivées partielles paraboliques peu diffusives

Abstract : Control theory is the branch of mathematics that is concerned in what extent the state of a system can be modified, depending in the intrinsic properties of the system and how we can act on it. For example, one may wonder if the temperature of a solid can be brought to a constant temperature in finite time by heating and cooling only a part of the solid. This problem, called the null-controllability of the heat equation, has been solved since 1995. But if we study degenerate parabolic equations, which looks like the heat equation but have a weaker diffusion, we know how to treat only a few particular examples, and the situation is more complicated: for the heat equation, the null-controllability is always true, even in arbitrarily small time; but for some degenerate parabolic equations there exists a minimum time for the null-controllability to hold. We study some degenerate parabolic equations, including the Grushin equation and some Kolmogorov-type equations, and partially complete existing results about the null-controllability on those equations. In particular, we make the relationship between the control domain and the minimum time of null-controllability more precise. We do this with a fine spectral analysis, which allows us to reduce the study of the Grushin and Kolmogorov-type equations to the study of the fractional heat equation. So we also study the fractional heat equation, with holomorphic functions techniques and geometric optics. We also study transport-heat systems, and prove that there exists a minimum control time of null-controllability, (almost) generalizing the existing results obtained on several examples of transport-heat systems. This study is based on a spectral analysis that separates the transport-heat systems into a transport system and a system of heat equations that are weakly coupled.
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Armand Koenig. Contrôlabilité de quelques équations aux dérivées partielles paraboliques peu diffusives. Equations aux dérivées partielles [math.AP]. COMUE Université Côte d'Azur (2015 - 2019), 2019. Français. ⟨NNT : 2019AZUR4066⟩. ⟨tel-02733032⟩

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