Tunneling estimates and approximate controllability for hypoelliptic equations

This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator L \mathcal {L} on compact manifold M"> mathvariant="script">M {M} assuming: alttext="left-parenthesis i right-parenthesis"> ( i stretchy="false">) encoding="application/x-tex">(i) the Chow-Rashevski-Hörmander condition ensuring hypoellipticity , and encoding="application/x-tex">(ii) analyticity coefficients . The first result tunneling estimate alttext="double-vertical-bar phi double-vertical-bar Subscript L squared left-parenthesis omega right-parenthesis Baseline greater-than-or-equal-to C e Superscript minus c lamda Super StartFraction k Over 2 EndFraction"> fence="false" stretchy="false">‖ φ L 2 ω<!-- ω </mml:msub> ≥<!-- ≥ <mml:mi>C e −<!-- − <mml:mi>c λ<!-- λ <mml:mfrac> k encoding="application/x-tex">\|\varphi \|_{L^2(\omega )} \geq Ce^{- c\lambda ^{\frac {k}{2}}} normalized eigenfunctions alttext="phi"> encoding="application/x-tex">\varphi from nonempty open set alttext="omega subset-of script ⊂<!-- ⊂ encoding="application/x-tex">\omega \subset \mathcal where alttext="k"> encoding="application/x-tex">k index alttext="lamda"> encoding="application/x-tex">\lambda eigenvalue. main stability solutions to hypoelliptic wave equation partial-differential t plus u equals 0"> mathvariant="normal">∂<!-- ∂ <mml:mi>t + u = 0 encoding="application/x-tex">(\partial _t^2+\mathcal {L})u=0 : alttext="upper T greater-than sup Underscript x element-of M Endscripts d s comma T &gt; movablelimits="true" form="prefix">sup x ∈<!-- ∈ </mml:munder> d s , encoding="application/x-tex">T&gt;2 \sup _{x \in {M}}(dist(x,\omega )) (here, alttext="d t"> encoding="application/x-tex">dist sub-Riemannian distance), observation solution 0 times omega"> ×<!-- × encoding="application/x-tex">(0,T)\times \omega determines data. constant involved in normal Lamda k"> mathvariant="normal">Λ<!-- Λ encoding="application/x-tex">Ce^{c\Lambda ^k} alttext="normal Lamda"> encoding="application/x-tex">\Lambda typical frequency We then prove approximate controllability heat v double-struck 1 f"> v 1 f _t+\mathcal {L})v=\mathbb {1}_\omega f any time, appropriate (exponential) cost, depending In case alttext="k 2"> encoding="application/x-tex">k=2 (Grushin, Heisenberg...), we further show trajectories polynomial cost large time. also explain how assumption can be relaxed, boundary alttext="partial-differential encoding="application/x-tex">\partial added some situations. Most results turn out optimal family Grushin-type operators. proof relies general strategy produce estimates, developed by authors Laurent-Léautaud (2019).