SIAM Journal on Control and Optimization, Volume 59, Issue 3, Page 1881-1902, January 2021.
We consider linear damped wave (resp., Schrödinger and plate) equations driven by a hypoelliptic “sum of squares” operator $\mathscr{L}$ on a compact manifold $\mathcal{M}$ and a damping function $b(x)$. We assume the Chow--Rashevski--Hörmander condition at rank $k$ (at most $k$ Lie brackets are needed to span the tangent space) together with analyticity of $\mathcal{M}$ and the coefficients of $\mathscr{L}$. We prove that the energy decays at rate $\log(t)^{-\frac{1}{k}}$ (resp., $\log(t)^{-\frac{2}{k}}$) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Baouendi--Grushin-type operators. This result follows from a perturbative argument (of independent interest) showing, in a general abstract setting, that quantitative approximate observability/controllability results for wave-type equations imply a priori decay rates for associated damped wave, Schrödinger, and plate equations. The adapted quantitative approximate observability/controllability theorem for hypoelliptic waves is obtained by the authors in [J. Eur. Math. Soc. (JEMS), 21 (2019), pp. 957--1069] and [Mem. Amer. Math. Soc., to appear].

中文翻译：

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线性阻尼次椭圆波和Schrödinger方程的对数衰减

SIAM控制与优化杂志，第59卷，第3期，第1881-1902页，2021年1月。
我们考虑由紧椭圆流形$ \ mathcal {M} $上的次椭圆“平方和”运算符$ \ mathscr {L} $驱动的线性阻尼波（分别为Schrödinger和板）方程，以及阻尼函数$ b（x ）$。我们假设排名为$ k $的Chow-Rashevski-Hörmander条件（最多需要$ k $个Lie括号来跨越切线空间）以及$ \ mathcal {M} $的解析度和$ \ mathscr的系数{L} $。我们证明能量以速率$ \ log（t）^ {-frac {1} {k}} $（resp。，$ \ log（t）^ {-\ frac {2} {k}} $ ）以获取相关组的生成器域中的数据。我们证明了这种衰减对于Baouendi-Grushin型算子族是最优的。该结果来自于（具有独立利益的）摄动论点，该论点在一般的抽象背景下显示出：波动型方程的定量近似可观测性/可控制性结果暗示了相关阻尼波，薛定ding和板方程的先验衰减率。作者在［J．欧元。数学。Soc。（JEMS），21（2019），957--1069页]和[Mem。阿米尔。数学。Soc。，出现]。