论文标题
在消失的粘度极限下梯度流的统一可观察性
On uniform observability of gradient flows in the vanishing viscosity limit
论文作者
论文摘要
我们考虑使用具有较小粘性扰动的梯度向量场的传输方程 - $εδ_g$。我们研究（单数）消失的粘度限制$ε\ rightarrow 0^+$的均匀可观察性（分别可控性）特性，即具有均匀界限的观测常数（分别控制成本）的可能性。我们证明了一系列示例，通常，均匀可观察性的最小时间可能比极限方程式$ε= 0 $所需的最小时间大得多。我们还证明，两个最小的时间重合阳性解决方案。这些证据依赖于该问题的半经典重新重新制定以及（a）关于经典禁止区域中本征函数衰减的（a）估计[HS84]（b）半经典热方程核的良好估计[ly86]。
We consider a transport equation by a gradient vector field with a small viscous perturbation --$εΔ_g$. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit $ε\rightarrow 0^+$, that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation $ε= 0$. We also prove that the two minimal times coincides for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY86].
