论文标题
$ l_p $ - 空间中抛物线方程的可观察性和无效的可控性
Observability and null-controllability for parabolic equations in $L_p$-spaces
论文作者
论文摘要
我们在$ l_p（\ mathbb {r}^d）$中研究抛物线方程（近似）的（近似），并在控制成本上提供明确的界限。特别是我们考虑表单的系统$ \ dot {x}（t）= -a_p x（t） + \ mathbf {1} _e u（t）$，$ x（t）$，$ x（0）= x_0 \ in l_p（\ mathbb {r}^d）$，在$ e e \ seps $ e e \ subset上，\ sebs n in $ p \ in [1，\ infty）$，其中$ a $是\ yathbb {n} $ in $ l_p（\ mathbb {r}^d）$ in \ mathbb {n} $的椭圆运算符。我们通过二元性证明了该系统的无效可控性，并且有足够的可观察性条件。这种情况是由不确定性原理和耗散估计给出的。我们的结果统一并概括了在Hilbert和Banach空间的背景下获得的早期结果。特别是，我们的结果适用于$ p = 1 $。
We study (approximate) null-controllability of parabolic equations in $L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In particular we consider systems of the form $\dot{x}(t) = -A_p x(t) + \mathbf{1}_E u(t)$, $x(0) = x_0\in L_p (\mathbb{R}^d)$, with interior control on a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, and where $A$ is an elliptic operator of order $m \in \mathbb{N}$ in $L_p(\mathbb{R}^d)$. We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the case $p=1$.
