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Sufficient Criteria and Sharp Geometric Conditions for Observability in Banach Spaces
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2020-08-31 , DOI: 10.1137/19m1266769
Dennis Gallaun , Christian Seifert , Martin Tautenhahn

SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2639-2657, January 2020.
Let $X,Y$ be Banach spaces, $(S_t)_{t \geq 0}$ a $C_0$-semigroup on $X$, $-A$ the corresponding infinitesimal generator on $X$, $C$ a bounded linear operator from $X$ to $Y$, and $T > 0$. We consider the system $\dot{x}(t) = -Ax(t), \quad y(t) = Cx(t), \quad t\in (0,T], \quad x(0) = x_0 \in X.$ We provide sufficient conditions such that this system satisfies a final state observability estimate in $L_r ((0,T) ; Y)$, $r \in [1,\infty]$. These sufficient conditions are given by an uncertainty relation and a dissipation estimate. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider the example where $A$ is an elliptic operator in $L_p(\mathbb{R}^d)$ for $1<p<\infty$ and where $C = {1}_{E}$ is the restriction onto a thick set ${E} \subset \mathbb{R}^d$. In this case, we show that the above system satisfies a final state observability estimate if and only if ${E} \subset \mathbb{R}^d$ is a thick set. Finally, we make use of the well-known relation between observability and null-controllability of the predual system and investigate bounds on the corresponding control costs.


中文翻译:

Banach空间中可观性的充分准则和尖锐几何条件

SIAM控制与优化杂志,第58卷,第4期,第2639-2657页,2020年1月。
令$ X,Y $为Banach空间,$(S_t)_ {t \ geq 0} $ $ X $上的$ C_0 $-半群,$ -A $ $ X $上的相应无穷小生成器,$ C $ a从$ X $到$ Y $的有界线性算子,并且$ T> 0 $。我们认为系统$ \ dot {x}(t)= -Ax(t),\ quad y(t)= Cx(t),\ quad t \ in(0,T],\ quad x(0)= x_0 \ in X. $我们提供了充分的条件,使得该系统满足$ L_r((0,T); Y)$,$ r \ in [1,\ infty] $中的最终状态可观察性估计。由不确定性关系和耗散估计给出。我们的方法统一并归纳了从希尔伯特空间上下文中获得的较早结果的各个优点。作为应用程序,我们考虑其中$ A $是$ L_p(\ mathbb的椭圆算子的示例{R} ^ d)$ for $ 1 <p < \ infty $,其中$ C = {1} _ {E} $是对厚集$ {E} \ subset \ mathbb {R} ^ d $的限制。在这种情况下,我们证明只有当$ {E} \ subset \ mathbb {R} ^ d $是一个粗集时,上述系统才能满足最终状态的可观察性估计。最后,我们利用惯常系统的可观测性与零可控制性之间的众所周知的关系,并研究了相应控制成本的界限。
更新日期:2020-09-01
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