Sufficient criteria and sharp geometric conditions for observability in Banach spaces

Abstract

Let $X,Y$ be Banach spaces, $(S_t{)}_{t\ge 0}$ 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 $C=1_{\omega }$ is the restriction onto a thick set $\omega \subset {\mathbb{R}}^d$. In this case, we show that the above system satisfies a final state observability estimate if and only if $\omega \subset {\mathbb{R}}^d$ 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.

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Cited by in corpus (11)

• An abstract Logvinenko-Sereda type theorem for spectral subspaces
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• Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus
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• Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups
• Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density
• Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials
• Uncertainty principles with error term in Gelfand-Shilov spaces
• Observability for Non-autonomous Systems
• Duality for Evolutionary Equations with Applications to Null Controllability
• A unified observability result for non-autonomous observation problems