45 research outputs found

    On measuring unboundedness of the H∞H^\infty-calculus for generators of analytic semigroups

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    We investigate the boundedness of the H∞H^\infty-calculus by estimating the bound b(Ξ΅)b(\varepsilon) of the mapping Hβˆžβ†’B(X)H^{\infty}\rightarrow \mathcal{B}(X): f↦f(A)T(Ξ΅)f\mapsto f(A)T(\varepsilon) for Ξ΅\varepsilon near zero. Here, βˆ’A-A generates the analytic semigroup TT and H∞H^{\infty} is the space of bounded analytic functions on a domain strictly containing the spectrum of AA. We show that b(Ξ΅)=O(∣log⁑Ρ∣)b(\varepsilon)=\mathcal{O}(|\log\varepsilon|) in general, whereas b(Ξ΅)=O(1)b(\varepsilon)=\mathcal{O}(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(Ξ΅)=O(∣log⁑Ρ∣)b(\varepsilon)=\mathcal{O}(\sqrt{|\log\varepsilon|}).Comment: Preprint of the final, published version. In comparison with previous version, Prop. 2.2 was added and Thm. 3.5 has been slightly adapted in order to point out the major assertio

    Functional calculus for C0C_0-semigroups using infinite-dimensional systems theory

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    In this short note we use ideas from systems theory to define a functional calculus for infinitesimal generators of strongly continuous semigroups on a Hilbert space. Among others, we show how this leads to new proofs of (known) results in functional calculus.Comment: 6 page

    Generators with a closure relation

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    Assume that a block operator of the form (A1A20)\left(\begin{smallmatrix}A_{1}\\A_{2}\quad 0\end{smallmatrix}\right), acting on the Banach space X1Γ—X2X_{1}\times X_{2}, generates a contraction C0C_{0}-semigroup. We show that the operator ASA_{S} defined by ASx=A1(xSA2x)A_{S}x=A_{1}\left(\begin{smallmatrix}x\\SA_{2}x\end{smallmatrix}\right) with the natural domain generates a contraction semigroup on X1X_{1}. Here, SS is a boundedly invertible operator for which \epsilon\ide-S^{-1} is dissipative for some Ο΅>0\epsilon>0. With this result the existence and uniqueness of solutions of the heat equation can be derived from the wave equation.Comment: 9 page

    On continuity of solutions for parabolic control systems and input-to-state stability

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    We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.Comment: 19 pages, final version of preprint, Prop. 6 and Thm 7 have been generalised to arbitrary Banach spaces, the assumption of boundedness of the semigroup in Thm 10 could be droppe

    Infinite-dimensional input-to-state stability

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    In this talk we discuss infinite-dimensional versions of well-known stability notions relating the external input uu and the state xx of a linear system governed by the equation xΛ™=Ax+Bu,x(0)=x0.\dot{x}=Ax+Bu, \quad x(0)=x_{0}. Here, AA and BB are unbounded operators. For instance, the system is called \textit{LpL^{p}-input-to-state stable} if u(β‹…)↦x(t)u(\cdot)\mapsto x(t) is bounded as a mapping from Lp(0,t)L^{p}(0,t) to the state space XX for all t3˘e0t\u3e0. In particular, we elaborate on the relation of this notion to \textit{integral input-to-state} stability and \textit{(zero-class) admissibility} with a special focus on the case p=∞p=\infty.\\ This is joint work with B.~Jacob, R.~Nabiullin and J.R.~Partington

    Funnel control for a moving water tank

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    We study tracking control for a moving water tank system, which is modelled using the Saint-Venant equations. The output is given by the position of the tank and the control input is the force acting on it. For a given reference signal, the objective is to achieve that the tracking error evolves within a prespecified performance funnel. Exploiting recent results in funnel control we show that it suffices to show that the operator associated with the internal dynamics of the system is causal, locally Lipschitz continuous and maps bounded functions to bounded functions. To show these properties we consider the linearized Saint-Venant equations in an abstract framework and show that it corresponds to a regular well-posed linear system, where the inverse Laplace transform of the transfer function defines a measure with bounded total variation.Comment: 11 page