Locally integrable

Auxiliary Results

Published in A.G. Ramm, A.I. Katsevich, The RADON TRANSFORM and LOCAL TOMOGRAPHY, 2020

Each locally integrable function f defines a distribution, i.e., a functional on C0∞(ℝn), such that the value of this functional on a test function ϕ∈C0∞(ℝn) is given by () <f,ϕ>=∫ℝnf(x)ϕ(x)¯dx.

Stability of forced higher-order continuous-time Lur'e systems: a behavioural input-output perspective

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Journal Information

Published in International Journal of Control, 2023

Chris Guiver, Hartmut Logemann

Obviously, interpreted in the classical sense, (17) and (18) are only meaningful if u, v and y are sufficiently often differentiable. Since it is desirable to allow for discontinuous inputs u and v (step functions, for example), it would be restrictive to impose any smoothness assumptions on u and v. However, in the absence of suitable differentiability properties, it is still possible to make sense of (17) and (18) by using basic ideas from distribution theory. To this end, let , where , and let denote the extension by zero of w to all of . In the following, the function and the regular -valued distribution on induced by will be identified. Locally integrable functions and the associated regular distributions will not be distinguished notationally. We say that the triple is a weak trajectory of (17) if there exist , , such that where denotes the distributional derivative and is the Dirac distribution. The behaviour of (17) is defined to be the set of all weak trajectories of (17). It is obvious that is a linear subspace of , and any ‘classical trajectory’ (in the sense that is sufficiently often differentiable for (18) to hold for almost every ) is a weak trajectory.

Locally integrable

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Auxiliary Results

Stability of forced higher-order continuous-time Lur'e systems: a behavioural input-output perspective