Abstract
The goal of this chapter is to present a different circle of ideas used in the study of systems with persistent memory. For this, we give a proof of controllability which relays on the direct and inverse inequalities of the wave equation. The key properties we shall encounter are the extension of the hidden regularity to systems with memory, a test for the solutions which shows explicitly the role of the region \({{\varOmega }}\) and propagation of singularities. Furthermore, we prove that the inverse inequality is equivalent to controllability both for systems with and without memory. So, we can conclude that the inverse inequality of the memoryless wave equation is inherited by the system with memory.
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Notes
- 1.
The equation is the same as (6.3). We rewrite and give a special number to stress that the initial conditions belong to \(\fancyscript{D}({{\varOmega }})\) and the affine term is zero.
- 2.
in fact for a suitable \(\varepsilon _n\rightarrow 0+\) and this is sufficient for the proof, but it is also possible to deduce that the limit for \(\varepsilon \rightarrow 0+\) exists.
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Pandolfi, L. (2014). Systems with Persistent Memory: The Observation Inequality. In: Distributed Systems with Persistent Memory. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-12247-2_6
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DOI: https://doi.org/10.1007/978-3-319-12247-2_6
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12246-5
Online ISBN: 978-3-319-12247-2
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