Abstract
In this book, we study certain results on the controllability of distributed systems with persistent memory (in the final section of this chapter, we show the derivation of the heat equation with memory and the equation of viscoelasticity). In this chapter, we define and give formulas for the solutions of the system under study and we derive their properties, using an operator approach. We define the notion of controllability and prove the key results relevant to the study of control problems. In particular, we prove that signal propagates with finite velocity, as in the case of the (memoryless) wave equation. A special and important case of the equation with persistent memory is the telegrapher’s equation. In Sect. 2.6.3, we use this important example to contrast the properties of the systems with memory and those of the (memoryless) wave and heat equations.
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- 1.
We introduced the velocity term \(2cw'\), which has a role in the application of moment methods. A term \(c_1w \) does not make any difference and we ignore it.
- 2.
\( B(x_0,r) \) denotes the ball of center \( x_0 \) and radius \( r \).
- 3.
the minus sign because the total internal energy \(\int _a^b e(x,t)\mathrm{{d}}x\) of a segment \( (a,b) \) decreases when the flux of heat is directed to the exterior of the segment.
- 4.
this is not realistic for an elastic or viscoelastic body. We should consider continuous initial conditions, which are not everywhere differentiable, but the qualitative facts described below are the same. For completeness, Fig. 2.2 (right) shows the case that the initial deformation is zero, with nonzero and discontinuous initial velocity.
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Pandolfi, L. (2014). The Model and Preliminaries. In: Distributed Systems with Persistent Memory. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-12247-2_2
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DOI: https://doi.org/10.1007/978-3-319-12247-2_2
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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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