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2014 | Buch

An Example Kapitel lesen Erstes Kapitel lesen

Distributed Systems with Persistent Memory

Control and Moment Problems

verfasst von: Luciano Pandolfi

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Electrical and Computer Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The subject of the book includes the study of control problems for systems which are encountered in viscoelasticity, non-Fickian diffusion and thermodynamic with memory. The common feature of these systems is that memory of the whole past history persists in the future. This class of systems is actively studied now, as documented in the recent book. This book will attract a diversified audience, in particular, engineers working on distributed systems, and applied mathematicians. Background of mathematics are the elements of functional analysis, which is now standard among people working on distributed systems, and the author describes very clearly the instruments which are used at every step.

Inhaltsverzeichnis

Frontmatter
Chapter 1. An Example
Abstract
In this preliminary chapter, in order to get a feeling of the properties of distributed systems with persistent memory and to introduce the key definitions, we study the very simple example of an unbounded viscoelastic string (laying on a half line) controlled at its end. We shall see in Chap. 2 that this example is simple but significant. Relying on known properties of the purely elastic string, we introduce the definition of the solutions and we study a control problem. In order to help the reader and to fix the notations, the final section of this chapter recalls the key definitions and notions that we shall use. Readers with a feeling of the subject can skip this chapter since this example is not used in the rest of the book (it is used in a few problems).
Luciano Pandolfi
Chapter 2. The Model and Preliminaries
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.
Luciano Pandolfi
Chapter 3. Moment Problems and Exact Controllability
Abstract
Controllability of linear systems can often be reduced to the solution of suitable, equivalent, moment problems. The moment problem in Hilbert spaces is studied in this chapter. This problem has many facets and we confine ourselves to those properties which are needed in the study of exact controllability for wave-like equations, stressing the relation of the moment operator with Riesz sequences. We present several examples both to clarify the properties of the moment operator and of the Riesz sequences and to show the applications to exact controllability. In particular, the last worked example presents in a simplified setting the crucial ideas used in the study of controllability of systems with persistent memory.
Luciano Pandolfi
Chapter 4. Controllability of the Wave Equation
Abstract
Results on controllability of systems with persistent memory have been derived appealing to the corresponding results of the (memoryless) wave equation. For this reason, in this short chapter, we review the key results on the controllability of wave type equations. First, we describe a “hidden regularity” of the normal derivative of the solutions of the uncontrolled system and the observation inequality. A consequence is that the “active part” \(\varGamma \) of the boundary must be “large” also in terms of the trace of the eigenvectors of \(A\) on \(\varGamma \). Finally, we characterize exact controllability in terms of suitable Riesz sequences.
Luciano Pandolfi
Chapter 5. Systems with Persistent Memory: Controllability via Moment Methods
Abstract
We study the controllability of the pair (deformation/velocity of deformation) for a viscoelastic body. The idea is that existing controllability results for the memoryless wave equation can be lifted to the system with memory. As we have seen, the component \( w \) can also be interpreted as the temperature of thermodynamic systems with memory, so that we get exact controllability of the temperature (at a suitable time \( T \)), a property that cannot hold for the standard heat equation, derived from Fourier law. In this chapter, we show the use of moment methods in the study of controllability of viscoelastic materials, or thermodynamical systems with memory (a different proof based on the observation inequality is in Chap. 6). The final section shows an application of controllability to a source identification problem.
Luciano Pandolfi
Chapter 6. Systems with Persistent Memory: The Observation Inequality
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.
Luciano Pandolfi
Backmatter
Metadaten
Titel
Distributed Systems with Persistent Memory
verfasst von
Luciano Pandolfi
Copyright-Jahr
2014
Electronic ISBN
978-3-319-12247-2
Print ISBN
978-3-319-12246-5
DOI
https://doi.org/10.1007/978-3-319-12247-2

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