Systems with Persistent Memory. Controllability, Stability, Identification

Preface
Contents
1 Preliminary Considerations and Examples
	1.1 The Systems with Persistent Memory Studied in This Book
	1.2 Heuristics on the Wave and Heat Equations with and Without Memory
	1.3 The Archetypal Model: The String Equation with Memory
		1.3.1 The String Equation Without Memory on a Half Line
			1.3.1.1 A Control Problem for the String Equation Without Memory
	1.4 The String with Persistent Memory on a Half Line
		1.4.1 Finite Propagation Speed
		1.4.2 A Formula for the Solutions and a Control Problem
			1.4.2.1 From the Regularity of the Input to the Interior Regularity
			1.4.2.2 The Response Operator
			1.4.2.3 From the Regularity of the Target to the Regularity of the Control
	1.5 Diffusion Processes and Viscoelasticity: the Derivationof the Equations with Persistent Memory
		1.5.1 Thermodynamics with Memory and Non-FickianDiffusion
		1.5.2 Viscoelasticity
		1.5.3 A Problem of Filtration
		1.5.4 Physical Constraints
	References
2 Operators and Semigroups for Systems with Boundary Inputs
	2.1 Preliminaries on Functional Analysis
		2.1.1 Continuous Operators
		2.1.2 The Complexification of Real Banach or Hilbert Spaces
		2.1.3 Operators and Resolvents
		2.1.4 Closed and Closable Operators
		2.1.5 The Transpose and the Adjoint Operators
			2.1.5.1 The Adjoint, the Transpose and the Imageof an Operator
		2.1.6 Compact Operators in Hilbert Spaces
	2.2 Integration, Volterra Integral Equations and Convolutions
	2.3 Laplace Transformation
		2.3.1 Holomorphic Functions in Banach Spaces
		2.3.2 Definition and Properties of the Laplace Transformation
		2.3.3 The Hardy Space H2(Π+;H ) and the LaplaceTransformation
	2.4 Graph Norm, Dual Spaces, and the Riesz Map of Hilbert spaces
		2.4.1 Relations of the Adjoint and the Transpose Operators
	2.5 Extension by Transposition and the Extrapolation Space
		2.5.1 Selfadjoint Operators with Compact Resolvent
			2.5.1.1 Fractional Powers of Positive Operatorswith Compact Resolvent
	2.6 Distributions
		2.6.1 Sobolev Spaces
			2.6.1.1 Sobolev Spaces of Hilbert Space ValuedFunctions
			2.6.1.2 Sobolev Spaces of Any Real Order
	2.7 The Laplace Operator and the Laplace Equation
		2.7.1 The Laplace Equation with Nonhomogeneous Dirichlet Boundary Conditions
		2.7.2 The Laplace Equation with Nonhomogeneous Neumann Boundary Conditions
	2.8 Semigroups of Operators
		2.8.1 Holomorphic Semigroups
	2.9 Cosine Operators and Differential Equations of the Second Order
	2.10 Extensions by Transposition and Semigroups
		2.10.1 Semigroups, Cosine Operators, and Boundary Inputs
	2.11 On the Terminology and a Final Observation
	References
3 The Heat Equation with Memory and Its Controllability
	3.1 The Abstract Heat Equation with Memory
	3.2 Preliminaries on the Associated Memoryless System
		3.2.1 Controllability of the Heat Equation (Without Memory)
	3.3 Systems with Memory, Semigroups, and Volterra IntegralEquations
		3.3.1 Projection on the Eigenfunctions
	3.4 The Definitions of Controllability for the Heat Equationwith Memory
	3.5 Memory Kernels of Class H1: Controllability via Semigroups
		3.5.1 Approximate Controllability Is Inherited by the System with Memory
		3.5.2 Controllability to the Target 0 Is Not Preserved
	3.6 Frequency Domain Methods for Systems with Memory
		3.6.1 Well Posedness via Laplace Transformation
	3.7 Controllability via Laplace Transformation
		3.7.1 Approximate Controllability Is Inherited by the System with Memory
		3.7.2 Controllability to the Target 0 Is Not Preserved
	3.8 Final Comments
	References
4 The Wave Equation with Memory and Its Controllability
	4.1 The Equations with and Without Memory
		4.1.1 Admissibility and the Direct Inequality for the System Without Memory
	4.2 The Solution of the Wave Equation with Memory
		4.2.1 Admissibility and the Direct Inequality for the Wave Equation with Memory
			4.2.1.1 Admissibility and Fourier Expansion
		4.2.2 Finite Speed of Propagation
		4.2.3 Memory Kernel of Class H2 and Compactness
	4.3 The Definitions of Controllability and Their Consequences
		4.3.1 Controllability of the Wave equation (Without Memory) with Dirichlet Boundary Controls
			4.3.1.1 Fourier Expansions
		4.3.2 Controllability of the Wave Equation (Without Memory) with Neumann Boundary Controls
			4.3.2.1 Fourier Expansions
		4.3.3 Controllability of the Wave Equation and Eigenvectors
	4.4 Controllability of the Wave Equation with Memory:The Definitions
		4.4.1 Computation of D,T* and N,T*
	4.5 Wave Equation with Memory: the Proof of Controllability
	4.6 Final Comments
	References
5 The Stability of the Wave Equation with Persistent Memory
	5.1 Introduction to Stability
	5.2 The Memory Kernel When the System Is Stable
		5.2.1 Consequent Properties of the Memory Kernel M(t)
			5.2.1.1 The Real and the Imaginary Parts of (λ)
			5.2.1.2 The Resolvent Kernel R(t) of -M(t)
		5.2.2 Positive Real (Transfer) Functions
	5.3 L2-Stability via Laplace Transform and Frequency Domain Techniques
		5.3.1 L2-Stability When =0
		5.3.2 The Memory Prior to the Time 0
	5.4 Stability via Energy Estimates
	5.5 Stability via the Semigroup Approach of Dafermos
		5.5.1 Generation of the Semigroup in the History Space
		5.5.2 Exponential Stability via Semigroups
	5.6 Final Comments
	References
6 Dynamical Algorithms for Identification Problems
	6.1 Introduction to Identification Problems
		6.1.1 Deconvolution and Numerical Computationof Derivatives
	6.2 Dynamical Algorithms for Kernel Identification
		6.2.1 A Linear Algorithm with Two Independent Measurements: Reduction to a Deconvolution Problem
		6.2.2 One Measurement: A Nonlinear Identification Algorithm
		6.2.3 Quasi Static Algorithms
	6.3 Dynamical Identification of an Elastic Coefficient
		6.3.1 From the Connecting Operator to the Identification of q(x)
		6.3.2 The Blagoveshchenskiǐ Equation and the Computationof the Connecting Operator
			6.3.2.1 Well Posedness of the Blagoveshchenskiǐ Equation
	6.4 Final Comments
	References
7 Final Miscellaneous Problems
	7.1 Solutions of a Nonlinear System with Memory: Galerkin's Method
	7.2  Asymptotics of Linear Heat Equations with Memory Perturbed by a Sector Nonlinearity
		7.2.1 Dafermos Method for the Heat Equation with Memory
		7.2.2 Asymptotic Properties of the Perturbed Equation
	7.3 Controllability and Small Perturbations
		7.3.1 The Belleni-Morante Method and Controllability
	7.4 Memory on the Boundary
	7.5 A Glimpse to Numerical Methods for Systems with Persistent Memory
	References
Index