This volume presents a timely overview of control theory and related topics, such as the reconstruction problem, the stability of PDEs, and the Calderón problem. The chapters are based on talks given at the conference "Control & Related Fields” held in Seville, Spain in March 2023. In addition to providing a snapshot of these areas, chapters also highlight breakthroughs on more specific topics, such as:
Stabilization of an acoustic system The Kramers-Fokker-Planck operator Control of parabolic equations Control of the wave equation Advances in Partial Differential Equations and Control will be a valuable resource for both established researchers as well as more junior members of the community.

lgrsnf/advances-partial-differential-equations-control.pdf

Springer International Publishing

Springer Nature Switzerland AG

Springer Basel AG

Switzerland, Switzerland

Preface
Contents
Part I Control of Partial Differential Equations
Energy Decay Estimate for a Wave-Plate Interface Transmission Problem with Only Two Dynamical Boundary Controls
1 Introduction
2 Well-Posedness of the Problem
3 Some Regularity Results
4 Strong Stability with Non-compact Resolvent
5 Polynomial Stability
6 Conclusion and Open Problems
References
Uniform Stabilization of an Acoustic System
1 Introduction
2 Preliminaries and Main Results
3 Tools
3.1 Pseudo-Differential Tools
Composition
Commutator
Adjoint
Operators Bounded on L2
Gårding Inequality
3.2 Uniform Stabilization
4 Introduction of a Semiclassical Measure
4.1 Semiclassical Measure
4.2 The Measure Is Not Identically Null
4.3 The Semiclassical Measure Null on Support of Damping
5 Propagation of the Support
6 Some Properties on Semiclassical Measure
6.1 Properties of Measure
References
Observability and Control of Parabolic Equations on Networks
1 Introduction
1.1 Basic Definitions
1.2 Controllability Results
1.3 Application to the Resolution of Inverse Problems
2 The Observability Problem
2.1 Construction of the Auxiliary Function
2.2 A New Carleman Inequality
References
Some Results on the Energy Decay of Solutions for a Wave Equation with a General Internal Feedback of Diffusive Type
1 Introduction
2 Preliminaries Results
3 Global Existence
4 Lack of Exponential Stability
5 Asymptotic Stability
5.1 Strong Stability of the System
5.2 General Decay Estimate
5.3 Spectral Analysis
5.4 Optimality of Energy Decay in One-Dimensional Case
5.5 Polynomial Stability (for η=0)
References
Numerical Approximation of the Boundary Control for the Wave Equation with Periodic Oscillating Coefficients
1 Introduction
2 Characterization of the Control
3 Matrix Formulation of the Control Problem
4 Numerical Approximation of the Control Problem
5 Numerical Experiments
References
Numerical Impulse Controllability for Parabolic Equations by a Penalized HUM Approach
1 Introduction and Main Results
2 Preliminary Results
2.1 Dirichlet Case
2.2 Neumann Case
2.3 Dynamic Case
3 Algorithm for Calculating HUM Impulse Controls
3.1 The HUM Impulse Controls
3.2 Numerical Experiments
3.3 Dirichlet Case
3.4 Neumann Case
3.5 Dynamic Case
References
Part II Related Fields
Decoding the Kramers–Fokker–Planck Operator: An Overview
1 Introduction
2 Analysis of the Kramers–Fokker–Planck Operator
2.1 Maximal Accretivity
2.2 Links with Witten Laplacians
3 Compactness Criteria for the Resolvent of Kramers–Fokker–Planck Operator
3.1 Initial Known Results
3.2 Case of Polynomial Potential
Degree Less than 3
Degree Greater than 2
3.3 Case of Homogeneous Potential
4 Conclusion and Perspectives
References
Exponential Decay of Solutions to Linear Evolution Equations with Time-Dependent Time Delay
1 Introduction
2 Well-Posedness
3 Exponential Stability
4 A Nonlinear Model
5 Examples
5.1 The Damped Wave Equation
5.2 A Damped Elasticity System
References
Study of the Numerical Method for an Inverse Problem of a Simplified Intestinal Crypt
1 The Simplified Intestinal Crypt Model
1.1 The Stem-Progenitor Interaction Model
2 Inverse Problem
2.1 The Problem of Identifying the Function ρdcs(z)
2.2 Example of ρdcs Shapes and δρdcs Directions
3 Numerical Approximation
3.1 BGK Schemes for the Direct Problem
3.2 Adjoint Scheme Associated with BGK Scheme
3.3 Discrete Inverse Problem
4 Numerical Examples
4.1 Inverse Problem Results
Appendix: Derivative of the Flux for 5-Point Schemes
Derivative of the Flux with Respect to ρsc and ρpc
Derivative of the Flux with Respect to ρdcs
Derivative of the Flux in Respect of the Maxwellian Distributions
Derivative of the Deep Crypt Secretory with Respect to Its Parameters
References
Solving Ill-Posed Inverse Problems via the Born Approximation
1 Introduction
2 An Abstract Setting for the Born Approximation to Ill-Posed Inverse Problems
3 The Born Approximation in Inverse Spectral Theory
4 The Born Approximation for the Calderón Problem
5 Toward a General Born Approximation for the Calderón Problem
References
Central Nervous System Action on Rolling Balance Board Robust Stabilization: Computer Algebra and MID-Based Feedback Design
1 Introduction
2 Mechanical Model
3 Background and Prerequisites
3.1 Cauchy's Argument Principle
3.2 The Stepan–Hassard Approach
3.3 Background on Gröbner Bases
3.4 Cell Decomposition in CAD Routine
3.5 The MID Paradigm: A Partial Pole-Placement Strategy
3.6 Algorithmic Investigation of the MID Property
4 Main Results
4.1 A Normalized Fourth-Order Friction-Free Model
4.2 Forcing Multiplicity
Real-Rooted Elimination-Produced Polynomial
Fredholm Representation of the Normalized Characteristic Function
Constancy Sign of the Integrand qυ,σ
Effective Admissible Region
4.3 Dominancy Sufficient Conditions
Frequency Bound
Dominancy
5 Conclusion
References
Index

Trends in Mathematicsis a series devoted to the publication of volumes arisingfrom conferences and lecture series focusing on a particular topic from any area ofmathematics. Its aim is to make current developments available to the community asrapidly as possible without compromise to quality and to archive these for reference.

Trends in Mathematics
Erscheinungsdatum: 28.07.2024

2024-07-29