Advances in Partial Differential Equations and Control: The 2023 Conference in Seville, Spain (Trends in Mathematics)

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. 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.