Fluids Under Control (Advances in Mathematical Fluid Mechanics)

Preface
Contents
1 On the Stabilization Problem by Feedback Control for Some Hydrodynamic Type Systems
	1.1 Introduction
	1.2 Setting of the Problem and the Main Idea of the Method
		1.2.1 Stabilization Problem for a Simple Parabolic Equation
		1.2.2 Setting of the Stabilization Problem
		1.2.3 Feedback Control: Previous Remarks
		1.2.4 The Main Idea of Construction
	1.3 Oseen Equations
		1.3.1 Preliminaries
		1.3.2 Structure of Rk with k<0
		1.3.3 Holomorphic Semigroups
		1.3.4 Unique Continuation Property
		1.3.5 On Linear Independence of (k)l(x,-λj)
	1.4 Stabilization of Oseen Equations
		1.4.1 Setting of the Problem
		1.4.2 Theorem on Extension
		1.4.3 Result on Stabilization
	1.5 Stabilization of 3D Navier–Stokes Equations
		1.5.1 Invariant Manifolds
		1.5.2 Extension Operator
		1.5.3 Theorem on Stabilization
	1.6 On Non-local Stabilization of Hydrodynamic Type System
		1.6.1 Navier–Stokes Equations
		1.6.2 Helmholtz Equations
		1.6.3 Derivation of Normal Parabolic Equations (NPEs)
		1.6.4 Explicit Formula for Solution of NPE
	1.7 Stabilization of Solution for NPE by Starting Control
		1.7.1 Formulation of the Main Result on Stabilization
		1.7.2 Formulation of the Main Preliminary Result
		1.7.3 Intermediate Control
		1.7.4  Proof of the Stabilization Result
	1.8 Burgers Equation and Corresponding Semi-linear Parabolic Equation of Normal Type
		1.8.1 Derivation of the Normal Parabolic Equation (NPE)
		1.8.2 Explicit Formula for a Solution of NPE
		1.8.3 Dynamic Structure Generated with NPE
		1.8.4 Formulation of the Main Result on Stabilization
	1.9 Differentiated Burgers Equation and Functional-Polar Coordinates (fpc)
		1.9.1 On Solvability of the Differential Burgers Equation for Small Initial Conditions
		1.9.2 Functional-Polar Coordinates (fpc)
		1.9.3 Contour Lines of the Functional  on the Sphere (1)
	1.10 Construction of a Stabilizing Impulsive Control for Differentiated Burgers Equation
	References
2 Unique Continuation Properties of Static Over-determined Eigenproblems: The Ignition Key for Uniform Stabilization of Dynamic Fluids by Feedback Controllers
	2.1 Introduction
	2.2 A First (Informal) Quantitative Description of the Strategy
	2.3 Part I: Kalman Rank Conditions Kal.1,Kal.2
		2.3.1 Second-Order Elliptic Operators. Linear Independence of Interior Localized Eigenfunctions
		2.3.2 Linear Independence of Boundary Traces of Eigenfunctions
		2.3.3 Implications of Linear Independence of Interior Localized Eigenfunctions to the Problem of Dirichlet Boundary Feedback Stabilization of Parabolic Problems. Verification of Kalman Rank Condition Kal.1,Kal.2
		2.3.4 Implications of Linear Independence of Localized Boundary Traces of Eigenfunctions to the Problem of Neumann Boundary Feedback Stabilization of Parabolic Problems. Verification of Kalman Rank Condition
	2.4 Relevant Unique Continuation Properties for Over-determined Oseen Eigenvalue Problems. Part IIA: Emphasis on the Localized Interior Case
		2.4.1 Oseen Eigenproblems. Unique Continuation of the Oseen Equations from an Arbitrary Interior Subdomain. Linear Independence of Interior Localized Eigenfunctions
		2.4.2 Proof of Theorem 2.4.1 T.5
		2.4.3 Implications of the UCP of Theorem 2.4.1: (i) Linear Independence of Interior Localized Eigenfunctions; (ii) Verification of the Corresponding Kalman Rank Condition
		2.4.4 Feedback Stabilization of the Unstable Oseen Dynamical System by Finite-Dimensional Localized Interior Controls
	2.5 Relevant Unique Continuation Properties for Over-determined Oseen Eigenvalue Problems. Part IIB: Emphasis on the Localized Boundary Case
		2.5.1 A First ``Minimally Invasive\'\' Attempt by Use of a Tangential Boundary Control Action on an Arbitrarily Small Portion \"0365 of the Boundary (Implementable by Jets of Air)
		2.5.2 Second Attempt: ``Minimal\'\' Extra Condition to be Added to the Boundary Control v Acting on \"0365
		2.5.3 Helmholtz Decomposition
		2.5.4 Implication of the UCP of Lemma 2.5.1 on the Validity of the Corresponding Kalman Rank Condition for Problem (2.144)
		2.5.5 Justification of (2.149)
	2.6 Part III: The Boussinesq Problem, d=2,3
		2.6.1 Introduction
		2.6.2 Localized Interior Feedback Control Pair {u,v}
		2.6.3 Main UCP Results for Both the Adjoint Eigenproblem, i.e., the Operator A*q; and the Original Eigenproblem, i.e., the Operator Aq
		2.6.4 Proof of Theorem 2.6.3 via Pointwise Carleman Estimates T-W.1
	2.7 Localized {interior = , boundary = v} pair of feedback controls
	References
3 Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction
	3.1 Introduction
		3.1.1 Motivation and Applications
		3.1.2 Flutter Basics
		3.1.3 More on Flutter
		3.1.4 Mathematical Challenges in Studying Flutter
		3.1.5 Overcoming the Challenges and Overall Strategy
	3.2 PDE Modeling
		3.2.1 Setup
		3.2.2 Obtaining the PDE Model
	3.3 Functional Setup and Well-posedness of Weak and Strong Solutions
		3.3.1 Notation and Conventions
		3.3.2 System Under Consideration
		3.3.3 Energies and Spaces
		3.3.4 Solutions
		3.3.5 Well-posedness Results
			Well-posedness of Weak and Strong Solutions
			Discussion of Well-posedness Proofs
	3.4 Long-Time Behavior of Weak Solutions
		3.4.1 Statement of the Main Result and Qualitative Demonstrations
		3.4.2 Further Discussion of Dissipation and the Effects of Rotational Inertia
	3.5 Outline of the Stability Proofs in Theorems 3.4.1 and 3.4.2
	3.6 Plate-to-Flow Mapping: Properties and Discussion
		3.6.1 The Neumann Wave Equation on the Half-space
		3.6.2 Flow Formulae
		3.6.3 Dynamical Systems Framework for the Plate
	3.7 Discussion of Past Stabilization Results
		3.7.1 Re-statement of Main Stabilization Result
		3.7.2 Weak Stability and Smooth Data for α=0
		3.7.3 Other Past Stability Results with α=0
			Large Static and Viscous Damping
			The Berger Nonlinearity
			Rotational Inertia and Thermal Effects—Velocity Smoothing
		3.7.4 General Approach to Stability in the Remainder of This Chapter
	3.8 α>0: Rotational Stabilization
		3.8.1 Existence of Attracting Set for the Structure: Statement
		3.8.2 Attractor Construction: Dissipativity of Dynamical System
		3.8.3 Attractor Construction: Asymptotic Smoothness Through Quasi-stability
			Attractor and its Properties
		3.8.4 Boundedness and Finiteness of Dissipation Integral
		3.8.5 Plate Convergence: Weak and Strong
		3.8.6 Lifting from Plate to Flow Directly
		3.8.7 Strong Convergence to Equilibria
	3.9 α=0; Non-Rotational Stabilization
		3.9.1 Existence of Structural Attractor: Statement
		3.9.2 Additional Preliminaries
		3.9.3 Attractor Construction: Dissipativity of Dynamical System
		3.9.4 Attractor Construction: Smoothness Through Compensated Compactness
		3.9.5 Exploiting Compactness of the Structural Attractor for Quasi-stability
		3.9.6 Proof Strategy for Stabilization to Equilibria: α=0
		3.9.7 Strong Plate Convergence and Weak Convergence for the Flow
		3.9.8 Weak Convergence to Equilibria
		3.9.9 Improving from Weak to Strong
			Convergence Through the Microlocal Regularity
			Converging Together
	3.10 Appendix
		3.10.1 Dynamical Systems and Attractors- Fundamental Notions
			Tools from Quasi-stability Theory
		3.10.2 Microlocal Regularity of the Hyperbolic Neumann-Dirichlet Map
			Proof of Theorem 3.10.10
			Trace Regularity
			Interior Regularity
			Change of Variables and Final Estimate
	References
4 Turbulence Control: From Model-Based to Machine Learned
	4.1 Introduction
	4.2 Fluidic Pinball
	4.3 Proximity Map
	4.4 Cluster-Based Network Modeling
	4.5 Machine Learning Control
	4.6 Conclusions and Perspectives
	References
5 Design Through Analysis
	5.1 Introduction
	5.2 A Spline Primer
		5.2.1 B-Splines
			Univariate B-Splines and Their Properties
			A Matrix Representation of B-Splines
			Efficient Evaluation of B-Splines
			Knot Insertion
			Multi-variate B-Splines
			Geometry Modeling with B-Splines
		5.2.2 Truncated Hierarchical B-Splines
		5.2.3 Non-uniform Rational B-Splines
		5.2.4 Multi-patch Splines
	5.3 Creation of Analysis-Suitable Parameterizations
		5.3.1 Problem Statement
		5.3.2 Classification of Parameterization Methods
			Algebraic Parameterization Methods
			Nonlinear Constrained Optimization Methods
			Nonlinear Unconstrained Optimization Methods
			Nonlinear Partial Differential Equation (PDE)-Based Methods
		5.3.3 Optimization-Based Parameterization Methods
			Barrier Function-Based Method
			Penalty Function-Based Method
		5.3.4 PDE-Based Methods
			Discretization in Sobolev Space H2
			Discretization in Sobolev Space H1
		5.3.5 Experiments and Comparisons
			Quality Metrics for Parameterizations
			Effectiveness and Quality Assessment
			Computational Time
			Volumetric Parameterizations
			Extension to Multi-patch Parameterizations
			Extension to THB-Spline Parameterizations
	5.4 Isogeometric Kirchhoff–Love Shell Analysis
		5.4.1 The Isogeometric Kirchhoff–Love Shell Element
			Geometry
			Kinematic Relation
			Constitutive Relation
			Variational Formulation
			Discretization
		5.4.2 Benchmark Problems
			Nonlinear Hyperelastic Shell Analysis
			Nonlinear Adaptive Shell Analysis
			Nonlinear Multi-patch Shell Analysis
	5.5 Conclusions and Outlook
	References