Optimal Control of Partial Differential Equations: Analysis, Approximation, and Applications (Applied Mathematical Sciences, 207)

✍ Scribed by Andrea Manzoni, Alfio Quarteroni, Sandro Salsa

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 507
• Category: Library

✦ Synopsis

This is a book on optimal control problems (OCPs) for partial differential equations (PDEs) that evolved from a series of courses taught by the authors in the last few years at Politecnico di Milano, both at the undergraduate and graduate levels. The book covers the whole range spanning from the setup and the rigorous theoretical analysis of OCPs, the derivation of the system of optimality conditions, the proposition of suitable numerical methods, their formulation, their analysis, including their application to a broad set of problems of practical relevance.

The first introductory chapter addresses a handful of representative OCPs and presents an overview of the associated mathematical issues. The rest of the book is organized into three parts: part I provides preliminary concepts of OCPs for algebraic and dynamical systems; part II addresses OCPs involving linear PDEs (mostly elliptic and parabolic type) and quadratic cost functions; part III deals with more general classes ofOCPs that stand behind the advanced applications mentioned above.

Starting from simple problems that allow a “hands-on” treatment, the reader is progressively led to a general framework suitable to face a broader class of problems. Moreover, the inclusion of many pseudocodes allows the reader to easily implement the algorithms illustrated throughout the text.

The three parts of the book are suitable to readers with variable mathematical backgrounds, from advanced undergraduate to Ph.D. levels and beyond. We believe that applied mathematicians, computational scientists, and engineers may find this book useful for a constructive approach toward the solution of OCPs in the context of complex applications.

✦ Table of Contents

Preface
List of Algorithms
Contents
About the Authors
Chapter 1 Introduction: Representative Examples, Mathematical Structure
1.1 Optimal Control Problems Governed by PDEs
1.2 An Intuitive Example: Optimal Control for Heat Transfer
1.3 Control of Pollutant Emissions from Chimneys
1.4 Control of Emissions from a Sewage System
1.5 Optimal Electrical Defibrillation of Cardiac Tissue
1.6 Optimal Flow Control for Drag Reduction
1.7 Optimal Shape Design for Drag Reduction
1.8 A General Setting for OCPs
1.9 Shape Optimization Problems
1.10 Parameter Estimation Problems
1.11 Theoretical Issues
1.12 Numerical Approximation of an OCP
Part I A Preview on Optimization and Control in Finite Dimensions
Chapter 2 Prelude on Optimization: Finite Dimension Spaces
2.1 Problem Setting and Analysis
2.1.1 Well Posedness Analysis
2.1.2 Convexity, Optimality Conditions, and Admissible Directions
2.2 Free (Unconstrained) Optimization
2.3 Constrained Optimization
2.3.1 Lagrange Multipliers: Equality Constraints
2.3.2 Karush-Kuhn-Tucker Multipliers: Inequality Constraints
2.3.3 Second Order Conditions
Chapter 3 Algorithms for Numerical Optimization
3.1 Free Minimization by Descent Methods
3.1.1 Choice of descent directions
3.1.2 Step Length Evaluation and Inexact Line-Search
3.1.3 Convergence of Descent Methods
3.2 Free optimization by trust region methods
3.3 Constrained Optimization by Projection Methods
3.4 Constrained Optimization for Quadratic Programming Problems
3.4.1 Equality Constraints: a Saddle-Point Problem
3.4.2 Inequality Constraints: Active Set Method
3.5 Constrained Optimization for More General Problems
3.5.1 Penalty and Augmented Lagrangian Methods
3.5.2 Sequential Quadratic Programming
Chapter 4 Prelude on Control: The Case of Algebraic and ODE Systems
4.1 Algebraic Optimal Control Problems
4.1.1 Existence and Uniqueness of the Solution
4.1.2 Optimality conditions
4.1.3 Gradient, Sensitivity and Minimum Principle
4.1.4 Direct vs. Adjoint Approach
4.2 Formulation as a Constrained Optimization Problem
4.2.1 Lagrange Multipliers
4.2.2 Control Constraints: Karush-Kuhn-Tucker Multipliers
4.3 Control Problems Governed by ODEs
4.4 Linear Payoff, Free End Point
4.4.1 Uniqueness of Optimal Control. Normal Systems
4.5 Minimum Time Problems
4.5.1 Controllability
4.5.2 Observability
4.5.3 Optimality conditions
4.6 Quadratic Cost
4.6.1 First-order conditions
4.6.2 The Riccati equation
4.6.3 The Algebraic Riccati Equation
4.7 Hints on Numerical Approximation
4.8 Exercises
Part II Linear-Quadratic Optimal Control Problems
Chapter 5 Quadratic control problems governed by linear elliptic PDEs
5.1 Optimal Heat Source (1): an Unconstrained Case
5.1.1 Analysis of the State Problem
5.1.2 Existence and Uniqueness of an Optimal Pair. A First Optimality Condition
5.1.3 Use of the Adjoint State
5.1.4 The Lagrange Multipliers Approach
5.2 Optimal Heat Source (2): a Box Constrained Case
5.2.1 Optimality Conditions
5.2.2 Projections onto a Closed Convex Set of a Hilbert Space
5.2.3 Karush-Kuhn-Tucker Conditions
5.3 A General Framework for Linear-quadratic OCPs
5.3.1 The Mathematical Setting
5.3.2 A First Optimality Condition
5.3.3 Use of the Adjoint State
5.3.4 The Lagrange Multipliers Approach
5.3.5 Existence and Uniqueness of an Optimal Control (*)
5.4 Variational Formulation and Well-posedness of Boundary Value Problems
5.5 Distributed Observation and Control
5.5.1 Robin Conditions
5.5.2 Dirichlet Conditions, Energy Cost Functional
5.6 Distributed Observation, Neumann Boundary Control
5.7 Boundary Observation, Neumann Boundary Control
5.8 Boundary Observation, Distributed Control, Dirichlet Conditions
5.9 Dirichlet Problems with L2 Data. Transposition (or Duality) Method
5.10 Pointwise Observations
5.11 Distributed Observation, Dirichlet Control
5.11.1 Case U=H1/2(Γ)
5.11.2 Case U=U0=L2(Γ)
5.11.3 Case U=U0=H1(Γ)
5.12 A State-Constrained Control Problem
5.13 Control of Viscous Flows: the Stokes Case
5.13.1 Distributed Velocity Control
5.13.2 Boundary Velocity Control, Vorticity Minimization
5.14 Exercises
Chapter 6 Numerical Approximation of Linear-Quadratic OCPs
6.1 A Classification of Possible Approaches
6.2 Optimize & Discretize, or the Other Way Around?
6.2.1 Optimize Then Discretize
6.2.2 Discretize then Optimize
6.2.3 Pro's and Con's
6.2.4 The case of Advection Diffusion Equations with Dominating Advection
6.2.5 The case of Stokes Equations
6.3 Iterative Methods (I): Unconstrained OCPs
6.3.1 Relation with Solving the Reduced Hessian Problem
6.4 Iterative Methods (II): Control Constraints
6.5 Numerical Examples
6.5.1 OCPs governed by Advection-Diffusion Equations
6.5.2 OCPs governed by the Stokes Equations
6.6 All-at-once Methods (I)
6.6.1 OCPs Governed by Scalar Elliptic Equations
6.6.2 OCPs governed by Stokes Equations
6.7 Numerical Examples
6.8 All-at-once Methods (II): Control Constraints
6.9 Numerical Examples
6.9.1 OCPs Governed by the Laplace Equation
6.9.2 OCPs Governed by the Stokes Equations
6.10 A priori Error Estimates
6.11 A Posteriori Error Estimates
6.12 Exercises
Chapter 7 Quadratic Control Problems Governed by Linear Evolution PDEs
7.1 Optimal Heat Source (1): an Unconstrained Case
7.1.1 Analysis of the State System
7.1.2 Existence & Uniqueness of the Optimal Control. Optimality Condition
7.1.3 Use of the Adjoint State
7.1.4 The Lagrange Multipliers Approach
7.2 Optimal Heat Source (2): a Constrained Case
7.3 Initial control
7.4 General Framework for Linear-Quadratic OCPs Governed by Parabolic PDEs
7.4.1 Initial-boundary Value Problems for Parabolic Linear Equations
7.4.2 The Mathematical Setting for OCPs
7.4.3 Optimality Conditions
7.5 Further Applications to Equations in Divergence Form
7.5.1 Side Control, Final and Distributed Observation
7.5.2 Time-distributed Control, Side Observation
7.6 Optimal Control of Time-Dependent Stokes Equations
7.7 Optimal Control of the Wave Equation
E7.8 xercises
Chapter 8 Numerical Approximation of Quadratic OCPs Governed by Linear Evolution PDEs
8.1 Optimize & Discretize, or Discretize & Optimize, Revisited
8.1.1 Optimize Then Discretize
8.1.2 Discretize Then Optimize
8.2 Iterative Methods
8.3 Numerical Examples
8.4 All-at-once Methods
8.5 Exercises
Part III More general PDE-constrained optimization problems
Chapter 9 A Mathematical Framework for Nonlinear OCPs
9.1 Motivation
9.2 Optimization in Banach and Hilbert Spaces
9.2.1 Existence and Uniqueness of Minimizers
9.2.2 Convexity and Lower Semicontinuity
9.2.3 First Order Optimality Conditions
9.2.4 Second Order Optimality Conditions
9.3 Control Constrained OCPs
9.3.1 Existence
9.3.2 First Order Conditions. Adjoint Equation. Multipliers
9.3.3 Karush-Kuhn-Tucker Conditions
9.3.4 Second Order Conditions
9.4 Distributed Control of a Semilinear State Equation
9.5 Least squares approximation of a reaction coefficient
9.6 Numerical Approximation of Nonlinear OCPs
9.6.1 Iterative Methods
9.6.2 All-at-once Methods: Sequential Quadratic Programming
9.7 Numerical Examples
9.8 Numerical Treatment of Control Constraints
9.9 Exercises
Chapter 10 Advanced Selected Applications
10.1 Optimal Control of Steady Navier-Stokes Flows
10.1.1 Problem Formulation
10.1.2 Analysis of the State Problem
10.1.3 Existence of an Optimal Control
10.1.4 Differentiability of the Control-to-State Map
10.1.5 First Order Optimality Conditions
10.1.6 Numerical Approximation
10.2 Time optimal Control in Cardiac Electrophysiology
10.2.1 Problem Formulation
10.2.2 Analysis of the Monodomain System
10.2.3 Existence of an Optimal Control (u,t ,f)
10.2.4 Reduction to a Control Problem with Fixed End Time
10.2.5 First Order Optimality Conditions
10.2.6 Numerical Approximation
10.3 Optimal Dirichlet Control of Unsteady Navier-Stokes Flows
10.3.1 Problem Formulation
10.3.2 Optimality Conditions
10.3.3 Dynamic Boundary Action
10.3.4 Numerical Approximation
10.4 Exercises
Chapter 11 Shape Optimization Problems
11.1 Formulation
11.1.1 A Model Problem
11.2 Shape Functionals and Derivatives
11.2.1 Domain Deformations
11.2.2 Elements of Tangential (or Surface) Calculus
11.2.3 Shape Derivative of Functions
11.3 Shape Derivatives of Functionals and Solutions of Boundary Value Problems
11.3.1 Domain Functionals
11.3.2 Boundary Functionals
11.3.3 Chain Rules
11.3.4 Shape Derivative for the Solution of a Dirichlet Problem
11.3.5 Shape Derivative for the Solution of a Neumann Problem
11.4 Gradient and First-Order Necessary Optimality Conditions
11.4.1 A Model Problem
11.4.2 The Lagrange Multipliers Approach
11.5 Numerical Approximation of Shape Optimization Problems
11.5.1 Computational Aspects
11.6 Minimizing the Compliance of an Elastic Structure
11.6.1 Numerical Results
11.7 Drag Minimization in Navier-Stokes Flows
11.7.1 The State Equation
11.7.2 The Drag Functional
11.7.3 Shape Derivative of the State
11.7.3 Shape Derivative and Gradient of T (Ω)
11.7.5 Numerical Results
11.8 Exercises
Appendix A Appendix Toolbox of Functional Analysis
A.1 Metric, Banach and Hilbert Spaces
A.1.1 Metric Spaces
A.1.2 Banach Spaces
A1.3 Hilbert Spaces
A.2 Linear Operators and Duality
A.2.1 Bounded Linear Operators
A.2.2 Functionals, Dual space and Riesz Theorem
A.2.3 Hilbert Triplets
A.2.4 Adjoint Operators
A.3 Compactness and Weak Convergence
A.3.1 Compactness
A.3.2 Weak Convergence, Reflexivity and Weak Sequential Compactness
A.3.3 Compact Operators
A.3.4 Convexity and Weak Closure
A.3.5 Weak Convergence, Separability and Weak Sequential Compactness
A.4 Abstract Variational Problems
A.4.1 Coercive Problems
A.4.2 Saddle-Point Problems
A.5 Sobolev spaces
A.5.1 Lebesgue Spaces
A.5.2 Domain Regularity
A.5.3 Integration by Parts and Weak Derivatives
A.5.4 The Space H1(Ω)
A.5.5 The Space H01(Ω) and its Dual
A.5.6 The Spaces Hm(Ω) , m>1
A.5.7 Approximation by Smooth Functions. Traces
A.5.8 Compactness
A.5.9 The space Hdiv(Ω)
A.5.10 The Space H001/2 and Its Dual
A.5.11 Sobolev Embeddings
A.6 Functions with Values in Hilbert Spaces
A.6.1 Spaces of Continuous Functions
A.6.2 Integrals and Spaces of Integrable Functions
A.6.3 Sobolev Spaces Involving Time
A.6.4 The Gronwall Lemma
A.7 Differential Calculus in Banach Spaces
A.7.1 The Fréchet Derivative
A.7.2 The Gâteaux Derivative
A.7.3 Derivative of Convex Functionals
A.7.4 Second Order Derivatives
A.8 Fixed Points and Implicit Function Theorems
Appendix B Appendix Toolbox of Numerical Analysis
B.1 Basic Matrix Properties
B.1.1 Eigenvalues, Eigenvectors, Positive Definite Matrices
B.1.2 Singular Value Decomposition
B.1.3 Spectral Properties of Saddle-Point Systems
B.2 Numerical Approximation of Elliptic PDEs
B.2.1 Strongly Coercive Problems
B.2.2 Algebraic Form of (P1h)
B.2.3 Saddle-Point Problems
B.2.4 Algebraic Form of (P2h)
B.3 Numerical Approximation of Parabolic PDEs
B.4 Finite Element Spaces and Interpolation Operator
B.5 A Priori Error Estimation
B.5.1 Elliptic PDEs
B.5.2 Parabolic PDEs
B.6 Solution of Nonlinear PDEs
References
Index