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E-raamat: Optimal Control of Partial Differential Equations: Analysis, Approximation, and Applications

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  • Sari: Applied Mathematical Sciences 207
  • Ilmumisaeg: 01-Jan-2022
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030772260
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Optimal Control of Partial Differential Equations: Analysis, Approximation, and ApplicationsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Applied Mathematical Sciences 207
  • Ilmumisaeg: 01-Jan-2022
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030772260
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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 of OCPs 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.


Arvustused

The book unique within the large set of existing textbooks and monographs in the field of OCPs. This excellent book is suitable to people interested in mathematical and applied sciences. (Gheorghe Moroanu, zbMATH 1483.49001, 2022)

1 Introduction: Representative Examples, Mathematical Structure
1(24)
1.1 Optimal Control Problems Governed by PDEs
1(2)
1.2 An Intuitive Example: Optimal Control for Heat Transfer
3(2)
1.3 Control of Pollutant Emissions from Chimneys
5(3)
1.4 Control of Emissions from a Sewage System
8(1)
1.5 Optimal Electrical Defibrillation of Cardiac Tissue
9(3)
1.6 Optimal Flow Control for Drag Reduction
12(2)
1.7 Optimal Shape Design for Drag Reduction
14(1)
1.8 A General Setting for OCPs
15(3)
1.9 Shape Optimization Problems
18(1)
1.10 Parameter Estimation Problems
19(1)
1.11 Theoretical Issues
20(1)
1.12 Numerical Approximation of an OCP
20(5)
Part I A Preview on Optimization and Control in Finite Dimensions
2 Prelude on Optimization: Finite Dimension Spaces
25(14)
2.1 Problem Setting and Analysis
25(4)
2.1.1 Well Posedness Analysis
26(2)
2.1.2 Convexity, Optimality Conditions, and Admissible Directions
28(1)
2.2 Free (Unconstrained) Optimization
29(1)
2.3 Constrained Optimization
30(9)
2.3.1 Lagrange Multipliers: Equality Constraints
30(3)
2.3.2 Karush-Kuhn-Tucker Multipliers: Inequality Constraints
33(3)
2.3.3 Second Order Conditions
36(3)
3 Algorithms for Numerical Optimization
39(22)
3.1 Free Minimization by Descent Methods
40(7)
3.1.1 Choice of descent directions
41(4)
3.1.2 Step Length Evaluation and Inexact Line-Search
45(1)
3.1.3 Convergence of Descent Methods
45(2)
3.2 Free optimization by trust region methods
47(2)
3.3 Constrained Optimization by Projection Methods
49(2)
3.4 Constrained Optimization for Quadratic Programming Problems
51(4)
3.4.1 Equality Constraints: a Saddle-Point Problem
51(1)
3.4.2 Inequality Constraints: Active Set Method
52(3)
3.5 Constrained Optimization for More General Problems
55(6)
3.5.1 Penalty and Augmented Lagrangian Methods
55(2)
3.5.2 Sequential Quadratic Programming
57(4)
4 Prelude on Control: The Case of Algebraic and ODE Systems
61(42)
4.1 Algebraic Optimal Control Problems
61(7)
4.1.1 Existence and Uniqueness of the Solution
62(1)
4.1.2 Optimality conditions
63(2)
4.1.3 Gradient, Sensitivity and Minimum Principle
65(1)
4.1.4 Direct vs. Adjoint Approach
66(2)
4.2 Formulation as a Constrained Optimization Problem
68(4)
4.2.1 Lagrange Multipliers
68(2)
4.2.2 Control Constraints: Karush-Kuhn-Tucker Multipliers
70(2)
4.3 Control Problems Governed by ODEs
72(2)
4.4 Linear Payoff, Free End Point
74(5)
4.4.1 Uniqueness of Optimal Control. Normal Systems
77(2)
4.5 Minimum Time Problems
79(7)
4.5.1 Controllability
79(3)
4.5.2 Observability
82(1)
4.5.3 Optimality conditions
83(3)
4.6 Quadratic Cost
86(8)
4.6.1 First-order conditions
87(2)
4.6.2 The Riccati equation
89(3)
4.6.3 The Algebraic Riccati Equation
92(2)
4.7 Hints on Numerical Approximation
94(4)
4.8 Exercises
98(5)
Part II Linear-Quadratic Optimal Control Problems
5 Quadratic control problems governed by linear elliptic PDEs
103(64)
5.1 Optimal Heat Source (1): an Unconstrained Case
103(8)
5.1.1 Analysis of the State Problem
104(2)
5.1.2 Existence and Uniqueness of an Optimal Pair. A First Optimality Condition
106(2)
5.1.3 Use of the Adjoint State
108(2)
5.1.4 The Lagrange Multipliers Approach
110(1)
5.2 Optimal Heat Source (2): a Box Constrained Case
111(4)
5.2.1 Optimality Conditions
111(1)
5.2.2 Projections onto a Closed Convex Set of a Hilbert Space
112(1)
5.2.3 Karush-Kuhn-Tucker Conditions
113(2)
5.3 A General Framework for Linear-quadratic OCPs
115(8)
5.3.1 The Mathematical Setting
116(1)
5.3.2 A First Optimality Condition
117(1)
5.3.3 Use of the Adjoint State
118(2)
5.3.4 The Lagrange Multipliers Approach
120(1)
5.3.5 Existence and Uniqueness of an Optimal Control
121(2)
5.4 Variational Formulation and Well-posedness of Boundary Value Problems
123(6)
5.5 Distributed Observation and Control
129(5)
5.5.1 Robin Conditions
129(2)
5.5.2 Dirichlet Conditions, Energy Cost Functional
131(3)
5.6 Distributed Observation, Neumann Boundary Control
134(1)
5.7 Boundary Observation, Neumann Boundary Control
135(1)
5.8 Boundary Observation, Distributed Control, Dirichlet Conditions
136(3)
5.9 Dirichlet Problems with L2 Data. Transposition (or Duality) Method
139(3)
5.10 Pointwise Observations
142(1)
5.11 Distributed Observation, Dirichlet Control
143(6)
5.11.1 Case U = H1/2 (Γ)
143(4)
5.11.2 Case U = U0 = L2 (Γ)
147(2)
5.11.3 Case U = U0 = H1 (Γ)
149(1)
5.12 A State-Constrained Control Problem
149(4)
5.13 Control of Viscous Flows: the Stokes Case
153(9)
5.13.1 Distributed Velocity Control
153(4)
5.13.2 Boundary Velocity Control, Vorticity Minimization
157(5)
5.14 Exercises
162(5)
6 Numerical Approximation of Linear-Quadratic OCPs
167(62)
6.1 A Classification of Possible Approaches
167(1)
6.2 Optimize & Discretize, or the Other Way Around?
168(12)
6.2.1 Optimize Then Discretize
170(3)
6.2.2 Discretize then Optimize
173(1)
6.2.3 Pro's and Con's
174(2)
6.2.4 The case of Advection Diffusion Equations with Dominating Advection
176(3)
6.2.5 The case of Stokes Equations
179(1)
6.3 Iterative Methods (I): Unconstrained OCPs
180(6)
6.3.1 Relation with Solving the Reduced Hessian Problem
185(1)
6.4 Iterative Methods (II): Control Constraints
186(1)
6.5 Numerical Examples
187(5)
6.5.1 OCPs governed by Advection-Diffusion Equations
187(3)
6.5.2 OCPs governed by the Stokes Equations
190(2)
6.6 All-at-once Methods (I)
192(10)
6.6.1 OCPs Governed by Scalar Elliptic Equations
193(6)
6.6.2 OCPs governed by Stokes Equations
199(3)
6.7 Numerical Examples
202(4)
6.8 All-at-once Methods (II): Control Constraints
206(5)
6.9 Numerical Examples
211(7)
6.9.1 OCPs Governed by the Laplace Equation
211(4)
6.9.2 OCPs Governed by the Stokes Equations
215(3)
6.10 A priori Error Estimates
218(4)
6.11 A Posteriori Error Estimates
222(4)
6.12 Exercises
226(3)
7 Quadratic Control Problems Governed by Linear Evolution PDEs
229(32)
7.1 Optimal Heat Source (1): an Unconstrained Case
229(7)
7.1.1 Analysis of the State System
230(2)
7.1.2 Existence & Uniqueness of the Optimal Control. Optimality Condition
232(1)
7.1.3 Use of the Adjoint State
233(1)
7.1.4 The Lagrange Multipliers Approach
234(2)
7.2 Optimal Heat Source (2): a Constrained Case
236(1)
7.3 Initial control
236(1)
7.4 General Framework for Linear-Quadratic OCPs Governed by Parabolic PDEs
237(8)
7.4.1 Initial-boundary Value Problems for Parabolic Linear Equations
237(3)
7.4.2 The Mathematical Setting for OCPs
240(2)
7.4.3 Optimality Conditions
242(3)
7.5 Further Applications to Equations in Divergence Form
245(5)
7.5.1 Side Control, Final and Distributed Observation
246(2)
7.5.2 Time-distributed Control, Side Observation
248(2)
7.6 Optimal Control of Time-Dependent Stokes Equations
250(5)
7.7 Optimal Control of the Wave Equation
255(3)
7.8 Exercises
258(3)
8 Numerical Approximation of Quadratic OCPs Governed by Linear Evolution PDEs
261(20)
8.1 Optimize & Discretize, or Discretize & Optimize, Revisited
261(8)
8.1.1 Optimize Then Discretize
262(3)
8.1.2 Discretize Then Optimize
265(4)
8.2 Iterative Methods
269(1)
8.3 Numerical Examples
270(4)
8.4 All-at-once Methods
274(3)
8.5 Exercises
277(4)
Part III More general PDE-constrained optimization problems
9 A Mathematical Framework for Nonlinear OCPs
281(44)
9.1 Motivation
281(1)
9.2 Optimization in Banach and Hilbert Spaces
282(5)
9.2.1 Existence and Uniqueness of Minimizers
282(3)
9.2.2 Convexity and Lower Semicontinuity
285(1)
9.2.3 First Order Optimality Conditions
286(1)
9.2.4 Second Order Optimality Conditions
286(1)
9.3 Control Constrained OCPs
287(8)
9.3.1 Existence
288(1)
9.3.2 First Order Conditions. Adjoint Equation. Multipliers
289(2)
9.3.3 Karush-Kuhn-Tucker Conditions
291(2)
9.3.4 Second Order Conditions
293(2)
9.4 Distributed Control of a Semilinear State Equation
295(5)
9.5 Least squares approximation of a reaction coefficient
300(6)
9.6 Numerical Approximation of Nonlinear OCPs
306(9)
9.6.1 Iterative Methods
306(4)
9.6.2 All-at-once Methods: Sequential Quadratic Programming
310(5)
9.7 Numerical Examples
315(3)
9.8 Numerical Treatment of Control Constraints
318(4)
9.9 Exercises
322(3)
10 Advanced Selected Applications
325(48)
10.1 Optimal Control of Steady Navier-Stokes Flows
325(15)
10.1.1 Problem Formulation
326(1)
10.1.2 Analysis of the State Problem
327(2)
10.1.3 Existence of an Optimal Control
329(1)
10.1.4 Differentiability of the Control-to-State Map
330(1)
10.1.5 First Order Optimality Conditions
331(4)
10.1.6 Numerical Approximation
335(5)
10.2 Time optimal Control in Cardiac Electrophysiology
340(18)
10.2.1 Problem Formulation
340(2)
10.2.2 Analysis of the Monodomain System
342(3)
10.2.3 Existence of an Optimal Control (u, t f)
345(2)
10.2.4 Reduction to a Control Problem with Fixed End Time
347(1)
10.2.5 First Order Optimality Conditions
348(5)
10.2.6 Numerical Approximation
353(5)
10.3 Optimal Dirichlet Control of Unsteady Navier-Stokes Flows
358(12)
10.3.1 Problem Formulation
358(2)
10.3.2 Optimality Conditions
360(2)
10.3.3 Dynamic Boundary Action
362(1)
10.3.4 Numerical Approximation
363(7)
10.4 Exercises
370(3)
11 Shape Optimization Problems
373(50)
11.1 Formulation
373(3)
11.1.1 A Model Problem
375(1)
11.2 Shape Functionals and Derivatives
376(10)
11.2.1 Domain Deformations
377(1)
11.2.2 Elements of Tangential (or Surface) Calculus
378(5)
11.2.3 Shape Derivative of Functions
383(3)
11.3 Shape Derivatives of Functionals and Solutions of Boundary Value Problems
386(7)
11.3.1 Domain Functionals
387(1)
11.3.2 Boundary Functionals
388(3)
11.3.3 Chain Rules
391(1)
11.3.4 Shape Derivative for the Solution of a Dirichlet Problem
391(1)
11.3.5 Shape Derivative for the Solution of a Neumann Problem
392(1)
11.4 Gradient and First-Order Necessary Optimality Conditions
393(8)
11.4.1 A Model Problem
393(2)
11.4.2 The Lagrange Multipliers Approach
395(6)
11.5 Numerical Approximation of Shape Optimization Problems
401(4)
11.5.1 Computational Aspects
403(2)
11.6 Minimizing the Compliance of an Elastic Structure
405(4)
11.6.1 Numerical Results
407(2)
11.7 Drag Minimization in Navier-Stokes Flows
409(11)
11.7.1 The State Equation
410(1)
11.7.2 The Drag Functional
411(1)
11.7.3 Shape Derivative of the State
412(1)
11.7.4 Shape Derivative and Gradient of T (Ω)
413(3)
11.7.5 Numerical Results
416(4)
11.8 Exercises
420(3)
Appendix A Toolbox of Functional Analysis
423(42)
A.1 Metric, Banach and Hilbert Spaces
423(4)
A.1.1 Metric Spaces
423(1)
A.1.2 Banach Spaces
424(2)
A.1.3 Hilbert Spaces
426(1)
A.2 Linear Operators and Duality
427(3)
A.2.1 Bounded Linear Operators
427(1)
A.2.2 Functionals, Dual space and Riesz Theorem
428(1)
A.2.3 Hilbert Triplets
429(1)
A.2.4 Adjoint Operators
429(1)
A.3 Compactness and Weak Convergence
430(4)
A.3.1 Compactness
430(1)
A.3.2 Weak Convergence, Reflexivity and Weak Sequential Compactness
431(1)
A.3.3 Compact Operators
432(1)
A.3.4 Convexity and Weak Closure
433(1)
A.3.5 Weak Convergence, Separability and Weak Sequential Compactness
433(1)
A.4 Abstract Variational Problems
434(3)
A.4.1 Coercive Problems
434(1)
A.4.2 Saddle-Point Problems
435(2)
A.5 Sobolev spaces
437(12)
A.5.1 Lebesgue Spaces
437(2)
A.5.2 Domain Regularity
439(1)
A.5.3 Integration by Parts and Weak Derivatives
440(1)
A.5.4 The Space H1 (Ω)
440(1)
A.5.5 The Space H10 (Ω)and its Dual
441(2)
A.5.6 The Spaces Hm (Ω), m > 1
443(1)
A.5.7 Approximation by Smooth Functions. Traces
444(2)
A.5.8 Compactness
446(1)
A.5.9 The space Hdiv(Ω)
447(1)
A.5.10 The Space H1/200 and Its Dual
447(1)
A.5.11 Sobolev Embeddings
448(1)
A.6 Functions with Values in Hilbert Spaces
449(4)
A.6.1 Spaces of Continuous Functions
449(1)
A.6.2 Integrals and Spaces of Integrable Functions
450(1)
A.6.3 Sobolev Spaces Involving Time
451(2)
A.6.4 The Gronwall Lemma
453(1)
A.7 Differential Calculus in Banach Spaces
453(10)
A.7.1 The Frechet Derivative
453(3)
A.7.2 The Gateaux Derivative
456(1)
A.7.3 Derivative of Convex Functionals
457(3)
A.7.4 Second Order Derivatives
460(3)
A.8 Fixed Points and Implicit Function Theorems
463(2)
Appendix B Toolbox of Numerical Analysis
465(18)
B.1 Basic Matrix Properties
465(5)
B.1.1 Eigenvalues, Eigenvectors, Positive Definite Matrices
465(1)
B.1.2 Singular Value Decomposition
466(1)
B.1.3 Spectral Properties of Saddle-Point Systems
467(3)
B.2 Numerical Approximation of Elliptic PDEs
470(4)
B.2.1 Strongly Coercive Problems
470(2)
B.2.2 Algebraic Form of (Ph1)
472(1)
B.2.3 Saddle-Point Problems
472(2)
B.2.4 Algebraic Form of (Ph2)
474(1)
B.3 Numerical Approximation of Parabolic PDEs
474(3)
B.4 Finite Element Spaces and Interpolation Operator
477(2)
B.5 A Priori Error Estimation
479(1)
B.5.1 Elliptic PDEs
479(1)
B.5.2 Parabolic PDEs
480(1)
B.6 Solution of Nonlinear PDEs
480(3)
References 483(10)
Index 493
Andrea Manzoni, PhD, is an Associate Professor of Numerical Analysis at Politecnico of Milan. He is the author of 2 books and of approximately 50 papers. In 2012 he won the ECCOMAS Award for the best PhD thesis in Europe about Computational Methods in Applied Sciences and Engineering and the Biannual SIMAI prize (Italian Society of Applied and Industrial Mathematics) in 2017. His research interests include the development of reduced-order modelling techniques for PDEs, PDE-constrained optimization, uncertainty quantification, computational statistics, and machine/deep learning.





Alfio Quarteroni is a Professor of Numerical Analysis at Politecnico of Milan and Professor Emeritus at EPFL, Lausanne. He is the author of 25 books, editor of 12 books, author of about 400 papers. He is the recipient of two ERC Advanced Grants. He is a member of the Italian Academy of Science, the European Academy of Science, Academia Europaea, and the Lisbon Academy of Science. His research Group at EPFL has carried out the mathematical simulation for the Alinghi sailing boat, the winner of two editions (2003 and 2007) of Americas Cup. His research interests include mathematical modeling and its applications at large.





Sandro Salsa is a Professor of Mathematical Analysis at the Department of Mathematics of the Politecnico of Milan, where he has been one of the main founders of the educational program in Mathematical Engineering. His research interest ranges over diverse aspects of nonlinear, nonlocal, singular or degenerate elliptic and parabolic equations, with particular emphasis on free boundary problems. He is an author of 13 books and several papers in the most prestigious scientific mathematical journals.
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