Numerical Control: Part A

1st Edition, Volume 23 - February 15, 2022
Editors: Emmanuel Trélat, Enrique Zuazua
Language: English
Hardback ISBN:
9 7 8 - 0 - 3 2 3 - 8 5 0 5 9 - 9
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 8 5 3 3 9 - 2

Numerical Control: Part A, Volume 23 in the Handbook of Numerical Analysis series, highlights new advances in the field, with this new volume presenting interesting chapters writ… Read more

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Numerical Control: Part A, Volume 23 in the Handbook of Numerical Analysis series, highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. Chapters in this volume include Numerics for finite-dimensional control systems, Moments and convex optimization for analysis and control of nonlinear PDEs, The turnpike property in optimal control, Structure-Preserving Numerical Schemes for Hamiltonian Dynamics, Optimal Control of PDEs and FE-Approximation, Filtration techniques for the uniform controllability of semi-discrete hyperbolic equations, Numerical controllability properties of fractional partial differential equations, Optimal Control, Numerics, and Applications of Fractional PDEs, and much more.

Cover image
Title page
Table of Contents
Copyright
Dedication
Contributors
Preface
Chapter 1: Control and numerical approximation of fractional diffusion equations
Abstract
1. Introduction
2. Finite Element approximation of the fractional Laplace operator
3. Interior controllability properties of the fractional heat equation
4. Exterior controllability properties of the fractional heat equation
5. Simultaneous control of parameter-dependent fractional heat equations
6. Conclusion and open problems
Acknowledgements
Appendix A. Fractional order Sobolev spaces and the fractional Laplacian
Appendix B. The fractional Laplace operator with exterior conditions
References
Chapter 2: Modeling, control, and numerics of gas networks
Abstract
1. Introduction
2. Modeling of gas flow
3. Well-posedness of mathematical models for fixed control action
4. Control and controllability
5. Uncertainty quantification
6. Numerical methods for simulation and control
7. Open problems
References
Chapter 3: Optimal control, numerics, and applications of fractional PDEs
Abstract
1. Introduction and applications of fractional operators
2. Two fractional operators and their properties
3. Fractional diffusion equation: analysis and numerical approximation
4. Exterior optimal control of fractional parabolic PDEs with control constraints
5. Distributed optimal control of fractional PDEs with state and control constraints
6. Fractional deep neural networks – FDNNs
7. Some open problems
Acknowledgements
References
Chapter 4: Optimal control of PDEs and FE-approximation
Abstract
Introduction
1. The L2 framework
2. Controlling with measures
3. Related topics
References
Chapter 5: Numerical solution of multi-objective optimal control and hierarchic controllability problems
Abstract
1. Introduction
2. Bi-objective control problems for heat and wave equations
3. Stackelberg strategies and hierarchical control problems
4. Additional comments and conclusions
References
Chapter 6: Numerics for stochastic distributed parameter control systems: a finite transposition method
Abstract
1. Introduction
2. Dual equations for stochastic distributed parameter control problems
3. The space of finite transposition
4. Finite transposition method for backward stochastic evolution equations
5. Numerical method for optimal controls
References
Chapter 7: Numerical solutions of stochastic control problems: Markov chain approximation methods
Abstract
1. Stochastic control problems
2. Methods of Markov chain approximation
3. Application to insurance
4. Application to mathematical biology
5. Final remarks
References
Chapter 8: Control of parameter dependent systems
Abstract
1. Introduction
2. Parameter invariant controls
3. Parameter dependent controls
4. Conclusion
Acknowledgements
Appendix A. Proof of technical results related to Section 2
References
Chapter 9: Space-time POD-Galerkin approach for parametric flow control
Abstract
1. Motivations and historical background
2. Introduction
3. Nonlinear time dependent parametrized optimal flow control problems
4. ROMs for nonlinear space-time OCP(μ)s
5. Application to shallow waters equations
6. Conclusions
Acknowledgements
References
Chapter 10: Moments and convex optimization for analysis and control of nonlinear PDEs
Abstract
1. Introduction
2. Problem statement (analysis)
3. Occupation measures for nonlinear PDEs
4. Computable bounds using SDP relaxations
5. Problem statement (control)
6. Linear representation (control)
7. Control design using SDP relaxations
8. Higher-order PDEs
9. Numerical examples
10. Conclusion
Acknowledgements
References
Chapter 11: Turnpike properties in optimal control
Abstract
1. Introduction and historical origins
2. Definition and taxonomy of turnpike properties
3. Generating mechanisms
4. Exploitation of turnpikes in numerics and receding-horizon control
5. Topics not discussed and open problems
Acknowledgement
References
Chapter 12: Some challenging optimization problems for logistic diffusive equations and their numerical modeling
Abstract
1. Introduction and bio-mathematical background
2. Optimal eigenvalue problem
3. Maximizing the total population size
4. Generalization and perspectives
References
Chapter 13: Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctions
Abstract
1. Introduction
2. Convex analysis and nonlinear eigenvalue problems
3. Gradient flows and decrease of Rayleigh quotients
4. Flows for solving nonlinear eigenproblems
5. Nonlinear power methods for homogeneous functionals
6. Γ-convergence implies convergence of ground states
7. Applications
Appendices
Appendix A. Exact reconstruction time
Appendix B. Extinction time
Appendix C. Remaining proofs
References
Chapter 14: Dynamic Programming versus supervised learning
Abstract
1. Introduction
2. A model problem
3. Brute force solution of the non-dynamic control problem by Monte-Carlo
4. Solution of the non-dynamic control problem by supervised learning
5. Bellman's Stochastic Dynamic Programming for the dynamic problem
6. Solution with the Hamilton-Jacobi-Bellman partial differential equations
7. Solution with the Kolmogorov equation
8. Solution by Itô calculus
9. Limit with vanishing volatility
10. Conclusion
Acknowledgements
Appendix A. A model with fishing quota
Appendix B. Reformulation
Appendix C. An analytical solution for a similar problem
References
Chapter 15: Data-driven modeling and control of large-scale dynamical systems in the Loewner framework
Abstract
1. Introduction: data-driven modeling and control
2. The Loewner framework for data-driven modeling: an overview
3. Model reduction examples (large-scale systems)
4. Control in the Loewner framework
5. Summary and conclusions
References
Chapter 16: Machine learning and control theory
Abstract
1. Introduction
2. Reinforcement learning
3. Control theory and deep learning
4. Stochastic gradient descent and control theory
5. Machine learning approach of stochastic control problems
6. Focus on the deterministic case
7. Convergence results
8. Numerical results
Acknowledgement
References
Index

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Emmanuel Trélat

Enrique Zuazua