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# 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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Request a sales quote*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.

- Provides the authority and expertise of leading contributors from an international board of authors
- Presents the latest release in the Handbook of Numerical Analysis series
- Updated release includes the latest information on Numerical Control

Mathematically trained research scientists and engineers with basic knowledge in numerical control systems

- 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

- No. of pages: 594
- Language: English
- Edition: 1
- Volume: 23
- Published: February 15, 2022
- Imprint: North Holland
- Hardback ISBN: 9780323850599
- eBook ISBN: 9780323853392

ET

### Emmanuel Trélat

Emmanuel Trelat works at Sorbonne Universite in Laboratoire Jacques-Louis Lions, CNRS, Inria, equipe CAGE, Paris, France.

Affiliations and expertise

Sorbonne Universite, Laboratoire Jacques-Louis Lions, CNRS, Inria, équipe CAGE, Paris, FranceEZ

### Enrique Zuazua

Enrique Zuazua works in the Department of Mathematics at Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen in Germany.

Affiliations and expertise

Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, GermanyRead

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