Numerical Control: Part B, Volume 24 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 Control problems in the coefficients and the domain for linear elliptic equations, Computational approaches for extremal geometric eigenvalue problems, Non-overlapping domain decomposition in space and time for PDE-constrained optimal control problems on networks, Feedback Control of Time-dependent Nonlinear PDEs with Applications in Fluid Dynamics, Stabilization of the Navier-Stokes equations - Theoretical and numerical aspects, Reconstruction algorithms based on Carleman estimates, and more.
Other sections cover Discrete time formulations as time discretization strategies in data assimilation, Back and forth iterations/Time reversal methods, Unbalanced Optimal Transport: from Theory to Numerics, An ADMM Approach to the Exact and Approximate Controllability of Parabolic Equations, Nonlocal balance laws -- an overview over recent results, Numerics and control of conservation laws, Numerical approaches for simulation and control of superconducting quantum circuits, and much more.
1. 1 Introduction - p. 2 2. 2 Statement of the problem and relaxed formulation - p. 5 3. 3 Discrete formulation and convergence - p. 14 4. 4 A numerical algorithm - p. 18 5. 5 Some numerical experiments - p. 22 6. Acknowledgements - p. 26 7. References - p. 26 8. Chapter 2: Structure-preserving numerical schemes for Hamiltonian dynamics - p. 29 9. Abstract - p. 30 10. 1 Overview - p. page29 11. 2 Classical properties of Hamiltonian systems - p. 35 12. 3 Symplectic numerical schemes for nonstiff systems - p. 44 13. 4 Backward analysis, stiff systems and numerical resonances - p. 53 14. References - p. 57 15. Chapter 3: Relaxation methods for optimal switching control of PDE-dynamical systems - p. 61 16. Chapter 4: Feedback control of time-dependent nonlinear PDEs with applications in fluid dynamics - p. 77 17. Abstract - p. 78 18. 1 Introduction: feedback control of time-dependent PDEs - p. page78 19. 2 Feedback control in fluid dynamics - p. 87 20. 3 Feedback control of nonlinear PDEs via Riccati equations - p. 109 21. 4 Conclusions and outlook - p. 123 22. References - p. 124 23. Chapter 5: Computation of invariant sets for complex systems - p. 131 24. Abstract - p. 131 25. 1 Introduction - p. 132 26. 2 Set programming formulations for the control of complex systems - p. 133 27. 3 Template-independent set programming - p. page140 28. 4 Four numerical templates - p. 146 29. 5 Comparison of performance for convex bodies - p. 159 30. 6 Conclusion - p. 162 31. References - p. 180 32. Chapter 6: Nonlocal balance laws – an overview over recent results - p. 183 33. Abstract - p. 183 34. 1 Introduction - p. 184 35. 2 Existence & uniqueness of solutions - p. 191 36. 3 Related models – extensions - p. 195 37. 4 Approximating “local” entropy solutions by solutions of nonlocal conservation laws or the singular limit problem - p. 203 38. 5 A numerical scheme without dissipation - p. 208 39. 6 Open problems - p. 210 40. Acknowledgements - p. 212 41. References - p. 212 42. Chapter 7: Nonoverlapping domain decomposition and virtual controls for optimal control problems of p-type on metric graphs - p. 217 43. Abstract - p. 217 44. 1 Introduction - p. 218 45. 2 Optimal control problems - p. 225 46. 3 Nonoverlapping domain decomposition: the concept of virtual controls - p. 235 47. 4 Domain decomposition for the static optimal control problem - p. 246 48. 5 Time-domain decomposition - p. 252 49. 6 Concluding remarks and open problems - p. 256 50. References - p. 257 51. Chapter 8: Filtration techniques for the uniform controllability of semidiscrete hyperbolic equations - p. 261 52. Abstract - p. 262 53. 1 Introduction - p. page261 54. 2 Problem of moments and biorthogonals - p. 263 55. 3 Semidiscrete hyperbolic control problems - p. 267 56. 4 A case study: finite elements for the one dimensional wave equation - p. 272 57. 5 Numerical experiments - p. page291 58. References - p. 295 59. Chapter 9: Discrete-time formulations as time discretization strategies in data assimilation - p. 297 60. Abstract - p. 297 61. 1 Introduction - p. 298 62. 2 Least squares estimation and associated discretization for linear parabolic cases - p. 301 63. 3 Luenberger observer strategies and discretization for linear hyperbolic cases - p. 314 64. 4 Least squares estimation and associated discretization for nonlinear models - p. 326 65. 5 Conclusion - p. 335 66. References - p. page335 67. Chapter 10: Approximation of exact controls for semilinear wave and heat equations through space-time methods - p. 341 68. Abstract - p. 342 69. 1 Introduction - p. page342 70. 2 The wave equation - p. 343 71. 3 The heat equation - p. 360 72. 4 Perspectives - p. 371 73. References - p. 372 74. Chapter 11: Computational approaches for extremal geometric eigenvalue problems - p. 377 75. Abstract - p. 378 76. 1 Introduction - p. page378 77. 2 Maximizing Laplace–Beltrami eigenvalues on closed surfaces - p. 379 78. 3 Maximizing Steklov eigenvalues on compact surfaces with boundary - p. 393 79. 4 Discussion and future directions - p. 402 80. Acknowledgements - p. page403 81. References - p. page403 82. Chapter 12: Unbalanced Optimal Transport, from theory to numerics - p. 407 83. Abstract - p. 407 84. 1 Introduction - p. 408 85. 2 φ-divergences, MMD and Optimal Transport - p. 410 86. 3 Unbalanced Optimal Transport - p. 422 87. 4 Entropic regularization - p. 434 88. 5 Gromov-Wasserstein distances - p. 452 89. 6 Conclusion - p. 462 90. References - p. 463 91. Chapter 13: Numerics and control of conservation laws - p. 473 92. Abstract - p. 474 93. 1 Introduction - p. page473 94. 2 Notation and variational calculus - p. 476 95. 3 Numerical analysis - p. 490 96. 4 Outlook - p. 504 97. Acknowledgements - p. 505 98. References - p. 505 99. Chapter 14: Extensions of ADMM for separable convex optimization problems with linear equality or inequality constraints - p. 511 100. Abstract - p. 512 101. 1 Introduction - p. page512 102. 2 Variational inequality characterization - p. 516 103. 3 Prototypical algorithmic framework - p. 518 104. 4 Primal-dual extension of the ADMM (3) for the model (8) - p. 527 105. 5 Dual-primal extension of ADMM (3) for the model (8) - p. 533 106. 6 Extensions to multiple-block separable convex optimization problems with linear equality or inequality constraints - p. 537 107. 7 Primal-dual extension of the ADMM (3) for the model (67) - p. 541 108. 8 Dual-primal extension of the ADMM (3) for the model (67) - p. 546 109. 9 Panorama - p. 551 110. 10 Conclusions - p. 553 111. References - p. 555 112. Chapter 15: An algorithmic guide for finite-dimensional optimal control problems - p. 559 113. Abstract - p. 560 114. 1 Introduction and statement of the problem - p. page560 115. 2 Direct methods: nonlinear programming - p. page566 116. 3 Indirect approaches: the shooting method - p. 571 117. 4 Hamilton–Jacobi–Bellman approach - p. 587 118. 5 Optimistic planning algorithms - p. 616 119. References - p. 623 120. Index - p. 626
Numerical Control: Part B, Volume 24 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 Control problems in the coefficients and the domain for linear elliptic equations, Computational approaches for extremal geometric eigenvalue problems, Non-overlapping domain decomposition in space and time for PDE-constrained optimal control problems on networks, Feedback Control of Time-dependent Nonlinear PDEs with Applications in Fluid Dynamics, Stabilization of the Navier-Stokes equations - Theoretical and numerical aspects, Reconstruction algorithms based on Carleman estimates, and more.
Other sections cover Discrete time formulations as time discretization strategies in data assimilation, Back and forth iterations/Time reversal methods, Unbalanced Optimal Transport: from Theory to Numerics, An ADMM Approach to the Exact and Approximate Controllability of Parabolic Equations, Nonlocal balance laws -- an overview over recent results, Numerics and control of conservation laws, Numerical approaches for simulation and control of superconducting quantum circuits, and much more.
1. 1 Introduction - p. 2 2. 2 Statement of the problem and relaxed formulation - p. 5 3. 3 Discrete formulation and convergence - p. 14 4. 4 A numerical algorithm - p. 18 5. 5 Some numerical experiments - p. 22 6. Acknowledgements - p. 26 7. References - p. 26 8. Chapter 2: Structure-preserving numerical schemes for Hamiltonian dynamics - p. 29 9. Abstract - p. 30 10. 1 Overview - p. page29 11. 2 Classical properties of Hamiltonian systems - p. 35 12. 3 Symplectic numerical schemes for nonstiff systems - p. 44 13. 4 Backward analysis, stiff systems and numerical resonances - p. 53 14. References - p. 57 15. Chapter 3: Relaxation methods for optimal switching control of PDE-dynamical systems - p. 61 16. Chapter 4: Feedback control of time-dependent nonlinear PDEs with applications in fluid dynamics - p. 77 17. Abstract - p. 78 18. 1 Introduction: feedback control of time-dependent PDEs - p. page78 19. 2 Feedback control in fluid dynamics - p. 87 20. 3 Feedback control of nonlinear PDEs via Riccati equations - p. 109 21. 4 Conclusions and outlook - p. 123 22. References - p. 124 23. Chapter 5: Computation of invariant sets for complex systems - p. 131 24. Abstract - p. 131 25. 1 Introduction - p. 132 26. 2 Set programming formulations for the control of complex systems - p. 133 27. 3 Template-independent set programming - p. page140 28. 4 Four numerical templates - p. 146 29. 5 Comparison of performance for convex bodies - p. 159 30. 6 Conclusion - p. 162 31. References - p. 180 32. Chapter 6: Nonlocal balance laws – an overview over recent results - p. 183 33. Abstract - p. 183 34. 1 Introduction - p. 184 35. 2 Existence & uniqueness of solutions - p. 191 36. 3 Related models – extensions - p. 195 37. 4 Approximating “local” entropy solutions by solutions of nonlocal conservation laws or the singular limit problem - p. 203 38. 5 A numerical scheme without dissipation - p. 208 39. 6 Open problems - p. 210 40. Acknowledgements - p. 212 41. References - p. 212 42. Chapter 7: Nonoverlapping domain decomposition and virtual controls for optimal control problems of p-type on metric graphs - p. 217 43. Abstract - p. 217 44. 1 Introduction - p. 218 45. 2 Optimal control problems - p. 225 46. 3 Nonoverlapping domain decomposition: the concept of virtual controls - p. 235 47. 4 Domain decomposition for the static optimal control problem - p. 246 48. 5 Time-domain decomposition - p. 252 49. 6 Concluding remarks and open problems - p. 256 50. References - p. 257 51. Chapter 8: Filtration techniques for the uniform controllability of semidiscrete hyperbolic equations - p. 261 52. Abstract - p. 262 53. 1 Introduction - p. page261 54. 2 Problem of moments and biorthogonals - p. 263 55. 3 Semidiscrete hyperbolic control problems - p. 267 56. 4 A case study: finite elements for the one dimensional wave equation - p. 272 57. 5 Numerical experiments - p. page291 58. References - p. 295 59. Chapter 9: Discrete-time formulations as time discretization strategies in data assimilation - p. 297 60. Abstract - p. 297 61. 1 Introduction - p. 298 62. 2 Least squares estimation and associated discretization for linear parabolic cases - p. 301 63. 3 Luenberger observer strategies and discretization for linear hyperbolic cases - p. 314 64. 4 Least squares estimation and associated discretization for nonlinear models - p. 326 65. 5 Conclusion - p. 335 66. References - p. page335 67. Chapter 10: Approximation of exact controls for semilinear wave and heat equations through space-time methods - p. 341 68. Abstract - p. 342 69. 1 Introduction - p. page342 70. 2 The wave equation - p. 343 71. 3 The heat equation - p. 360 72. 4 Perspectives - p. 371 73. References - p. 372 74. Chapter 11: Computational approaches for extremal geometric eigenvalue problems - p. 377 75. Abstract - p. 378 76. 1 Introduction - p. page378 77. 2 Maximizing Laplace–Beltrami eigenvalues on closed surfaces - p. 379 78. 3 Maximizing Steklov eigenvalues on compact surfaces with boundary - p. 393 79. 4 Discussion and future directions - p. 402 80. Acknowledgements - p. page403 81. References - p. page403 82. Chapter 12: Unbalanced Optimal Transport, from theory to numerics - p. 407 83. Abstract - p. 407 84. 1 Introduction - p. 408 85. 2 φ-divergences, MMD and Optimal Transport - p. 410 86. 3 Unbalanced Optimal Transport - p. 422 87. 4 Entropic regularization - p. 434 88. 5 Gromov-Wasserstein distances - p. 452 89. 6 Conclusion - p. 462 90. References - p. 463 91. Chapter 13: Numerics and control of conservation laws - p. 473 92. Abstract - p. 474 93. 1 Introduction - p. page473 94. 2 Notation and variational calculus - p. 476 95. 3 Numerical analysis - p. 490 96. 4 Outlook - p. 504 97. Acknowledgements - p. 505 98. References - p. 505 99. Chapter 14: Extensions of ADMM for separable convex optimization problems with linear equality or inequality constraints - p. 511 100. Abstract - p. 512 101. 1 Introduction - p. page512 102. 2 Variational inequality characterization - p. 516 103. 3 Prototypical algorithmic framework - p. 518 104. 4 Primal-dual extension of the ADMM (3) for the model (8) - p. 527 105. 5 Dual-primal extension of ADMM (3) for the model (8) - p. 533 106. 6 Extensions to multiple-block separable convex optimization problems with linear equality or inequality constraints - p. 537 107. 7 Primal-dual extension of the ADMM (3) for the model (67) - p. 541 108. 8 Dual-primal extension of the ADMM (3) for the model (67) - p. 546 109. 9 Panorama - p. 551 110. 10 Conclusions - p. 553 111. References - p. 555 112. Chapter 15: An algorithmic guide for finite-dimensional optimal control problems - p. 559 113. Abstract - p. 560 114. 1 Introduction and statement of the problem - p. page560 115. 2 Direct methods: nonlinear programming - p. page566 116. 3 Indirect approaches: the shooting method - p. 571 117. 4 Hamilton–Jacobi–Bellman approach - p. 587 118. 5 Optimistic planning algorithms - p. 616 119. References - p. 623 120. Index - p. 626
