Handbook of numerical analysis: Volume 23 Numerical control: Part A
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| Beschreibung: | xxii, 572 Seiten Illustrationen, Diagramme |
| ISBN: | 9780323850599 |
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| adam_text | Contents Contributors Preface 1. xvii xxi Control and numerical approximation of fractional diffusion equations Umberto Biccari, Mahamadi Warna, and Enrique Zuazua 1 Introduction 2 Finite Element approximation of the fractional Laplace operator 2.1 Computation of the stiffness matrix 2.2 Error analysis 2.3 Numerical experiments 3 Interior controllability properties of the fractional heat equation 3.1 Review of theoretical controllability results 3.2 The penalized Hilbert Uniqueness Method 3.3 Numerical experiments 4 Exterior controllability properties of the fractional heat equation 4.1 Review of theoretical controllability results 4.2 Numerical experiments 5 Simultaneous control of parameter-dependent fractional heat equations 5.1 Problem formulation 5.2 CD and SCD approaches 5.3 Practical considerations on the implementation of GD and CG 5.4 Numerical experiments 6 Conclusion and open problems Acknowledgments Appendix A Fractional order Sobolev spaces and the fractional Laplacian Appendix В The fractional Laplace operator with exterior conditions References 2 4 6 11 12 12 13 15 19 26 27 30 33 34 35 41 41 44 47 47 51 54 vii
viii 2. Contents Contents Modeling, control, and numerics of gas networks 4. Martin Cugat and Michael Herty 1 Introduction 2 Modeling of gas flow 2.1 Models for gas flow on edges 2.2 Models for the flow through vertices 3 Well-posedness of mathematical models for fixed control action 3.1 Classical solutions 3.2 Weak solutions 4 Control and controllability 4.1 Optimal control 4.2 Controllability 4.3 Feedback stabilization 5 Uncertainty quantification 6 Numerical methods for simulation and control 6.1 Discretization of coupling conditions for finite volume schemes 6.2 Discretization of stabilization problems 6.3 Numerical methods for optimal control problems 7 Open problems References 3. Optimal control of PDEs and FE-approximation Eduardo Casas, Karl Kunisch, and Fredi Tröltzsch 60 61 61 63 66 66 68 70 70 72 73 75 77 77 78 80 80 81 Optimal control, numerics, and applications of fractional PDEs Harbir Antil, Thomas Brown, Ratna Khatri, Akwum Onwunta, Deepanshu Verma, and Mahamadi Warna 1 Introduction and applications of fractional operators 2 Two fractional operators and their properties 2.1 Fractional Laplacian (֊Δ)-5 and nonlocal normal derivative Ms 2.2 Fractional time derivative: çdj 3 Fractional diffusion equation: analysis and numerical approximation 3.1 Non-homogeneous diffusion equations:analytic results 3.2 Non-homogeneous diffusion equations: numerical approximation 3.3 Fractional time derivative: 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 Acknowledgments References ix 88 92 92 94 94 94 96 100 101 103 107 111 111 111 Introduction 1 The L1 23 framework 456 1.1 The control problem 1.2 Existence of solutions and first order optimality conditions 1.3 Second order optimality conditions 1.4 Finite element approximation of (P) 2 Controlling with measures 2.1 The control problem 2.2 Finite element approximation of measure-valued control problem 3 Related topics References 5. 116 117 117 121 128 134 146 146 149 155 158 Numerical solution of multi-objective optimal control and hierarchic controllability problems Enrique Fernández-Cara 1 Introduction 2 Bi-objective control problems for heat and wave equations Equilibria for linear heat equations 2.1 Computation of Nash equilibria 2.2 Computation of Pareto equilibria 2.3 Nash and Pareto bi-objective optimal control for 2.4 semilinear heat equations Equilibria for linear and semilinear wave equations 2.5 A numerical experiment concerning bi-objective optimal 2.6 control 3 Stackelberg strategies and hierarchical control problems A strategy for the computation of null controls 3.1 Mixed formulations and numerical approximations of 3.2 Stackelberg-Nash control problems A numerical experiment concerning Stackelberg-Nash 3.3 control 4 Additional comments and conclusions Stackelberg-Nash controllability for semilinear and 4.1 nonlinear equations On the numerical approximation of (47) 4.2 More extensions 4.3 References 166 168 169 171 175 176 179 181 184 186 189 193 196 196 196 197 197 Numerics
for stochastic distributed parameter control systems: a finite transposition method Qi LÜ, Penghui Wang, Yanqing Wang, and Xu Zhang 1 Introduction 202
x Contents 2 Dual equations for stochastic distributed parameter control problems 3 The space of finite transposition 4 Finite transposition method for backward stochastic evolution equations 4.1 Spacial discretization 4.2 Temporal discretization 5 Numerical method for optimalcontrols References 7. Contents xi 2 Introduction 3 Nonlinear time dependent parametrized optimal flow control problems 3.1 Problem formulation 3.2 The space-time approximation 4 ROMs for nonlinear space-time OCP(/t)s 4.1 General reduction strategy 4.2 Offline and online phase: from space-time POD algorithm for OCP(/¿)s to Galerkin projection 5 Application to shallow waters equations 5.1 Main motivations and problem formulation 5.2 Numerical results 6 Conclusions Acknowledgments References 203 209 214 215 216 226 230 Numerical solutions of stochastic control problems: Markov chain approximation methods Zhuo Jin, Ky Tran, and George Yin 1 Stochastic control problems 2 Methods of Markov chain approximation 2.1 Controlled switching diffusions 2.2 Singular control 3 Application to insurance 4 Application to mathematical biology 5 Final remarks References 8. 234 238 238 244 247 255 262 262 10. 9. Space-time POD-Galerkin approach for parametric flow control Francesco Balların, Gianluigi Rozza, and Maria Strazzullo 1 Motivations and historical background 308 320 324 324 327 333 334 334 Moments and convex optimization for analysis and control of nonlinear PDEs 1 Introduction 2 Problem statement (analysis) 3 Occupation measures for nonlinear PDEs 3.1 Linear representation 3.2 Infinite-dimensional linear program
4 Computable bounds using SDP relaxations 5 Problem statement (control) 6 Linear representation (control) 6.1 Infinite dimensional LP (control) 7 Control design using SDP relaxations 7.1 Controller extraction 8 Higher-order PDEs 9 Numerical examples 9.1 Burgers equation ֊ analysis 9.2 Burgers equation ֊ control 10 Conclusion Acknowledgments References Martin Lazar and Jérôme Lohéac 266 269 271 279 284 286 292 296 298 300 300 300 301 302 303 310 310 313 318 319 Milan Korda, Didier Henrion, and Jean Bernard Lasserre Control of parameter dependent systems Λ Introduction 2 Parameter invariant controls 2.1 Averaged controllability 2.2 Ensemble controllability 3 Parameter dependent controls 3.1 Selection of a reduced basis 3.2 The online procedure 3.3 Other approaches 4 Conclusion Acknowledgments Appendix A Proof of technical results related to Section 2 A.1 Proof of Lemma 2.6 A.2 Estimates related to Example 2.11 A.3 Proof of Lemma 2.15 References 309 11. 340 343 346 347 350 353 354 355 357 358 359 360 361 361 362 363 364 364 Turnpike properties in optimal control Timm Faulwasser and Lars Grüne 1 Introduction and historical origins From Ramsey and von Neumann to aircraft design Notation 1.1 Discrete-time and continuous-time setting 1.2 Optimality conditions 368 368 370 370 371
xii Contents Contents 2 Definition and taxonomy of turnpike properties 2.1 2.2 Motivating example Turnpike definitions 3 Generating mechanisms 3.1 3.2 Necessary and sufficient conditions for turnpikes Finite and infinite-horizon cases 4 Exploitation of turnpikes in numerics and receding-horizon control 5 Topics not discussed and open problems Acknowledgment References 12. 374 375 376 380 384 387 390 393 396 396 1.1 General introduction 2.1 2.2 Qualitative analysis Numerical investigations 3 Maximizing the total population size 3.1 Qualitative analysis Numerical investigations 3.2 4 Generalization and perspectives 4.1 4.2 References 13. Spectral optimization for the biased movement of species Open problems 402 402 408 409 412 414 415 420 420 421 423 425 Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctions Leon Bungert and Martin Burger 1 Introduction 2 Convex analysis and nonlinear eigenvalue problems 3 Gradient flows and decrease of Rayleigh quotients 3.1 3.2 The p-gradient flow and minimizing movements Decrease of the Rayleigh quotients 4 Flows for solving nonlinear eigenproblems 4.1 4.2 4.3 428 432 434 435 436 438 Rescaled p-flows Rescaled gradient flows Normalized gradient flows Rayleigh quotient minimizing flows 439 439 441 Normalized gradient flows and power methods Analysis of nonlinear power methods 445 446 6 Г-convergence implies convergence of ground states 449 4.4 5 Nonlinear power methods for homogeneous functionals 5.1 5.2 457 457 458 460 462 Dynamic Programming versus supervised learning 1 Introduction 2 A model problem
Idriss Mazarí, Grégoire Nadin, and Yannick Privat 2 Optimal eigenvalue problem Appendices Appendix A Exact reconstruction time Appendix В Extinction time Appendix C Remaining proofs References 453 453 454 455 Gilles Pagès and Olivier Pironneau Some challenging optimization problems for logistic diffusive equations and their numerical modeling 1 Introduction and bio-mathematical background 7 Applications 7.1 Calibrable sets 7.2 Graph clustering 7.3 Geodesic distance functions on graphs xiii 442 444 2.1 2.2 2.3 2.4 Existence of solution: the deterministic case Existence of solution: the stochasticnon-dynamic case Existence of solution: the dynamiccase Discretization 3 Brute force solution of the non-dynamic control problem by Monte-Carlo 3.1 3.2 Solution with a gradient method Implementation and examples 4 Solution of the non-dynamic control problem by supervised learning 4.1 A numerical test 5 Bellman s Stochastic Dynamic Programming for the dynamic problem 6 Solution with the Hamilton-Jacobi-Bellman partial differential equations 6.1 6.2 6.3 Numerical results Analysis when ß = 0 Discretization 7 Solution with the Kolmogorov equation 7.1 7.2 Computation of gradients Results 8 Solution by Itô calculus 8.1 Gradient computation 9 Limit with vanishing volatility 10 Conclusion Acknowledgments Appendix A A model with fishing quota Appendix В Reformulation B.1 More constraints on quotas Appendix C An analytical solution for a similar problem References 468 469 470 471 472 473 473 474 475 475 477 478 481 483 484 485 485 488 489 490 491 492 492 493 493 494 495 496 497
XIV 15. Contents Contents Data-driven modeling and control of large-scale dynamical systems in the Loewner framework Non-parametric approach for the value function and its gradient 6 Focus on the deterministic case 6.1 Problem and algorithm 6.2 Splitting up method 7 Convergence results 7.1 Setting of the problem 7.2 Preliminaries 7.3 Main result 7.4 Algorithm 7.5 Linear quadratic case 8 Numerical results Acknowledgment References 5.3 Ion Victor Cosea, Charles Poussot-Vassal, and Athanasios C. Antonias 1 Introduction: data-driven modeling and control 2 The Loewner framework for data-driven modeling: an overview 2.1 Generalities on the Loewner framework and model structures 2.2 The Loewner framework in the LTI case 2.3 Generalizations to parametric linear systems 2.4 Generalization to modeling from time-domain data 2.5 Extensions to nonlinear systems 3 Model reduction examples (large-scale systems) 3.1 Gust load oriented generic business jet aircraft model 3.2 Ground vibration tests on business jet aircraft 3.3 Hydroelectricity open-channel benchmark 4 Control in the Loewner framework 4.1 Data-driven control, virtual reference model and L-DDC rationale 4.2 Pulsed fluidic actuator control 4.3 Transport phenomena benchmark 5 Summary and conclusions References 16. 500 502 502 504 507 510 514 518 518 520 521 523 523 524 525 527 528 Machine learning and control theory Alain Bensoussan, Ylqun Li, Dinh Phan Cao Nguyen, Minh-Binh Tran, Sheung Chi Phillip Yam, and Xiang Zhou 1 Introduction 2 Reinforcement learning 2.1 General concepts 2.2 Mathematical model without action 2.3
Approximation 2.4 Mathematical model with action 3 Control theory and deep learning 3.1 Supervised learning 3.2 Deep learning 3.3 Control theory approach 4 Stochastic gradient descent and control theory 4.1 Comments 4.2 Stochastic gradient and MDP 4.3 Continuous version 5 Machine learning approach of stochastic control problems 5.1 General theory 5.2 Parametric approach for the feedback xv 532 535 535 535 536 537 540 540 540 541 542 542 542 543 544 544 545 Index 546 546 547 548 549 549 549 550 554 554 554 556 557 559
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| adam_txt |
Contents Contributors Preface 1. xvii xxi Control and numerical approximation of fractional diffusion equations Umberto Biccari, Mahamadi Warna, and Enrique Zuazua 1 Introduction 2 Finite Element approximation of the fractional Laplace operator 2.1 Computation of the stiffness matrix 2.2 Error analysis 2.3 Numerical experiments 3 Interior controllability properties of the fractional heat equation 3.1 Review of theoretical controllability results 3.2 The penalized Hilbert Uniqueness Method 3.3 Numerical experiments 4 Exterior controllability properties of the fractional heat equation 4.1 Review of theoretical controllability results 4.2 Numerical experiments 5 Simultaneous control of parameter-dependent fractional heat equations 5.1 Problem formulation 5.2 CD and SCD approaches 5.3 Practical considerations on the implementation of GD and CG 5.4 Numerical experiments 6 Conclusion and open problems Acknowledgments Appendix A Fractional order Sobolev spaces and the fractional Laplacian Appendix В The fractional Laplace operator with exterior conditions References 2 4 6 11 12 12 13 15 19 26 27 30 33 34 35 41 41 44 47 47 51 54 vii
viii 2. Contents Contents Modeling, control, and numerics of gas networks 4. Martin Cugat and Michael Herty 1 Introduction 2 Modeling of gas flow 2.1 Models for gas flow on edges 2.2 Models for the flow through vertices 3 Well-posedness of mathematical models for fixed control action 3.1 Classical solutions 3.2 Weak solutions 4 Control and controllability 4.1 Optimal control 4.2 Controllability 4.3 Feedback stabilization 5 Uncertainty quantification 6 Numerical methods for simulation and control 6.1 Discretization of coupling conditions for finite volume schemes 6.2 Discretization of stabilization problems 6.3 Numerical methods for optimal control problems 7 Open problems References 3. Optimal control of PDEs and FE-approximation Eduardo Casas, Karl Kunisch, and Fredi Tröltzsch 60 61 61 63 66 66 68 70 70 72 73 75 77 77 78 80 80 81 Optimal control, numerics, and applications of fractional PDEs Harbir Antil, Thomas Brown, Ratna Khatri, Akwum Onwunta, Deepanshu Verma, and Mahamadi Warna 1 Introduction and applications of fractional operators 2 Two fractional operators and their properties 2.1 Fractional Laplacian (֊Δ)-5 and nonlocal normal derivative Ms 2.2 Fractional time derivative: çdj 3 Fractional diffusion equation: analysis and numerical approximation 3.1 Non-homogeneous diffusion equations:analytic results 3.2 Non-homogeneous diffusion equations: numerical approximation 3.3 Fractional time derivative: 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 Acknowledgments References ix 88 92 92 94 94 94 96 100 101 103 107 111 111 111 Introduction 1 The L1 23 framework 456 1.1 The control problem 1.2 Existence of solutions and first order optimality conditions 1.3 Second order optimality conditions 1.4 Finite element approximation of (P) 2 Controlling with measures 2.1 The control problem 2.2 Finite element approximation of measure-valued control problem 3 Related topics References 5. 116 117 117 121 128 134 146 146 149 155 158 Numerical solution of multi-objective optimal control and hierarchic controllability problems Enrique Fernández-Cara 1 Introduction 2 Bi-objective control problems for heat and wave equations Equilibria for linear heat equations 2.1 Computation of Nash equilibria 2.2 Computation of Pareto equilibria 2.3 Nash and Pareto bi-objective optimal control for 2.4 semilinear heat equations Equilibria for linear and semilinear wave equations 2.5 A numerical experiment concerning bi-objective optimal 2.6 control 3 Stackelberg strategies and hierarchical control problems A strategy for the computation of null controls 3.1 Mixed formulations and numerical approximations of 3.2 Stackelberg-Nash control problems A numerical experiment concerning Stackelberg-Nash 3.3 control 4 Additional comments and conclusions Stackelberg-Nash controllability for semilinear and 4.1 nonlinear equations On the numerical approximation of (47) 4.2 More extensions 4.3 References 166 168 169 171 175 176 179 181 184 186 189 193 196 196 196 197 197 Numerics
for stochastic distributed parameter control systems: a finite transposition method Qi LÜ, Penghui Wang, Yanqing Wang, and Xu Zhang 1 Introduction 202
x Contents 2 Dual equations for stochastic distributed parameter control problems 3 The space of finite transposition 4 Finite transposition method for backward stochastic evolution equations 4.1 Spacial discretization 4.2 Temporal discretization 5 Numerical method for optimalcontrols References 7. Contents xi 2 Introduction 3 Nonlinear time dependent parametrized optimal flow control problems 3.1 Problem formulation 3.2 The space-time approximation 4 ROMs for nonlinear space-time OCP(/t)s 4.1 General reduction strategy 4.2 Offline and online phase: from space-time POD algorithm for OCP(/¿)s to Galerkin projection 5 Application to shallow waters equations 5.1 Main motivations and problem formulation 5.2 Numerical results 6 Conclusions Acknowledgments References 203 209 214 215 216 226 230 Numerical solutions of stochastic control problems: Markov chain approximation methods Zhuo Jin, Ky Tran, and George Yin 1 Stochastic control problems 2 Methods of Markov chain approximation 2.1 Controlled switching diffusions 2.2 Singular control 3 Application to insurance 4 Application to mathematical biology 5 Final remarks References 8. 234 238 238 244 247 255 262 262 10. 9. Space-time POD-Galerkin approach for parametric flow control Francesco Balların, Gianluigi Rozza, and Maria Strazzullo 1 Motivations and historical background 308 320 324 324 327 333 334 334 Moments and convex optimization for analysis and control of nonlinear PDEs 1 Introduction 2 Problem statement (analysis) 3 Occupation measures for nonlinear PDEs 3.1 Linear representation 3.2 Infinite-dimensional linear program
4 Computable bounds using SDP relaxations 5 Problem statement (control) 6 Linear representation (control) 6.1 Infinite dimensional LP (control) 7 Control design using SDP relaxations 7.1 Controller extraction 8 Higher-order PDEs 9 Numerical examples 9.1 Burgers'equation ֊ analysis 9.2 Burgers'equation ֊ control 10 Conclusion Acknowledgments References Martin Lazar and Jérôme Lohéac 266 269 271 279 284 286 292 296 298 300 300 300 301 302 303 310 310 313 318 319 Milan Korda, Didier Henrion, and Jean Bernard Lasserre Control of parameter dependent systems Λ Introduction 2 Parameter invariant controls 2.1 Averaged controllability 2.2 Ensemble controllability 3 Parameter dependent controls 3.1 Selection of a reduced basis 3.2 The online procedure 3.3 Other approaches 4 Conclusion Acknowledgments Appendix A Proof of technical results related to Section 2 A.1 Proof of Lemma 2.6 A.2 Estimates related to Example 2.11 A.3 Proof of Lemma 2.15 References 309 11. 340 343 346 347 350 353 354 355 357 358 359 360 361 361 362 363 364 364 Turnpike properties in optimal control Timm Faulwasser and Lars Grüne 1 Introduction and historical origins From Ramsey and von Neumann to aircraft design Notation 1.1 Discrete-time and continuous-time setting 1.2 Optimality conditions 368 368 370 370 371
xii Contents Contents 2 Definition and taxonomy of turnpike properties 2.1 2.2 Motivating example Turnpike definitions 3 Generating mechanisms 3.1 3.2 Necessary and sufficient conditions for turnpikes Finite and infinite-horizon cases 4 Exploitation of turnpikes in numerics and receding-horizon control 5 Topics not discussed and open problems Acknowledgment References 12. 374 375 376 380 384 387 390 393 396 396 1.1 General introduction 2.1 2.2 Qualitative analysis Numerical investigations 3 Maximizing the total population size 3.1 Qualitative analysis Numerical investigations 3.2 4 Generalization and perspectives 4.1 4.2 References 13. Spectral optimization for the biased movement of species Open problems 402 402 408 409 412 414 415 420 420 421 423 425 Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctions Leon Bungert and Martin Burger 1 Introduction 2 Convex analysis and nonlinear eigenvalue problems 3 Gradient flows and decrease of Rayleigh quotients 3.1 3.2 The p-gradient flow and minimizing movements Decrease of the Rayleigh quotients 4 Flows for solving nonlinear eigenproblems 4.1 4.2 4.3 428 432 434 435 436 438 Rescaled p-flows Rescaled gradient flows Normalized gradient flows Rayleigh quotient minimizing flows 439 439 441 Normalized gradient flows and power methods Analysis of nonlinear power methods 445 446 6 Г-convergence implies convergence of ground states 449 4.4 5 Nonlinear power methods for homogeneous functionals 5.1 5.2 457 457 458 460 462 Dynamic Programming versus supervised learning 1 Introduction 2 A model problem
Idriss Mazarí, Grégoire Nadin, and Yannick Privat 2 Optimal eigenvalue problem Appendices Appendix A Exact reconstruction time Appendix В Extinction time Appendix C Remaining proofs References 453 453 454 455 Gilles Pagès and Olivier Pironneau Some challenging optimization problems for logistic diffusive equations and their numerical modeling 1 Introduction and bio-mathematical background 7 Applications 7.1 Calibrable sets 7.2 Graph clustering 7.3 Geodesic distance functions on graphs xiii 442 444 2.1 2.2 2.3 2.4 Existence of solution: the deterministic case Existence of solution: the stochasticnon-dynamic case Existence of solution: the dynamiccase Discretization 3 Brute force solution of the non-dynamic control problem by Monte-Carlo 3.1 3.2 Solution with a gradient method Implementation and examples 4 Solution of the non-dynamic control problem by supervised learning 4.1 A numerical test 5 Bellman's Stochastic Dynamic Programming for the dynamic problem 6 Solution with the Hamilton-Jacobi-Bellman partial differential equations 6.1 6.2 6.3 Numerical results Analysis when ß = 0 Discretization 7 Solution with the Kolmogorov equation 7.1 7.2 Computation of gradients Results 8 Solution by Itô calculus 8.1 Gradient computation 9 Limit with vanishing volatility 10 Conclusion Acknowledgments Appendix A A model with fishing quota Appendix В Reformulation B.1 More constraints on quotas Appendix C An analytical solution for a similar problem References 468 469 470 471 472 473 473 474 475 475 477 478 481 483 484 485 485 488 489 490 491 492 492 493 493 494 495 496 497
XIV 15. Contents Contents Data-driven modeling and control of large-scale dynamical systems in the Loewner framework Non-parametric approach for the value function and its gradient 6 Focus on the deterministic case 6.1 Problem and algorithm 6.2 Splitting up method 7 Convergence results 7.1 Setting of the problem 7.2 Preliminaries 7.3 Main result 7.4 Algorithm 7.5 Linear quadratic case 8 Numerical results Acknowledgment References 5.3 Ion Victor Cosea, Charles Poussot-Vassal, and Athanasios C. Antonias 1 Introduction: data-driven modeling and control 2 The Loewner framework for data-driven modeling: an overview 2.1 Generalities on the Loewner framework and model structures 2.2 The Loewner framework in the LTI case 2.3 Generalizations to parametric linear systems 2.4 Generalization to modeling from time-domain data 2.5 Extensions to nonlinear systems 3 Model reduction examples (large-scale systems) 3.1 Gust load oriented generic business jet aircraft model 3.2 Ground vibration tests on business jet aircraft 3.3 Hydroelectricity open-channel benchmark 4 Control in the Loewner framework 4.1 Data-driven control, virtual reference model and L-DDC rationale 4.2 Pulsed fluidic actuator control 4.3 Transport phenomena benchmark 5 Summary and conclusions References 16. 500 502 502 504 507 510 514 518 518 520 521 523 523 524 525 527 528 Machine learning and control theory Alain Bensoussan, Ylqun Li, Dinh Phan Cao Nguyen, Minh-Binh Tran, Sheung Chi Phillip Yam, and Xiang Zhou 1 Introduction 2 Reinforcement learning 2.1 General concepts 2.2 Mathematical model without action 2.3
Approximation 2.4 Mathematical model with action 3 Control theory and deep learning 3.1 Supervised learning 3.2 Deep learning 3.3 Control theory approach 4 Stochastic gradient descent and control theory 4.1 Comments 4.2 Stochastic gradient and MDP 4.3 Continuous version 5 Machine learning approach of stochastic control problems 5.1 General theory 5.2 Parametric approach for the feedback xv 532 535 535 535 536 537 540 540 540 541 542 542 542 543 544 544 545 Index 546 546 547 548 549 549 549 550 554 554 554 556 557 559 |
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| genre | (DE-588)4143413-4 Aufsatzsammlung gnd-content |
| genre_facet | Aufsatzsammlung |
| id | DE-604.BV047891246 |
| illustrated | Illustrated |
| index_date | 2024-07-03T19:26:15Z |
| indexdate | 2024-07-10T09:24:22Z |
| institution | BVB |
| isbn | 9780323850599 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-033273355 |
| oclc_num | 1310260606 |
| open_access_boolean | |
| owner | DE-384 DE-706 DE-739 |
| owner_facet | DE-384 DE-706 DE-739 |
| physical | xxii, 572 Seiten Illustrationen, Diagramme |
| publishDate | 2022 |
| publishDateSearch | 2022 |
| publishDateSort | 2022 |
| publisher | North-Holland |
| record_format | marc |
| spelling | Handbook of numerical analysis Volume 23 Numerical control: Part A general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions Amsterdam North-Holland [2022] xxii, 572 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf (DE-588)4143413-4 Aufsatzsammlung gnd-content Numerische Mathematik (DE-588)4042805-9 s DE-604 Trélat, Emmanuel (DE-588)1256438812 edt Ciarlet, Philippe G. 1938- Sonstige (DE-588)143368362 oth Zuazua, Enrique 1961- (DE-588)1231285508 edt Lions, Jacques-Louis 1928-2001 Sonstige (DE-588)124055397 oth Du, Qiang 1964- (DE-588)1188249320 edt (DE-604)BV002745459 23 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033273355&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Handbook of numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4143413-4 |
| title | Handbook of numerical analysis |
| title_auth | Handbook of numerical analysis |
| title_exact_search | Handbook of numerical analysis |
| title_exact_search_txtP | Handbook of numerical analysis |
| title_full | Handbook of numerical analysis Volume 23 Numerical control: Part A general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_fullStr | Handbook of numerical analysis Volume 23 Numerical control: Part A general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_full_unstemmed | Handbook of numerical analysis Volume 23 Numerical control: Part A general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_short | Handbook of numerical analysis |
| title_sort | handbook of numerical analysis numerical control part a |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerische Mathematik Aufsatzsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033273355&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002745459 |
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