Exact and approximate controllability for distributed parameter systems: a numerical approach
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2008
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
117 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [430] - 449 |
| Physical Description: | XII, 458 S. graph. Darst. |
| ISBN: | 9780521885720 0521885728 |
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| 650 | 4 | |a Distributed parameter systems |x Mathematical models | |
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| adam_text | EXACT AND APPROXIMATE CONTROLLABILITY FOR DISTRIBUTED PARAMETER SYSTEMS
A NUMERICAL APPROACH ROLANDGLOWINSKI UNIVERSITY OF HOUSTON JACQUES-LOUIS
LIONS COLLEGE DE FRANCE, PARIS * JIWENHE. UNIVERSITY OF HOUSTON
CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE PAGE XI INTRODUCTION 1 1.1
WHAT IT IS ALL ABOUT? 1 1.2 MOTIVATION 2 1.3 TOPOLOGIES AND NUMERICAL
METHODS 3 1.4 CHOICE OF THE CONTROL 4 1.5 RELAXATION OF THE
CONTROLLABILITY NOTION 4 1.6 VARIOUS REMARKS 5 PART I DIFFUSION MODELS 1
DISTRIBUTED AND POINTWISE CONTROL FOR LINEAR DIFFUSION EQUATIONS 9 1.1
FIRST EXAMPLE , -*.. 9 1.2 APPROXIMATE CONTROLLABILITY 12 1.3
FORMULATION OF THE APPROXIMATE CONTROLLABILITY PROBLEM . 14 1.4 DUAL
PROBLEM 15 1.5 DIRECT SOLUTION TO THE DUAL PROBLEM 17 1.6 PENALTY
ARGUMENTS 19 1.7 L 00 COST FUNCTIONS AND BANG-BANG CONTROLS 22 1.8
NUMERICAL METHODS . 2 8 1.9 RELAXATION OF CONTROLLABILITY * 57 1.10
POINTWISE CONTROL . : : 62 1.11 FURTHER REMARKS (I): ADDITIONAL
CONSTRAINTSON THE STATE FUNCTION 96 1.12 FURTHER REMARKS (II): A
BISECTION BASED MEMORY SAVING METHOD FOR THE SOLUTION OF TIME DEPENDENT
CONTROL PROBLEMS BY ADJOINT EQUATION BASED METHODOLOGIES 112 1.13
FURTHER REMARKS (III): A BRIEF INTRODUCTION TO RICCATI EQUATIONS BASED
CONTROL METHODS * * . . * . 117 VIII CONTENTS 2 BOUNDARY CONTROL 124
2.1 DIRICHLET CONTROL (I): FORMULATION OF THE CONTROL PROBLEM 124 2.2
DIRICHLET CONTROL (II): OPTIMALITY CONDITIONS AND DUAL FORMULATIONS 126
2.3 DIRICHLET CONTROL (III): ITERATIVE SOLUTION OF THE CONTROL PROBLEMS
128 2.4 DIRICHLET CONTROL (IV): APPROXIMATION OF THE CONTROL PROBLEMS
133 2.5 DIRICHLET CONTROL (V): ITERATIVE SOLUTION OF THE FULLY DISCRETE
DUAL PROBLEM (2.124) 143 2.6 DIRICHLET CONTROL (VI): NUMERICAL
EXPERIMENTS 146 2.7 NEUMANN CONTROL (I): FORMULATION OF THE CONTROL
PROBLEMS AND SYNOPSIS 155 2.8 NEUMANN CONTROL (II): OPTIMALITY
CONDITIONS AND DUAL FORMULATIONS 163 2.9 NEUMANN CONTROL (III):
CONJUGATE GRADIENT SOLUTION OF THE DUAL PROBLEM (2.192) 176 2.10 NEUMANN
CONTROL (IV): ITERATIVE SOLUTION OF THE , DUAL PROBLEM (2.208), (2.209)
178 2.11 NEUMANN CONTROL OF UNSTABLE PARABOLIC SYSTEMS: A NUMERICAL
APPROACH . 178 2.12 CLOSED-LOOP NEUMANN CONTROL OF UNSTABLE PARABOLIC
SYSTEMS VIA THE RICCATI EQUATION APPROACH , 223 3 CONTROL OF THE
STOKES SYSTEM 231 3.1 GENERALITIES, SYNOPSIS * 231 3.2 FORMULATION OF
THE STOKES SYSTEM. A FUNDAMENTAL CONTROLLABILITY RESULT ***-_.* : .
231 3.3 TWO APPROXIMATE CONTROLLABILITY PROBLEMS , 234 3.4 OPTIMALITY
CONDITIONS AND DUAL PROBLEMS * * * * , * 234 3.5 ITERATIVE SOLUTION OF
THE CONTROL PROBLEM (3.19) ** * * * 236 3.6 TIME DISCRETIZATION OF
THE CONTROL PROBLEM (3.19) 238 3.7 NUMERICAL EXPERIMENTS ,, 239 4
CONTROL OF NONLINEAR DIFFUSION SYSTEMS ,* I 1 243 4.1 GENERALITIES.
SYNOPSIS **** * ** ****/ , ** . 2 4 3 4.2 EXAMPLE OF A
NONCONTROLLABLE NONLINEAR SYSTEM * * * * * * 243 4.3 POINTWISE
CONTROL OF THE VISCOUS BURGER S EQUATION * *** 245 4.4 ON THE
CONTROLLABILITY AND THE STABILIZATION/OF THE ! KURAMOTO-SIVASHINSKY
EQUATION IN ONE SPACE DIMENSION * * 259 5 DYNAMIC PROGRAMMING FOR LINEAR
DIFFUSION EQUATIONS (..;..*:*. 277 5.1 INTRODUCTION. SYNOPSIS 277 5.2
DERIVATION OF THE HAMILTON-JACOBI-BELLMAN EQUATION 278 5.3 SOME REMARKS
279 CONTENTS IX PART II WAVE MODELS 6 WAVE EQUATIONS 283 6.1 WAVE
EQUATIONS: DIRICHLET BOUNDARY CONTROL 283 6.2 APPROXIMATE
CONTROLLABILITY 285 . 6.3 FORMULATION OF THE APPROXIMATE CONTROLLABILITY
PROBLEM 286 6.4 DUAL PROBLEMS 287 6.5 DIRECT SOLUTION OF THE DUAL
PROBLEM 288 6.6 EXACT CONTROLLABILITY AND NEW FUNCTIONAL SPACES 289 6.7
ON THE STRUCTURE OF SPACE E 291 6.8 NUMERICAL METHODS FOR THE DIRICHLET
BOUNDARY CONTROLLABILITY OF THE WAVE EQUATION 291 6.9 EXPERIMENTAL
VALIDATION OF THE FILTERING PROCEDURE OF SECTION 6.8.7 VIA THE SOLUTION
OF THE TEST PROBLEM OF SECTION 6.8.5 315 6.10 SOME REFERENCES ON
ALTERNATIVE APPROXIMATION METHODS 319 6.11 OTHER BOUNDARY CONTROLS 320
6.12 DISTRIBUTED CONTROLS FOR WAVE EQUATIONS 328 6.13 DYNAMIC
PROGRAMMING 329 7 ON THE APPLICATION OF CONTROLLABILITY METHODS TO THE
SOLUTION OF THE HELMHOLTZ EQUATION AT LARGE WAVE NUMBERS 332 7.1
INTRODUCTION 332 7.2 THE HELMHOLTZ EQUATION AND ITS EQUIVALENT WAVE
PROBLEM 332 7.3 EXACT CONTROLLABILITY METHODS FOR THE CALCULATION OF
TIME-PERIODIC SOLUTIONS TO THE WAVE EQUATION 334 7.4 LEAST-SQUARES
FORMULATION OF THE PROBLEM (7.8)-(7.11) 334 7.5 CALCULATION OF J 336
7.6 CONJUGATE GRADIENT SOLUTION OF THE LEAST-SQUARES PROBLEM (7.14) 337
7.7 A FINITE ELEMENT*FINITE DIFFERENCE IMPLEMENTATION 340 7.8 NUMERICAL
EXPERIMENTS 341 7.9 FURTHER COMMENTS. DESCRIPTION OF A MIXED FORMULATION
BASED VARIANT OF THE CONTROLLABILITY METHOD 349 7.10 A FINAL COMMENT 355
OTHER WAVE AND VIBRATION PROBLEMS. COUPLED SYSTEMS 356 8.1 GENERALITIES
AND FURTHER REFERENCES 356 8.2 COUPLED SYSTEMS (I): A PROBLEM FROM
THERMO-ELASTICITY 359 8.3 COUPLED SYSTEMS (II): OTHER SYSTEMS 367 X
CONTENTS PART III FLOW CONTROL 9 OPTIMAL CONTROL OF SYSTEMS MODELLED BY
THE NAVIER-STOKES EQUATIONS: APPLICATION TO DRAG REDUCTION 371 , 9.1
INTRODUCTION. SYNOPSIS 371 9.2 FORMULATION OF THE CONTROL PROBLEM 373
9.3 TIME DISCRETIZATION OF THE CONTROL PROBLEM 377 9.4 FULL
DISCRETIZATION OF THE CONTROL PROBLEM 379 9.5 GRADIENT CALCULATION 384
9.6 A BFGS ALGORITHM FOR SOLVING THE DISCRETE CONTROL PROBLEM 388 9.7
VALIDATION OF THE FLOW SIMULATOR . 389 9.8 ACTIVE CONTROL BY ROTATION ,
394 9.9 ACTIVE CONTROL BY BLOWING AND SUCTION 408 9.10 FURTHER COMMENTS
ON FLOW CONTROL AND CONCLUSION 419 EPILOGUE 426 FURTHER ACKNOWLEDGEMENTS
429 REFERENCES 430 INDEX OF NAMES 450 INDEX OF SUBJECTS 454
|
| adam_txt |
EXACT AND APPROXIMATE CONTROLLABILITY FOR DISTRIBUTED PARAMETER SYSTEMS
A NUMERICAL APPROACH ROLANDGLOWINSKI UNIVERSITY OF HOUSTON JACQUES-LOUIS
LIONS COLLEGE DE FRANCE, PARIS * JIWENHE.' UNIVERSITY OF HOUSTON
CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE PAGE XI INTRODUCTION 1 1.1
WHAT IT IS ALL ABOUT? 1 1.2 MOTIVATION 2 1.3 TOPOLOGIES AND NUMERICAL
METHODS 3 1.4 CHOICE OF THE CONTROL 4 1.5 RELAXATION OF THE
CONTROLLABILITY NOTION 4 1.6 VARIOUS REMARKS 5 PART I DIFFUSION MODELS 1
DISTRIBUTED AND POINTWISE CONTROL FOR LINEAR DIFFUSION EQUATIONS 9 1.1
FIRST EXAMPLE , -*. 9 1.2 APPROXIMATE CONTROLLABILITY 12 1.3
FORMULATION OF THE APPROXIMATE CONTROLLABILITY PROBLEM . 14 1.4 DUAL
PROBLEM 15 1.5 DIRECT SOLUTION TO THE DUAL PROBLEM 17 1.6 PENALTY
ARGUMENTS 19 1.7 L 00 COST FUNCTIONS AND BANG-BANG CONTROLS 22 1.8
NUMERICAL METHODS . 2 8 1.9 RELAXATION OF CONTROLLABILITY * 57 1.10
POINTWISE CONTROL . : : 62 1.11 FURTHER REMARKS (I): ADDITIONAL
CONSTRAINTSON THE STATE FUNCTION 96 1.12 FURTHER REMARKS (II): A
BISECTION BASED MEMORY SAVING METHOD FOR THE SOLUTION OF TIME DEPENDENT
CONTROL PROBLEMS BY ADJOINT EQUATION BASED METHODOLOGIES 112 1.13
FURTHER REMARKS (III): A BRIEF INTRODUCTION TO RICCATI EQUATIONS BASED
CONTROL METHODS * * . ". * . 117 VIII CONTENTS 2 BOUNDARY CONTROL 124
2.1 DIRICHLET CONTROL (I): FORMULATION OF THE CONTROL PROBLEM 124 2.2
DIRICHLET CONTROL (II): OPTIMALITY CONDITIONS AND DUAL FORMULATIONS 126
2.3 DIRICHLET CONTROL (III): ITERATIVE SOLUTION OF THE CONTROL PROBLEMS
128 2.4 DIRICHLET CONTROL (IV): APPROXIMATION OF THE CONTROL PROBLEMS
133 2.5 DIRICHLET CONTROL (V): ITERATIVE SOLUTION OF THE FULLY DISCRETE
DUAL PROBLEM (2.124) 143 2.6 DIRICHLET CONTROL (VI): NUMERICAL
EXPERIMENTS 146 2.7 NEUMANN CONTROL (I): FORMULATION OF THE CONTROL
PROBLEMS AND SYNOPSIS 155 2.8 NEUMANN CONTROL (II): OPTIMALITY
CONDITIONS AND DUAL FORMULATIONS 163 2.9 NEUMANN CONTROL (III):
CONJUGATE GRADIENT SOLUTION OF THE DUAL PROBLEM (2.192) 176 2.10 NEUMANN
CONTROL (IV): ITERATIVE SOLUTION OF THE , DUAL PROBLEM (2.208), (2.209)
178 2.11 NEUMANN CONTROL OF UNSTABLE PARABOLIC SYSTEMS: A NUMERICAL
APPROACH . 178 2.12 CLOSED-LOOP NEUMANN CONTROL OF UNSTABLE PARABOLIC
SYSTEMS VIA THE RICCATI EQUATION APPROACH , 223 3 CONTROL OF THE
STOKES SYSTEM 231 3.1 GENERALITIES, SYNOPSIS * 231 3.2 FORMULATION OF
THE STOKES SYSTEM. A FUNDAMENTAL CONTROLLABILITY RESULT ***-_.* : ' .
231 3.3 TWO APPROXIMATE CONTROLLABILITY PROBLEMS , 234 3.4 OPTIMALITY
CONDITIONS AND DUAL PROBLEMS * * * * , * 234 3.5 ITERATIVE SOLUTION OF
THE CONTROL PROBLEM (3.19) ' ** * *'* ' 236 3.6 TIME DISCRETIZATION OF
THE CONTROL PROBLEM (3.19) ' 238 3.7 NUMERICAL EXPERIMENTS '' ,, 239 4
CONTROL OF NONLINEAR DIFFUSION SYSTEMS ,* I 1 243 4.1 GENERALITIES.
SYNOPSIS '**** * ** ****/', **". ' 2 4 3 4.2 EXAMPLE OF A
NONCONTROLLABLE NONLINEAR SYSTEM * ' * '*'* *" ''* 243 4.3 POINTWISE
CONTROL OF THE VISCOUS BURGER'S EQUATION ' '*''*** ' 245 4.4 ON THE
CONTROLLABILITY AND THE STABILIZATION/OF THE ! ' ' KURAMOTO-SIVASHINSKY
EQUATION IN ONE SPACE DIMENSION *'* 259 5 DYNAMIC PROGRAMMING FOR LINEAR
DIFFUSION EQUATIONS '(.;.*:*. 277 5.1 INTRODUCTION. SYNOPSIS 277 5.2
DERIVATION OF THE HAMILTON-JACOBI-BELLMAN EQUATION 278 5.3 SOME REMARKS
279 CONTENTS IX PART II WAVE MODELS 6 WAVE EQUATIONS 283 6.1 WAVE
EQUATIONS: DIRICHLET BOUNDARY CONTROL 283 6.2 APPROXIMATE
CONTROLLABILITY 285 . 6.3 FORMULATION OF THE APPROXIMATE CONTROLLABILITY
PROBLEM 286 6.4 DUAL PROBLEMS 287 6.5 DIRECT SOLUTION OF THE DUAL
PROBLEM 288 6.6 EXACT CONTROLLABILITY AND NEW FUNCTIONAL SPACES 289 6.7
ON THE STRUCTURE OF SPACE E 291 6.8 NUMERICAL METHODS FOR THE DIRICHLET
BOUNDARY CONTROLLABILITY OF THE WAVE EQUATION 291 6.9 EXPERIMENTAL
VALIDATION OF THE FILTERING PROCEDURE OF SECTION 6.8.7 VIA THE SOLUTION
OF THE TEST PROBLEM OF SECTION 6.8.5 315 ' 6.10 SOME REFERENCES ON
ALTERNATIVE APPROXIMATION METHODS 319 6.11 OTHER BOUNDARY CONTROLS 320
6.12 DISTRIBUTED CONTROLS FOR WAVE EQUATIONS 328 6.13 DYNAMIC
PROGRAMMING 329 7 ON THE APPLICATION OF CONTROLLABILITY METHODS TO THE
SOLUTION OF THE HELMHOLTZ EQUATION AT LARGE WAVE NUMBERS 332 7.1
INTRODUCTION 332 7.2 THE HELMHOLTZ EQUATION AND ITS EQUIVALENT WAVE
PROBLEM 332 7.3 EXACT CONTROLLABILITY METHODS FOR THE CALCULATION OF
TIME-PERIODIC SOLUTIONS TO THE WAVE EQUATION 334 7.4 LEAST-SQUARES
FORMULATION OF THE PROBLEM (7.8)-(7.11) 334 7.5 CALCULATION OF J' 336
7.6 CONJUGATE GRADIENT SOLUTION OF THE LEAST-SQUARES PROBLEM (7.14) 337
7.7 A FINITE ELEMENT*FINITE DIFFERENCE IMPLEMENTATION 340 7.8 NUMERICAL
EXPERIMENTS 341 7.9 FURTHER COMMENTS. DESCRIPTION OF A MIXED FORMULATION
BASED VARIANT OF THE CONTROLLABILITY METHOD 349 7.10 A FINAL COMMENT 355
OTHER WAVE AND VIBRATION PROBLEMS. COUPLED SYSTEMS 356 8.1 GENERALITIES
AND FURTHER REFERENCES 356 8.2 COUPLED SYSTEMS (I): A PROBLEM FROM
THERMO-ELASTICITY 359 8.3 COUPLED SYSTEMS (II): OTHER SYSTEMS 367 X
CONTENTS PART III FLOW CONTROL 9 OPTIMAL CONTROL OF SYSTEMS MODELLED BY
THE NAVIER-STOKES EQUATIONS: APPLICATION TO DRAG REDUCTION 371 , 9.1
INTRODUCTION. SYNOPSIS 371 9.2 FORMULATION OF THE CONTROL PROBLEM 373
9.3 TIME DISCRETIZATION OF THE CONTROL PROBLEM 377 9.4 FULL
DISCRETIZATION OF THE CONTROL PROBLEM 379 9.5 GRADIENT CALCULATION 384
9.6 A BFGS ALGORITHM FOR SOLVING THE DISCRETE CONTROL PROBLEM 388 9.7
VALIDATION OF THE FLOW SIMULATOR . 389 9.8 ACTIVE CONTROL BY ROTATION ,
394 9.9 ACTIVE CONTROL BY BLOWING AND SUCTION 408 9.10 FURTHER COMMENTS
ON FLOW CONTROL AND CONCLUSION 419 EPILOGUE 426 FURTHER ACKNOWLEDGEMENTS
429 REFERENCES 430 INDEX OF NAMES 450 INDEX OF SUBJECTS 454 |
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| author | Glowinski, Roland 1937-2022 Lions, Jacques-Louis 1928-2001 He, Jiwen |
| author_GND | (DE-588)120514737 (DE-588)124055397 |
| author_facet | Glowinski, Roland 1937-2022 Lions, Jacques-Louis 1928-2001 He, Jiwen |
| author_role | aut aut aut |
| author_sort | Glowinski, Roland 1937-2022 |
| author_variant | r g rg j l l jll j h jh |
| building | Verbundindex |
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| callnumber-label | QA402 |
| callnumber-raw | QA402.3 |
| callnumber-search | QA402.3 |
| callnumber-sort | QA 3402.3 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 880 |
| ctrlnum | (OCoLC)166357726 (DE-599)OBVAC06653614 |
| dewey-full | 515/.642 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.642 |
| dewey-search | 515/.642 |
| dewey-sort | 3515 3642 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. publ. |
| format | Book |
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| index_date | 2024-07-02T20:16:44Z |
| indexdate | 2024-07-09T21:13:27Z |
| institution | BVB |
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| oclc_num | 166357726 |
| open_access_boolean | |
| owner | DE-824 DE-706 DE-634 |
| owner_facet | DE-824 DE-706 DE-634 |
| physical | XII, 458 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Glowinski, Roland 1937-2022 Verfasser (DE-588)120514737 aut Exact and approximate controllability for distributed parameter systems a numerical approach Roland Glowinski ; Jacques-Louis Lions ; Jiwen He 1. publ. Cambridge Cambridge Univ. Press 2008 XII, 458 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 117 Literaturverz. S. [430] - 449 Mathematisches Modell Control theory Distributed parameter systems Mathematical models Kontrolltheorie (DE-588)4032317-1 gnd rswk-swf System mit verteilten Parametern (DE-588)4058803-8 gnd rswk-swf System mit verteilten Parametern (DE-588)4058803-8 s Kontrolltheorie (DE-588)4032317-1 s DE-604 Lions, Jacques-Louis 1928-2001 Verfasser (DE-588)124055397 aut He, Jiwen Verfasser aut Encyclopedia of mathematics and its applications 117 (DE-604)BV000903719 117 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016408677&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Glowinski, Roland 1937-2022 Lions, Jacques-Louis 1928-2001 He, Jiwen Exact and approximate controllability for distributed parameter systems a numerical approach Encyclopedia of mathematics and its applications Mathematisches Modell Control theory Distributed parameter systems Mathematical models Kontrolltheorie (DE-588)4032317-1 gnd System mit verteilten Parametern (DE-588)4058803-8 gnd |
| subject_GND | (DE-588)4032317-1 (DE-588)4058803-8 |
| title | Exact and approximate controllability for distributed parameter systems a numerical approach |
| title_auth | Exact and approximate controllability for distributed parameter systems a numerical approach |
| title_exact_search | Exact and approximate controllability for distributed parameter systems a numerical approach |
| title_exact_search_txtP | Exact and approximate controllability for distributed parameter systems a numerical approach |
| title_full | Exact and approximate controllability for distributed parameter systems a numerical approach Roland Glowinski ; Jacques-Louis Lions ; Jiwen He |
| title_fullStr | Exact and approximate controllability for distributed parameter systems a numerical approach Roland Glowinski ; Jacques-Louis Lions ; Jiwen He |
| title_full_unstemmed | Exact and approximate controllability for distributed parameter systems a numerical approach Roland Glowinski ; Jacques-Louis Lions ; Jiwen He |
| title_short | Exact and approximate controllability for distributed parameter systems |
| title_sort | exact and approximate controllability for distributed parameter systems a numerical approach |
| title_sub | a numerical approach |
| topic | Mathematisches Modell Control theory Distributed parameter systems Mathematical models Kontrolltheorie (DE-588)4032317-1 gnd System mit verteilten Parametern (DE-588)4058803-8 gnd |
| topic_facet | Mathematisches Modell Control theory Distributed parameter systems Mathematical models Kontrolltheorie System mit verteilten Parametern |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016408677&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000903719 |
| work_keys_str_mv | AT glowinskiroland exactandapproximatecontrollabilityfordistributedparametersystemsanumericalapproach AT lionsjacqueslouis exactandapproximatecontrollabilityfordistributedparametersystemsanumericalapproach AT hejiwen exactandapproximatecontrollabilityfordistributedparametersystemsanumericalapproach |