Verfügbarkeit: Viability theory

Viability theory: new directions

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Hauptverfasser:	Aubin, Jean-Pierre (VerfasserIn), Bayen, Alexandre M. (VerfasserIn), Saint-Pierre, Patrick (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin [u.a.] Springer 2011
Ausgabe:	2nd. ed.
Schlagworte:	Mengenwertige Abbildung Regelungssystem Unsicherheit Evolution Differentialinklusion
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XXI, 803 S. graph. Darst.
ISBN:	9783642166839

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Datensatz im Suchindex

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adam_text	Contents Overview and Organization .............................. 1 1.1 Motivations .......................................... 2 1.1.1 Chance and Necessity .......................... 2 1.1.2 Motivating Applications ........................ 6 1.1.3 Motivations of Viability Theory from Living Systems ...................................... 7 1.1.4 Applications of Viability Theory to... Mathematics .................................. 10 1.2 Main Concepts of Viability Theory ...................... 11 1.2.1 Viability Kernels and Capture Basins Under Regulated Systems ............................. 12 1.2.2 Viability Kernel and Capture Basin Algorithms .... 18 1.2.3 Restoring Viability ............................. 19 1.3 Examples of Simple Viability Problems .................. 19 1.3.1 Engineering Applications ........................ 20 1.3.2 Environmental Illustrations ..................... 25 1.3.3 Strange Attractors and Fractals .................. 28 1.4 Organization of the Book .............................. 33 1.4.1 Overall Organization ........................... 33 1.4.2 Qualitative Concepts ........................... 34 1.4.3 Quantitative Concepts .......................... 36 1.4.4 First-Order Partial Differential Equations ......... 37 1.4.5 Social Sciences Illustrations ..................... 37 1.4.6 Engineering Applications ........................ 38 1.4.7 The Four Parts of the Monograph ................ 39 x Contents Part I Viability Kernels and Examples 2 Viability and Capturability .............................. 43 2.1 Introduction .......................................... 43 2.2 Evolutions ........................................... 45 2.2.1 Stationary and Periodic Evolutions ............... 46 2.2.2 Transient Evolutions ........................... 48 2.2.3 Viable and Capturing Evolutions ................. 49 2.3 Discrete Systems ...................................... 50 2.4 Differential Equations .................................. 53 2.4.1 Determinism and Predictability .................. 53 2.4.2 Example of Differential Equations: The Lorenz System ............................. 56 2.5 Régulons and Tyches .................................. 58 2.6 Discrete Nondeterministic Systems ...................... 62 2.7 Retroactions of Parameterized Dynamical Systems ........ 64 2.7.1 Parameterized Dynamical Systems ............... 64 2.7.2 Retroactions .................................. 65 2.7.3 Differential Inclusions .......................... 66 2.8 Evolutionary Systems .................................. 68 2.9 Viability Kernels and Capture Basins for Discrete Time Systems .............................. 71 2.9.1 Definitions .................................... 71 2.9.2 Viability Kernel Algorithms ..................... 74 2.9.3 Julia and Mandelbrot Sets ...................... 75 2.9.4 Viability Kernels under Disconnected Discrete Systems and Fractals ........................... 79 2.10 Viability Kernels and Capture Basins for Continuous Time Systems ........................................ 85 2.10.1 Definitions .................................... 85 2.10.2 Viability Kernels under the Lorenz System ........ 87 2.11 Invariance Kernel under a Tychastic System .............. 88 2.12 Links between Kernels and Basins ....................... 92 2.13 Local Viability and Invariance .......................... 94 2.14 Discretization Issues ................................... 96 2.15 A Viability Survival Kit ................................ 98 2.15.1 Bilateral Fixed Point Characterization ............ 99 2.15.2 Viability Characterization ....................... 100 2.15.3 Tangential Characterization ..................... 101 3 Viability Problems in Robotics ........................... 105 3.1 Introduction .......................................... 105 3.2 Fields Experiment with the Pioneer ..................... 105 Contents xi 3.3 Safety Envelopes in Aeronautics .........................108 3.3.1 Landing Aircraft: Application of the Capture Basin Algorithm ...............................109 3.3.2 Computation of the Safe Flight Envelope and the Viability Feedback ......................115 3.4 Path Planing for a Light Autonomous Underwater Vehicle ....................................119 3.4.1 Dynamics, Environment and Target ..............119 3.4.2 Experimental Results ...........................122 4 Viability and Dynamic Intertemporal Optimality ........125 4.1 Introduction .......................................... 125 4.2 Intertemporal Criteria .................................129 4.2.1 Examples of Functionals ........................129 4.2.2 Extended Functions and Hidden Constraints ......131 4.3 Exit and Minimal Time Functions .......................132 4.3.1 Viability and Capturability Tubes ................ 132 4.3.2 Exit and Minimal Time Functions ................ 134 4.4 Minimal Length Function .............................. 139 4.5 Attracting and Exponential Lyapunov for Asymptotic Stability ...............................142 4.5.1 Attracting Functions ...........................142 4.5.2 Exponential Lyapunov Functions .................145 4.5.3 The Montagnes Russes Algorithm for Global Optimization ..................................149 4.6 Safety and Transgression Functions ......................152 4.7 Infinite Horizon Optimal Control and Viability Kernels .... 155 4.8 Intergenerational Optimization ..........................158 4.9 Finite Horizon Optimal Control and Capture Basins .......162 4.10 Occupational Costs and Measures .......................164 4.11 Optimal Control Survival Kit ...........................168 4.12 Links Between Viability Theory and Other Problems ......171 4.12.1 Micro Macro Dynamic Consistency ............. . 171 4.12.2 The Epigraphical Miracle .......................172 4.12.3 First-Order Systems of Partial Differential Equations .....................................174 4.12.4 Intertemporal Optimization Under State Constraints ....................................1.76 4.12.5 The Viability Strategy for Dynamic Optimization ..................................177 5 Avoiding Skylla and Charybdis ..........................179 5.1 Introduction: The Zermelo Navigation Problem ...........179 5.2 The Odyssey .........................................181 xii Contents 5.3 Examples of Concepts and Numerical Computations .......182 5.3.1 Dynamical Morphological Structure of the Environment and its Behavior .............182 5.3.2 Behavior in the Environment: Minimal Length and Persistent Evolutions ....................... 184 5.3.3 Behavior In and Out the Harbor: Minimal Time, Lyapunov and Optimal Evolutions ............... 185 5.4 Viewgraphs Describing Five Value Functions .............. 187 6 Inertia Functions, Viability Oscillators and Hysteresis .... 199 6.1 Introduction ..........................................199 6.2 Prerequisites for Intertemporal Optimization .............202 6.3 Punctuated Equilibria and the Inertia Principle ...........203 6.4 Inertia Functions. Metasystems and Viability Niches .......207 6.4.1 The Inertia Cascade ............................208 6.4.2 First-Order Inertia Functions ....................210 6.4.3 Inert Regulation and Critical Maps ...............216 6.4.4 Heavy Evolutions ..............................217 6.4.5 The Adjustment Map and Heavy Evolutions .......221 6.4.6 High-Order Metasystems ........................226 6.4.7 The Transition Cost Function ...................231 6.5 Viability Oscillators and Hysterons ......................233 6.5.1 Persistent Oscillator and Preisach Hysteron .......235 6.5.2 Viability Oscillators and Hysterons ...............238 6.6 Controlling Constraints and Targets .....................242 7 Management of Renewable Resources ....................247 7.1 Introduction ..........................................247 7.2 Evolution of the Biomass of a Renewable Resource ........248 7.2.1 From Malthus to Verhuist and Beyond ............248 7.2.2 The Inverse Approach ..........................252 7.2.3 The Inert Hysteresis Cycle ......................257 7.3 Management of Renewable Resources ....................262 7.3.1 Inert Evolutions ...............................266 7.3.2 Heavy Evolutions ..............................267 7.3.3 The Crisis Function ............................269 Part II Mathematical Properties of Viability Kernels 8 Connection Basins .......................................273 8.1 Introduction ..........................................273 8.2 Past and Future Evolutions .............................275 8.3 Bilateral Viability Kernels ..............................279 8.3.1 Forward and Backward Viability Kernels under the Lorenz System ........................282 Contents xiii 8.4 Detection Tubes ......................................284 8.4.1 Reachable Maps ...............................284 8.4.2 Detection and Cournot Tubes ...................286 8.4.3 Volterra Inclusions .............................289 8.5 Connection Basins and Eupalinian Kernels ...............291 8.5.1 Connection Basins .............................291 8.6 Collision Kernels ......................................298 8.7 Particular Solutions to a Differential Inclusion ............301 8.8 Visiting Kernels and Chaos A la Saari ...................302 8.9 Contingent Temporal Logic .............................305 8.10 Time Dependent Evolutionary Systems ..................309 8.10.1 Links Between Time Dependent and Independent Systems .......................309 8.10.2 Reachable Maps and Detectors ..................312 8.11 Observation. Measurements and Identification Tubes ......316 8.11.1 Anticipation Tubes .............................317 9 Local and Asymptotic Properties of Equilibria ...........319 9.1 Introduction ..........................................319 9.2 Permanence and Fluctuation ...........................321 9.2.1 Permanence Kernels and Fluctuation Basins ....... 322 9.2.2 The Crisis Function ............................ 326 9.2.3 Cascades of Environments ....................... 328 9.2.4 Fluctuation Between Monotonie Cells ............ 331 9.2.5 Hofbauer-Sigmund Permanence .................. 338 9.2.6 Heteroclines and Homoclines .................... 341 9.3 Asymptotic Behavior: Limit Sets, Attractors and Viability Kernels ..................................344 9.3.1 Limit Sets and Attractor ........................344 9.3.2 Attraction and Cluster Basins ...................346 9.3.3 Viability Properties of Limit Sets and Attractors ... 351 9.3.4 Nonemptyness of Viability Kernels ...............353 9.4 Lyapunov Stability and Sensitivity to Initial Conditions .... 354 9.4.1 Lyapunov Stability ............................. 354 9.4.2 The Sensitivity Functions ....................... 355 9.4.3 Topologically Transitive Sets and Dense Trajectories ................................... 358 9.5 Existence of Equilibria ................................. 360 9.6 Newton s Methods for Finding Equilibria ................. 361 9.6.1 Behind the Newton Method .....................361 9.6.2 Building Newton Algorithms ....................363 9.7 Stability: The Inverse Set-Valued Map Theorem ...........365 9.7.1 The Inverse Set-Valued Map Theorem ............365 9.7.2 .Metric Regularity ..............................367 XIV Contents 9.7.3 Local and Pointwise Norms of Graphical Derivatives ....................................370 9.7.4 Norms of Derivatives and Metric Regularity .......371 10 Viability and Capturability Properties of Evolutionary Systems ..................................................375 10.1 Introduction ..........................................375 10.2 Bilateral Fixed Point Characterization of Kernels and Basins ...........................................377 10.2.1 Bilateral Fixed Point Characterization of Viability Kernels ............................379 10.2.2 Bilateral Fixed Point Characterization of Invariance Kernels ...........................382 10.3 Topological Properties .................................382 10.3.1 Continuity Properties of Evolutionary Systems .....382 10.3.2 Topological Properties of Viability Kernels and Capture Basins ............................387 10.4 Persistent Evolutions and Exit Sets ......................392 10.4.1 Persistent and Minimal Time Evolutions ..........392 10.4.2 Temporal Window .............................396 10.4.3 Exit Sets and Local Viability ....................396 10.5 Viability Characterizations of Kernels and Basins .........399 10.5.1 Subsets Viable Outside a Target .................399 10.5.2 Relative Invariance .............................401 10.5.3 Isolated Subsets ...............................403 10.5.4 The Second Fundamental Characterization Theorem ......................................405 10.5.5 The Barrier Property ...........................407 10.6 Other Viability Characterizations .......................411 10.6.1 Characterization of Invariance Kernels ............411 10.6.2 Characterization of Connection Basins ............413 10.7 Stability of Viability and Invariance Kernels ..............416 10.7.1 Kernels and Basins of Limits of Environments .....416 10.7.2 Invariance and Viability Envelopes ...............420 10.8 The Hard Version of the Inertia Principle ................422 10.9 Parameter Identification: Inverse Viability and Invariance Maps ..................................427 10.9.1 Inverse Viability and Invariance ..................427 10.9.2 Level Tubes of Extended Functions ...............429 10.10 Stochastic and Tychastic Viability .......................433 11 Regulation of Control Systems ...........................437 11.1 Introduction ..........................................437 11.2 Tangential Conditions for Viability and Invariance .........440 11.2.1 The Nagumo Theorem ..........................443 11.2.2 Integrating Differential Inequalities ...............445 Contents xv 11.2.3 Characterization of the Viability Tangential Condition .....................................447 11.2.4 Characterization of the Invariance Tangential Condition .....................................451 11.3 Fundamental Viability and Invariance Theorems for Control Systems ...................................453 11.3.1 The Regulation Map for Control Systems .........453 11.3.2 The Fundamental Viability Theorem .............454 11.3.3 The Fundamental Invariance Theorem ............457 11.4 Regulation Maps of Kernels and Basins ..................460 11.5 Lax-Hopf Formula for Capture Basins ...................465 11.5.1 P-Convex and Exhaustive Sets ..................465 11.5.2 Lax-Hopf Formulas ............................468 11.5.3 Viability Kernels Under P-Valued Micro—Macro Systems ..........................472 11.6 Hamiltonian Characterization of the Regulation Map ......475 11.7 Deriving Viable Feedbacks .............................480 11.7.1 Static Viable Feedbacks .........................480 11.7.2 Dynamic Viable Feedbacks ......................483 12 Restoring Viability .......................................485 12.1 Introduction ..........................................485 12.2 Viability Multipliers ...................................485 12.2.1 Definition of Viability Multipliers ................485 12.2.2 Viability Multipliers with Minimal Norm ..........486 12.2.3 Viability Multipliers and Variational Inequalities . . . 488 12.2.4 Slow Evolutions of Variational Inequalities ........491 12.2.5 Case of Explicit Constraints .....................491 12.2.6 Meta Variational Inequalities ....................494 12.2.7 Viability Multipliers for Bilateral Constraints ......495 12.2.8 Connectionnist Complexity ......................497 12.2.9 Hierarchical Viability ...........................501 12.3 Impulse and Hybrid Systems ...........................503 12.3.1 Runs of Impulse Systems .......................503 12.3.2 Impulse Evolutionary Systems ...................507 12.3.3 Hvbrid Control Systems and Differential Inclusions ..................................... 510 12.3.4 Substratum of an Impulse System ................ 512 12.3.5 Impulse Viability Kernels ....................... 516 12.3.6 Stability Properties ............................ 518 12.3.7 Cadenced Runs ................................ 519 xvi Contents Part III First-Order Partial Differential Equations 13 Viability Solutions to Hamilton—Jacobi Equations ........ 523 13.1 Introduction .......................................... 523 13.2 From Duality to Trinity ................................ 524 13.3 Statement of the Problem .............................. 527 13.3.1 Lagrangian and Hamiltonian .................... 527 13.3.2 The Viability Solution .......................... 530 13.4 Variational Principle ................................... 531 13.4.1 Lagrangian Microsystems ....................... 531 13.4.2 The Variational Principle ....................... 532 13.5 Viability Implies Optimality ............................ 538 13.5.1 Optimal Evolutions ............................ 538 13.5.2 Dynamic Programming under Viability Constraints ................................... 539 13.6 Regulation of Optimal Evolutions ....................... 541 13.7 Aggregation of Hamiltonians ............................ 544 13.7.1 Aggregation ................................... 544 13.7.2 Composition of a Hamiltonian by a Linear Operator ........................... 546 13.8 Hamilton—Jacobi—Cournot Equations .................... 551 13.9 Lax— Hopf Formula .................................... 554 13.10 Barron-Jensen/Frankowska Viscosity Solution ............ 557 14 Regulation of Traffic ..................................... 563 14.1 Introduction .......................................... 563 14.2 The Transportation Problem and its Viability Solution ..... 566 14.3 Density and Celerity Flux Functions ..................... 571 14.4 The Viability Traffic Function .......................... 579 14.5 Analytical Properties of the Viability Traffic Functions ..... 584 14.5.1 Cauchy Initial Conditions ....................... 586 14.5.2 Lagrangian Traffic Conditions ................... 587 14.5.3 Combined Traffic Conditions .................... 590 14.6 Decreasing Envelopes of Functions ...................... 592 14.7 The Min-Inf Convolution Morphism Property and Decreasing Traffic Functions ........................ 595 14.8 The Traffic Regulation Map ............................ 599 14.9 From Density to Celerity: Shifting Paradigms ............. 601 15 Illustrations in Finance and Economics ................... 603 15.1 Introduction .......................................... 603 15.2 Uncovering Implicit Volatility in Macroscopic Portfolio Properties ................................... 605 15.2.1 Managing a Portfolio under Unknown Volatility. . . . 607 15.2.2 Implicit Portfolio Problem ...................... 608 Contents xvii 15.2.3 Emergence of the Tychastic Domain and the Cost Function ..........................611 15.2.4 The Viability Portfolio Value ....................612 15.2.5 Managing the Portfolio .........................615 15.3 Bridging the Micro—Macro Economic Information Divide . . . 620 15.3.1 Two Stories for the Same Problem ...............620 15.3.2 The Viability Economic Value ...................624 16 Viability Solutions to Conservation Laws .................631 16.1 Introduction ..........................................631 16.2 Viability Solutions to Conservation Laws .................631 16.2.1 Viabili tv Solution as Solution to Burgers Equation ......................................635 16.2.2 Viability and Tracking Solutions .................637 16.2.3 Piecewise Cauchy Conditions ....................641 16.2.4 Dirichlet Boundary Conditions ...................648 16.2.5 Piecewise Dirichlet Conditions ...................653 16.2.6 Cauchy/Dirichlet Condition .....................654 16.2.7 Additional Eulerian Conditions ..................659 16.2.8 Viability Constraints ...........................662 16.2.9 Lagrangian Conditions ..........................664 16.2.10 Regulating the Burgers Controlled Problem .......667 16.3 The Invariant Manifold Theorem ........................675 17 Viability Solutions to Hamilton—Jacobi—Bellman Equations ................................................681 17.1 Introduction ..........................................681 17.2 Viability Solutions to Hamilton—Jacobi—Bellman Problems .............................................682 17.3 Valuation Functions and Viability Solutions ..............685 17.3.1 A Class of Intertemporal Functionals ............. 685 17.3.2 Valuation Functions and Viability Solutions ....... 687 17.3.3 Existence of Optimal Evolution .................. 692 17.3.4 Dynamic Programming Equation ................ 696 17.4 Solutions to Hamilton-Jacobi Equation and Viability Solutions .................................698 17.4.1 Contingent Solution to the Hamilton—Jacobi Equation ......................................698 17.4.2 В ar ron— Jensen/ Franko ws ka Solution to the Hamilton--Jacobi Equation ............... . 701 17.4.3 The Regulation Map of Optimal Evolutions .......704 17.5 Other Intertemporal Optimization Problems ..............704 17.5.1 Maximization Problems .........................704 17.5.2 Two Other Classes of Optimization Problems ......705 17.6 Summary: Valuation Functions of Four Classes of Intertemporal Optimization Problems .................708 xviii Contents Part IV Appendices 18 Set-Valued Analysis at a Glance .........................713 18.1 Introduction ..........................................713 18.2 Notations ............................................713 18.3 Set-Valued Maps ......................................719 18.3.1 Graphical Approach ............................722 18.3.2 Amalgams of Set-Valued Maps ...................724 18.4 Limits of Sets .........................................727 18.4.1 Upper and Lower Limits ........................727 18.4.2 Upper Semicompact and Upper and Lower Semi-Continuous Maps .........................729 18.4.3 Tangent Cones ................................731 18.4.4 Polar Cones ...................................734 18.4.5 Projectors .....................................735 18.4.6 Normals ......................................737 18.5 Graphical Analysis ....................................738 18.5.1 Graphical Convergence of Maps .................. 738 18.5.2 Derivatives of Set-Valued Maps .................. 739 18.5.3 Derivative of Numerical Set-Valued Maps ......... 741 18.6 Epigraphical Analysis .................................. 742 18.6.1 Extended Functions ............................742 18.6.2 Epidifferential Calculus .........................747 18.6.3 Generalized Gradients ..........................751 18.6.4 Tangent and Normal Cones to Epigraphs .........753 18.7 Convex Analysis: Moreau—Rockafellar Subdifferentials ......755 18.8 Weighted Inf- Convolution ..............................762 18.9 The Graal of the Ultimate Derivative ....................765 19 Convergence and Viability Theorems ....................769 19.1 Introduction ..........................................769 19.2 The Fundamental Convergence Theorems ................770 19.3 Upper Convergence of Viability Kernels and Regulation Maps ..................................774 19.3.1 Prolongation of Discrete Time Evolutions .........774 19.3.2 Upper Convergence ............................776 19.4 Proofs of Viability Theorems ...........................781 19.4.1 Necessary Condition for Viability ................ 781 19.4.2 The General Viability Theorem .................. 782 19.4.3 The Local Viability Theorem .................... 783 19.4.4 Proof of the Viability Theorem 11.3.4............ 785 19.4.5 Pointwise Characterization of Differential Inclusions .....................................786 References ...................................................789 Index ........................................................797
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author	Aubin, Jean-Pierre Bayen, Alexandre M. Saint-Pierre, Patrick
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id	DE-604.BV039539315
illustrated	Illustrated
indexdate	2024-07-10T00:05:48Z
institution	BVB
isbn	9783642166839
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-024391438
oclc_num	748696496
open_access_boolean
owner	DE-11 DE-19 DE-BY-UBM DE-706 DE-739
owner_facet	DE-11 DE-19 DE-BY-UBM DE-706 DE-739
physical	XXI, 803 S. graph. Darst.
publishDate	2011
publishDateSearch	2011
publishDateSort	2011
publisher	Springer
record_format	marc
spelling	Aubin, Jean-Pierre Verfasser aut Viability theory new directions Jean-Pierre Aubin ; Alexandre M. Bayen ; Patrick Saint-Pierre 2nd. ed. Berlin [u.a.] Springer 2011 XXI, 803 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mengenwertige Abbildung (DE-588)4270772-9 gnd rswk-swf Regelungssystem (DE-588)4134712-2 gnd rswk-swf Unsicherheit (DE-588)4186957-6 gnd rswk-swf Evolution (DE-588)4071050-6 gnd rswk-swf Differentialinklusion (DE-588)4149777-6 gnd rswk-swf Regelungssystem (DE-588)4134712-2 s Evolution (DE-588)4071050-6 s Unsicherheit (DE-588)4186957-6 s Differentialinklusion (DE-588)4149777-6 s Mengenwertige Abbildung (DE-588)4270772-9 s DE-604 Bayen, Alexandre M. Verfasser aut Saint-Pierre, Patrick Verfasser (DE-588)135610389 aut 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=024391438&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Aubin, Jean-Pierre Bayen, Alexandre M. Saint-Pierre, Patrick Viability theory new directions Mengenwertige Abbildung (DE-588)4270772-9 gnd Regelungssystem (DE-588)4134712-2 gnd Unsicherheit (DE-588)4186957-6 gnd Evolution (DE-588)4071050-6 gnd Differentialinklusion (DE-588)4149777-6 gnd
subject_GND	(DE-588)4270772-9 (DE-588)4134712-2 (DE-588)4186957-6 (DE-588)4071050-6 (DE-588)4149777-6
title	Viability theory new directions
title_auth	Viability theory new directions
title_exact_search	Viability theory new directions
title_full	Viability theory new directions Jean-Pierre Aubin ; Alexandre M. Bayen ; Patrick Saint-Pierre
title_fullStr	Viability theory new directions Jean-Pierre Aubin ; Alexandre M. Bayen ; Patrick Saint-Pierre
title_full_unstemmed	Viability theory new directions Jean-Pierre Aubin ; Alexandre M. Bayen ; Patrick Saint-Pierre
title_short	Viability theory
title_sort	viability theory new directions
title_sub	new directions
topic	Mengenwertige Abbildung (DE-588)4270772-9 gnd Regelungssystem (DE-588)4134712-2 gnd Unsicherheit (DE-588)4186957-6 gnd Evolution (DE-588)4071050-6 gnd Differentialinklusion (DE-588)4149777-6 gnd
topic_facet	Mengenwertige Abbildung Regelungssystem Unsicherheit Evolution Differentialinklusion
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024391438&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT aubinjeanpierre viabilitytheorynewdirections AT bayenalexandrem viabilitytheorynewdirections AT saintpierrepatrick viabilitytheorynewdirections

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