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2011 | Buch

Overview and Organization Kapitel lesen Erstes Kapitel lesen

Viability Theory

New Directions

verfasst von: Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Viability theory designs and develops mathematical and algorithmic methods for investigating the adaptation to viability constraints of evolutions governed by complex systems under uncertainty that are found in many domains involving living beings, from biological evolution to economics, from environmental sciences to financial markets, from control theory and robotics to cognitive sciences. It involves interdisciplinary investigations spanning fields that have traditionally developed in isolation. The purpose of this book is to present an initiation to applications of viability theory, explaining and motivating the main concepts and illustrating them with numerous numerical examples taken from various fields.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Overview and Organization
Abstract
Viability theory designs and develops mathematical and algorithmic methods for investigating the adaptation to viability constraints of evolutions governed by complex systems under uncertainty that are found in many domains involv- ing living beings, from biological evolution to economics, from environmental sciences to financial markets, from control theory and robotics to cognitive sciences. It involves interdisciplinary investigations spanning fields that have traditionally developed in isolation.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre

Viability Kernels and Examples

Frontmatter
Chapter 2. Viability and Capturability
Abstract
This rather long chapter is the central one. It is aimed at allowing the reader to grasp enough concepts and statements of the principal results proved later on in the book to read directly and independently most of the chapters of the book:
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 3. Viability Problems in Robotics
Abstract
This chapter studies three applications to robotics, one focussing on field experiments of the viability feedback allowing a robot to rally a target in a urban environment while avoiding obstacles, the second one dealing with the safety envelope of the landing of a plane as well as the regulation law governing the safe landing evolutions viable in this envelope, and the third one focused on navigation of submarines in rivers.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 4. Viability and Dynamic Intertemporal Optimality
Abstract
We consider throughout this chapter the parameterized system
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 5. Avoiding Skylla and Charybdis
Abstract
This chapter provides a same viability framework with two-dimensional nonsmooth environments and targets and a nonlinear control system for which we illustrate and compare basic concepts, such as minimal length and exit time functions, minimal time, Lyapunov and value function of an optimal control problem.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 6. Inertia Functions, Viability Oscillators and Hysteresis
Abstract
This chapter is devoted to the original motivation of viability theory, which was (and still is) an attempt to mathematically capture some central ideas of evolution of species of Darwinian type, turning around the concept of “punctuated equilibrium” of Eldredge and Gould.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 7. Management of Renewable Resources
Abstract
This chapter is devoted to some problems dealing with births, growth, and survival of populations.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre

Mathematical Properties of Viability Kernels

Frontmatter
Chapter 8. Connection Basins
Abstract
Until now, we presented and studied evolutions in positive time, or forward evolutions, and the associated concepts of (forward) viability kernels and basins. We were looking from the present to the future, without taking into account the past or the history of the evolution.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 9. Local and Asymptotic Properties of Equilibria
Abstract
This chapter expends to the case of control and regulated systems some central concepts of “dynamical systems”, dealing with local and asymptotic stability, attractors, Lyapunov stability and sensitivity to initial conditions, equilibria, Newton’s methods for finding them and the inverse function theorem for studying their stability.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 10. Viability and Capturability Properties of Evolutionary Systems
Abstract
This chapter presents properties proved at the level of evolutionary systems, whereas Chap.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 11. Regulation of Control Systems
Abstract
This chapter is devoted to viability properties specific to evolutionary 5 systems generated by control system of the form
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 12. Restoring Viability
Abstract
There is no reason why an arbitrary subset K should be viable under a given control system. The introduction of the concept of viability kernel does not exhaust the problem of restoring viability by keeping the same dynamics of the control system and “shrinking” the environment to its viability kernel. We devote this chapter to two other methods for restoring viability without changing the environment.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre

First-Order Partial Differential Equations

Frontmatter
Chapter 13. Viability Solutions to Hamilton–Jacobi Equations
Abstract
This chapter presents the viability approach to a class of Hamilton-Jacobi equations. We assume not only that the solution depends on time, but on “structured” or “causal” variables. They include age-structured Hamilton-Jacobi-McKendrick equations, useful in population dynamics as well as in transport management (the age variable being replaced by the travel time), as well as Hamilton-Jacobi-Cournot equation, where the “structured” or “causal” variable is the initial state of the underlying control system. Chapters 14, p. 565 and 15, p. 605 apply the results of this chapter to transportation management, finance and economics.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 14. Regulation of Traffic
Abstract
The advent of techniques to measure velocities of probe vehicles using GPS technology, for instance, complementing or replacing fixed sensing infras- tructures such as density sensors of the road traffic sensors, motivates the revision of conceptual, mathematical algorithms and software based models used by the transportation engineering community.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 15. Illustrations in Finance and Economics
Abstract
This chapter describes two problems motivated by financial mathematics (implicit evaluation of the volatility of portfolios) and of economic theory (bridging the gap between micro and macro economics). These are selected examples chosen for their intrinsic interest and for illustrating how viability concepts and theorems can be used to solve these questions. The focus of this chapter is not the place to expose and develop more examples.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 16. Viability Solutions to Conservation Laws
Abstract
Several chapters (Chaps. 13,p. 525, 14,p. 525, 17,p. 525,and Sects. 15,p. and 525) are devoted to first-order Hamilton-Jacobi-Bellman partial differential equations. This chapter presets a viability approach to another class of partial differential equations, conservation laws. We restrict our study to the Burgers equation (the canonical example of conservation laws) in Sect. 16.2,p. 525 for illustrating this approach.We also include in this chapter the short Sect. 16.3, p.569 to a generalization of the Invariant Manifold Theorem to control problems which plays an important of control theory. It is an addendum to Chap. 8 of the first edition of Viability Theory [18,A ubin] (1991) which is not repeated in this second edition.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 17. Viability Solutions to Hamilton–Jacobi–Bellman Equations
Abstract
We summarized the main results of the Hamilton-Jacobi-Bellman strategy to study intertemporalo ptimization in the “optimalco ntrol survival kit”, Sect. 4.11, p. 168. Chapters 4, p. 125 and 14, p. 525 and Sects. 15.3, p. 527 and 15.4, p. 542 provided many examples of value functions of a series of optimization problems over state-control pairs solutions to control systems which were characterized in terms of viability kernels and capture basins of auxiliary systems, which we referred to as viability episolutions.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre

Appendices

Frontmatter
Chapter 18. Set-Valued Analysis at a Glance
Abstract
The purpose of this chapter is to present a short introduction to the main concepts of Set-Valued Analysis used throughout this book, and regarded here as a “Toolbox” for viability theory, optimal control, differential games and their applications to mathematical economics and finance.
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Chapter 19. Convergence and Viability Theorems
Abstract
This chapter is mainly devoted to the proof of a series of Viability Theorems leading to Theorem 11.3.4, p.393. This is done by approximating the differential inclusion by the simple explicit finite-difference scheme (the Euler method). In this setting, being a discrete scheme, the viability characterization of an environment is trivial (see Theorem 2.9.3, p.72).
Jean-Pierre Aubin, Alexandre M. Bayen, Patrick Saint-Pierre
Backmatter
Metadaten
Titel
Viability Theory
verfasst von
Jean-Pierre Aubin
Alexandre M. Bayen
Patrick Saint-Pierre
Copyright-Jahr
2011
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-16684-6
Print ISBN
978-3-642-16683-9
DOI
https://doi.org/10.1007/978-3-642-16684-6

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