Viability Theory
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| Author | : Jean Pierre Aubin |
| Publisher | : |
| Total Pages | : 584 |
| Release | : 1991 |
| Genre | : Differential inclusions |
| ISBN | : |
This work examines viability theory and its applications to control theory and differential games. The emphasis is on the construction of feedbacks and dynamical systems by myopic optimization methods. Systems of first-order partial differential inclusions, whose solutions are feedbacks, are constructed and investigated. Basic results are then extended to the case of fuzzy control problems, distributed control problems, and control systems with delays and memory. Aimed at graduate students and research mathematicians, both pure and applied, this book offers specialists in control and nonlinear systems tools to take into account general state constraints. Viability theory also allows researchers in other disciplinesâartificial intelligence, economics, game theory, theoretical biology, population genetics, cognitive sciencesâto go beyond deterministic models by studying them in a dynamical or evolutionary perspective in an uncertain environment. "The book is a compendium of the state of knowledge about viability...Mathematically, the book should be accessible to anyone who has had basic graduate courses in modern analysis and functional analysisâ¦The concepts are defined and many proofs of the requisite results are reproduced here, making the present book essentially self-contained." (Bulletin of the AMS) "Because of the wide scope, the book is an ideal reference for people encountering problems related to viability theory in their researchâ¦It gives a very thorough mathematical presentation. Very useful for anybody confronted with viability constraints." (Mededelingen van het Wiskundig Genootschap)
| Author | : Jean-Pierre Aubin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 558 |
| Release | : 2009-05-28 |
| Genre | : Science |
| ISBN | : 0817649107 |
"The book is a compendium of the state of knowledge about viability...Mathematically, the book should be accessible to anyone who has had basic graduate courses in modern analysis and functional analysis...The concepts are defined and many proofs of the requisite results are reproduced here, making the present book essentially self-contained." —Bulletin of the AMS "Because of the wide scope, the book is an ideal reference for people encountering problems related to viability theory in their research...It gives a very thorough mathematical presentation. Very useful for anybody confronted with viability constraints." —Mededelingen van het Wiskundig Genootschap
| Author | : J.-P. Aubin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 353 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642695124 |
A great impetus to study differential inclusions came from the development of Control Theory, i.e. of dynamical systems x'(t) = f(t, x(t), u(t)), x(O)=xo "controlled" by parameters u(t) (the "controls"). Indeed, if we introduce the set-valued map F(t, x)= {f(t, x, u)}ueu then solutions to the differential equations (*) are solutions to the "differen tial inclusion" (**) x'(t)EF(t, x(t)), x(O)=xo in which the controls do not appear explicitely. Systems Theory provides dynamical systems of the form d x'(t)=A(x(t)) dt (B(x(t))+ C(x(t)); x(O)=xo in which the velocity of the state of the system depends not only upon the x(t) of the system at time t, but also on variations of observations state B(x(t)) of the state. This is a particular case of an implicit differential equation f(t, x(t), x'(t)) = 0 which can be regarded as a differential inclusion (**), where the right-hand side F is defined by F(t, x)= {vlf(t, x, v)=O}. During the 60's and 70's, a special class of differential inclusions was thoroughly investigated: those of the form X'(t)E - A(x(t)), x (0) =xo where A is a "maximal monotone" map. This class of inclusions contains the class of "gradient inclusions" which generalize the usual gradient equations x'(t) = -VV(x(t)), x(O)=xo when V is a differentiable "potential". 2 Introduction There are many instances when potential functions are not differentiable
| Author | : Guillaume Deffuant |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 227 |
| Release | : 2011-08-03 |
| Genre | : Social Science |
| ISBN | : 3642204228 |
One common characteristics of a complex system is its ability to withstand major disturbances and the capacity to rebuild itself. Understanding how such systems demonstrate resilience by absorbing or recovering from major external perturbations requires both quantitative foundations and a multidisciplinary view on the topic. This book demonstrates how new methods can be used to identify the actions favouring the recovery from perturbations. Examples discussed include bacterial biofilms resisting detachment, grassland savannahs recovering from fire, the dynamics of language competition and Internet social networking sites overcoming vandalism. The reader is taken through an introduction to the idea of resilience and viability and shown the mathematical basis of the techniques used to analyse systems. The idea of individual or agent-based modelling of complex systems is introduced and related to analytically tractable approximations of such models. A set of case studies illustrates the use of the techniques in real applications, and the final section describes how one can use new and elaborate software tools for carrying out the necessary calculations. The book is intended for a general scientific audience of readers from the natural and social sciences, yet requires some mathematics to gain a full understanding of the more theoretical chapters. It is an essential point of reference for those interested in the practical application of the concepts of resilience and viability
| Author | : Jean-Pierre Aubin |
| Publisher | : Springer |
| Total Pages | : 510 |
| Release | : 2013-11-13 |
| Genre | : Business & Economics |
| ISBN | : 9783642645426 |
This book is intended to provide economists with mathematical tools necessary to handle the concepts of evolution under uncertainty and adaption arising in economics, pursuing the Arrow-Debreu-Hahn legacy. It applies the techniques of viability theory to the study of economic systems evolving under contingent uncertainty, faced with scarcity constraints, and obeying various implementation of the inertia principle. The book illustrates how new tools can be used to move from static analysis, built on concepts of optima, equilibria and attractors to a contingent dynamic framework.
| Author | : William F. Morris |
| Publisher | : Sinauer Associates Incorporated |
| Total Pages | : 480 |
| Release | : 2002-01-01 |
| Genre | : Science |
| ISBN | : 9780878935468 |
The goal of this book is to provide practical, intelligible, and intuitive explanations of population modelling to empirical ecologists and conservation biologists. Modelling methods that do not require large amounts of data (typically unavailable for endangered species) are emphasised. As such, the book is appropriate for undergraduate and graduate students interested in quantitative conservation biology, managers charged with preserving endangered species, and, in short, for any conservation biologist or ecologist seeking to better understand the analysis and modelling of population data.
| Author | : Girol Karacaoglu |
| Publisher | : Springer |
| Total Pages | : 257 |
| Release | : 2019-01-28 |
| Genre | : Political Science |
| ISBN | : 9811361045 |
The distinctive contribution of this book is the formulation of an integrated social, environmental, and economic framework for public policy. This contribution is realised through investigations and conclusions in the following four domains: a formal stylised model that provides a platform for an integrated approach to public policy; a policy-informing simulation model that can be used to operationalise the public policy insights proposed in the stylised model; the implications of introducing fundamental (or radical) uncertainty and complexity into the policy framework; and the use of viability theory to demonstrate how one can think of and implement public policy in an uncertain and complex world, when the focus of policy needs to shift to building resilience to systemic risks. The book’s stylised model is constructed by weaving together threads from the wellbeing, human needs, complex systems, sustainable development, endogenous economic growth, directed technical change, and credit-based-money literatures. Throughout, the perspective is that of a policy adviser to a "wellbeing state", as distinct from a "welfare state". The key linkages or relationships in the model are supported by empirical evidence that draws on the wider literature in related fields.
| Author | : Steven R. Beissinger |
| Publisher | : University of Chicago Press |
| Total Pages | : 594 |
| Release | : 2002-05-04 |
| Genre | : Nature |
| ISBN | : 0226041786 |
Many of the world's leading conservation and population biologists evaluate what has become a key tool in estimating extinction risk and evaluating potential recovery strategies - population viability analysis, or PVA.
| Author | : Georgi V. Smirnov |
| Publisher | : American Mathematical Society |
| Total Pages | : 226 |
| Release | : 2022-02-22 |
| Genre | : Mathematics |
| ISBN | : 1470468549 |
A differential inclusion is a relation of the form $dot x in F(x)$, where $F$ is a set-valued map associating any point $x in R^n$ with a set $F(x) subset R^n$. As such, the notion of a differential inclusion generalizes the notion of an ordinary differential equation of the form $dot x = f(x)$. Therefore, all problems usually studied in the theory of ordinary differential equations (existence and continuation of solutions, dependence on initial conditions and parameters, etc.) can be studied for differential inclusions as well. Since a differential inclusion usually has many solutions starting at a given point, new types of problems arise, such as investigation of topological properties of the set of solutions, selection of solutions with given properties, and many others. Differential inclusions play an important role as a tool in the study of various dynamical processes described by equations with a discontinuous or multivalued right-hand side, occurring, in particular, in the study of dynamics of economical, social, and biological macrosystems. They also are very useful in proving existence theorems in control theory. This text provides an introductory treatment to the theory of differential inclusions. The reader is only required to know ordinary differential equations, theory of functions, and functional analysis on the elementary level. Chapter 1 contains a brief introduction to convex analysis. Chapter 2 considers set-valued maps. Chapter 3 is devoted to the Mordukhovich version of nonsmooth analysis. Chapter 4 contains the main existence theorems and gives an idea of the approximation techniques used throughout the text. Chapter 5 is devoted to the viability problem, i.e., the problem of selection of a solution to a differential inclusion that is contained in a given set. Chapter 6 considers the controllability problem. Chapter 7 discusses extremal problems for differential inclusions. Chapter 8 presents stability theory, and Chapter 9 deals with the stabilization problem.
| Author | : Jean-Pierre Aubin |
| Publisher | : Cambridge University Press |
| Total Pages | : 306 |
| Release | : 1996-03-29 |
| Genre | : Computers |
| ISBN | : 9780521445320 |
This book is devoted to some mathematical methods that arise in two domains of artificial intelligence: neural networks and qualitative physics. Professor Aubin makes use of control and viability theory in neural networks and cognitive systems, regarded as dynamical systems controlled by synaptic matrices, and set-valued analysis that plays a natural and crucial role in qualitative analysis and simulation. This allows many examples of neural networks to be presented in a unified way. In addition, several results on the control of linear and nonlinear systems are used to obtain a "learning algorithm" of pattern classification problems, such as the back-propagation formula, as well as learning algorithms of feedback regulation laws of solutions to control systems subject to state constraints.
