By Jean-Pierre Aubin
Viability conception designs and develops mathematical and algorithmic tools for investigating the variation to viability constraints of evolutions ruled by means of advanced structures less than uncertainty which are present in many domain names related to residing beings, from organic evolution to economics, from environmental sciences to monetary markets, from keep an eye on concept and robotics to cognitive sciences. It comprises interdisciplinary investigations spanning fields that experience typically constructed in isolation. the aim of this ebook is to give an initiation to functions of viability conception, explaining and motivating the most options and illustrating them with a variety of numerical examples taken from a variety of fields.
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Additional resources for Viability Theory: New Directions
43, 4, p. 125 and 6, p. 199. They are devoted to the exposition of the main concepts and principal results of viability, dynamic intertemporal optimization (optimal control) and heavy evolutions (inertia functions). Most of the proofs are not provided, only the simple ones which are instructive. However, “survival kits” summarizing the main results are provided to be applied without waiting for the proofs and the underlying mathematical machinery behind. These three chapters should allow the reader to master the concepts of those three domains and use them for solving a manifold of problems: engineering, requiring both viability and intertemporal optimization problems, life sciences, requiring both viability and inertia concepts and...
These three ﬁgures display the domains of the minimal time function, which is the capture basin of the harbor, of the Lyapunov function, which is the attraction basin and of the value function of an optimal control problem (minimizing the cumulated squares of the velocity). These domains are empty north of the waterfall and have “holes” south of the two mains obstacles and of the harbor on the east. The presence of obstacles induces three lines of discontinuity, delimitating evolutions sailing north of the northern obstacle, between the two obstacles and south or east of the southern obstacle, as it is shown for the evolutions starting from A, B and C in the capture basin, domain of the minimal time function.
15 The “Touch down safety envelope”. The 3-d boundary of the safety envelope is displayed. The colorimetric scale indicates the altitude. This topic is developed in Sect. 3, p. 108. 1 Management of Renewable Resources This example is fully developed in Sects. 2, p. 3, p. 262. The management of renewable resources requires both dynamics governing the renewable resource (ﬁshes) and the economics (ﬁsheries), and biological and economic viability constraints. The following classical example (called Verhulst-Schaeﬀer example) assumes that the evolution of the population of a ﬁsh species x(t) is driven by the logistic Verhlust diﬀerential equation x (t) = r(t, x(t))x(t) := rx(t) 1 − x(t) (see Sect.
Viability Theory: New Directions by Jean-Pierre Aubin