Viability Theory: New Directions by Jean-Pierre Aubin

By Jean-Pierre Aubin

Viability concept designs and develops mathematical and algorithmic tools for investigating the difference to viability constraints of evolutions ruled by means of advanced platforms 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 regulate concept and robotics to cognitive sciences. It consists of interdisciplinary investigations spanning fields that experience usually constructed in isolation. the aim of this e-book is to offer an initiation to functions of viability conception, explaining and motivating the most suggestions and illustrating them with various numerical examples taken from a number of fields.

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In continuous time, discrete time, or a “hybrid” of the two when impulses are involved, 2. constrained to adapt to an environment, 3. evolving under contingent and/or tychastic uncertainty, 4. using for this purpose controls, regulons (regulation controls), subsets of regulons, and in the case of networks, connectionnist matrices, 5. , 6. co-evolving with their environment (mutational and morphological viability), 7. and corrected by introducing adequate controls (viability multipliers) when viability or capturability is at stakes.

The parameters range over a state-dependent “cybernetic” map U : x U (x), providing the system opportunities to adapt at each state to viability constraints (often, as slowly as possible) and/or to regulate intertemporal optimal evolutions. The nature of the parameters differs according to the problems and to questions asked: They can be: • “controls”, whenever a controller or a decision maker “pilots” the system by choosing the controls, as in engineering, • “regulons” or regulatory parameters in those natural systems where no identified or consensual agent acts on the system, • “tyches” or disturbances, perturbations under which nobody has any control.

Chapters 10, p. 375 and 11, p. 437 provide the proofs of the results stated in the viability survival kit of Chap. 2, p. 43, in the framework of general evolutionary systems, first, and next, of parameterized systems, where the tangential regulation maps governing the evolution of viable evolutions ares studied. 3 Quantitative Concepts We chose to interrupt the logical course of studying the qualitative properties first before tackling the quantitative ones. The reason is that they provide the most interesting application to different issues of intertemporal optimization.

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