Abstract
This chapter provides the mathematical characterization of averagers, Cournot, reachable and Eupalinian maps without using the evolutions governed by the differential inclusions involved in their definitions. Hence, the resolution of differential inclusion is bypassed and those maps are automatically triggered to associate their inputs and their outputs. This is possible by characterizing their graphs in terms of viable capture basins of adequate “characteristic” targets viable in “characteristic” environments under “characteristic” systems to be constructed in each case.
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Notes
- 1.
The contribution of Peano was unfortunately forgotten until it was uncovered by Szymon Dolecki and Gabriele Greco in 2007.
- 2.
If K is sleek, the tangent cone coincides with the Clarke (or circatangent) and adjacent cones (see Set-Valued Analysis, [42, Aubin and Frankowska]).
- 3.
Property (4.12) is a consequence of the inverse set-valued map Theorem.
- 4.
Since tangent cones to graphs are graphs of derivatives and tangent cones to epigraphs are epigraphs of “epiderivatives”, we can translate the viability solutions defined in terms of viable capture basins as “generalised solutions” to first-order partial differential equations, as summarized in Sect. 8.6, p. 233.
- 5.
Patrick Saint Pierre designed in 1992 the first viability algorithm in [204, Saint-Pierre], followed by [73, 74, Cardaliaguet, Quincampoix and Saint-Pierre] (with the contribution of Philippe Lacoude, Pierre-Olivier Vandanjon and Elena Dominguez). They have been also used for solving various problems by Chen Luxi, Anya Désilles, Olivier Dordan, Luc Doyen, Sophie Martin, Rodéric Moitié, Christian Mullon and Nicolas Seube. Other algorithms have been designed for approximating viability kernels ([191, Neznakhin and Ushakov], [59, Botkin and Turova] (bridges)), [163, Krawczyk], (VIKAASA: an application capable of computing and graphing viability kernels for simple viability problems), [111, Deffuant, Chapel and Martin], (support vector machine, which involve in the background Aronsjan’s reproducing kernel of a Hilbertian structure on sets, used also in Kriging estimation with the covariance function as a reproducing kernel (see [213, Vazquez and Walter], and the references therein)), [155, Jaulin, Ninin, Chabert, Le Ménec, Saad, LeDoze and Stancu], (interval analysis), [126, Fraichard and Asama], etc.
- 6.
They are defined by their graphs (graphical approach to maps), which are the actual subsets delivered as “control boards” embedded in the vehicles or read by drivers on a (future) dedicated velocity regulator. The viability algorithms compute subsets, and thus graphs of control boards.
- 7.
We refer to [56, 57, Bayen, Claudel, Saint-Pierre], [81, 82, Claudel, Bayen], [112, Désilles], Chap. 14, p. 563, of Viability Theory. New Directions, [31, Aubin, Bayen and Saint-Pierre] for available numerical results. A software for computing Cournot maps is presently under investigation and implementation by Anya Désilles (see Sect. 7.4.4, p. 195).
- 8.
See for instance A Posteriori Error Estimation in Finite Element Analysis, [2, Ainsworth and Oden], and Chap. 10, p. 284, of Approximation of Elliptic Boundary-Value Problems, [10, Aubin].
- 9.
In the same way, so to speak, than the difference between Riemann and Lebesgue integrals, where the Lebesgue integral uses the inverse images of a partition of the arrival space by the function to integrate, the viability algorithms use also the inverse images of the maps \(\varPhi _{n}\). Shooting methods and viability algorithm have been used to compute fractals, for example (see Sect. 2.9.3, p. 75 of Viability Theory. New Directions, [31, Aubin, Bayen and Saint-Pierre], and have been compared in this context).
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Aubin, JP., Désilles, A. (2017). Viability Characterizations and Construction of Celerity Regulators. In: Traffic Networks as Information Systems. Mathematical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54771-3_4
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DOI: https://doi.org/10.1007/978-3-642-54771-3_4
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