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Generalized Lax-Hopf Formulas for Cournot Maps and Hamilton-Jacobi-McKendrick Equations

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Analysis and Geometry in Control Theory and its Applications

Part of the book series: Springer INdAM Series ((SINDAMS,volume 11))

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Abstract

A generalization of Lax-Hopf formula for value functions has been proved by the second author in the case when the Lagrangian depends not only on the velocity, as for the original one, but also on time and velocity. She associated with the Lagrangian another function, called its moderation, for which the Lax-Hopf formula holds true. On the other hand, arrival maps have been introduced in control theory for associating with terminal time and duration a terminal state at which arrives at least one evolution governed by a differential inclusion. Cournot maps provide their departure time and state. They involve “temporal windows”, on which average evolutions and velocities can be measured. A temporal window is coded by a two-dimensional “time-duration” pair, the classical one-dimensional time being a temporal window of zero duration. We associate with a set-valued map another map, called its “Lax-Hopf hull”, generally contained in its closed convex hull, and its “moderation”. If such a set-valued map is the right hand side of differential inclusion governing evolutions, its Lax-Hopf hull governs their average states and velocities on evolving temporal windows and its moderation provides the set of average velocities. Knowing the terminal state, the average velocity provides the departure time and state provided by the Cournot map. A Cournot map, providing all departure states of evolutions arriving at a terminal state at a terminal time, describes mathematically the retrospective and dynamical concept of uncertainty introduced in 1843 by Augustin Cournot. In economics, the Lax-Hop hull associates with average transactions on an investment window the average profit involving the duration of the production process, quasi null for the century of construction of cathedrals to quasi infinite for the transactions performed on the high-frequency markets. Average profits could therefore be used as a basis for a taxation on shareholder value instead of added value. Time averaging processes of evolutions and their velocities on adequate temporal windows may be useful in many problems, such traffic congestion management.

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Notes

  1. 1.

    Prospective average velocities \(\nabla x(t):= \frac{x(t) - x(0)} {t}\) are distinguished by dropping the right arrow.

  2. 2.

    See [5, 36–38] among a myriad of references on time.

  3. 3.

    “What was God doing before He created the Heavens and the Earth?” asked Augustine of Hippo in his confessions. What was the universe behaving before the Big Bang, ask some physicists? Mathematically, one can introduce helicoidal durations which are concatenations of durations \(d_{(T_{i},T_{i}-T_{i-1})}\) on successive temporal windows [T i−1, T i ] (see La valeur n’existe pas. À moins que…, [12]). Introducing the concepts of temporal windows and duration function bypasses the question of origin of time.

  4. 4.

    The max-plus algebra structure on \(\mathbb{R}\) and the hyperspaces (of subsets of a given space) as well provides an adequate framework to study evolutionary problems (see [1–3] and Evaluation and Stratification of Sets, [14]).

  5. 5.

    See Sect. 14.12, p. 171, of Viability Theory. New Directions, [18].

  6. 6.

    See Sect. 3.3, p. 26.

  7. 7.

    See Theorem 10.2.5, p. 379, of Viability Theory. New Directions, [18].

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Acknowledgements

This work was partially supported by the Commission of the European Communities under the 7th Framework Programme Marie Curie Initial Training Network (FP7-PEOPLE-2010-ITN), project SADCO, contract number 264735, ANR-11-ASTRID-0041-04 and ANR-14-ASMA-0005-03.

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Authors and Affiliations

  1. VIMADES (Viabilité, Marchés Automatique et Décision), 14, rue Domat, 75005, Paris, France

    Jean-Pierre Aubin

  2. Euronext and Société VIMADES (Viabilité, Marchés, Automatique et Décision), Université Panthéon-Sorbonne, Paris, France

    Luxi Chen

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  1. Jean-Pierre Aubin
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Correspondence to Jean-Pierre Aubin .

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Editors and Affiliations

  1. Department of Mathematics, Université de Bretagne Occidentale, Brest, France

    Piernicola Bettiol

  2. Department of Mathematics, Università di Roma "Tor Vergata", Roma, Italy

    Piermarco Cannarsa

  3. Department of Mathematics, Università di Padova, Padova, Italy

    Giovanni Colombo

  4. Department of Mathematics, Università di Padova, Padova, Italy

    Monica Motta

  5. Department of Mathematics, Università di Padova, Padova, Italy

    Franco Rampazzo

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Aubin, JP., Chen, L. (2015). Generalized Lax-Hopf Formulas for Cournot Maps and Hamilton-Jacobi-McKendrick Equations. In: Bettiol, P., Cannarsa, P., Colombo, G., Motta, M., Rampazzo, F. (eds) Analysis and Geometry in Control Theory and its Applications. Springer INdAM Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-06917-3_1

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