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We present a complete analytic parametrization of constant width bodies in dimension 3 based on the median surface: more precisely, we define a bijection between some space of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic… (More)

Curves which can be rotated freely in an n-gon (that is, an regular polygon with n sides) so that they always remain in contact with every side of the n-gon are called rotors. Using optimal control theory, we prove that the rotor with minimal area consists of a finite union of arcs of circles. Moreover, the radii of these arcs are exactly the distances of… (More)

- Térence Bayen
- CDC
- 2009

- T. Bayen, J.-B. Hiriart-Urruty
- 2013

" When Minkowski's theory of convexity appeared, some mathematicians said that he discovered a nice mathematical joy which, unfortunately, is quite useless. About a century passed, and now the theory of convex sets is a very important applied branch of mathematics. " V. Boltyanski, in Geometric methods and optimization problems (1999). " La convexité dans… (More)

We consider the problem of minimizing a functional (like the area, perimeter, surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér-Riesz theorem on positive trigono-metric polynomials into a semidefinite programming problem. Several problems such as the… (More)

We study the minimum time control problem of a series of two interconnected chemostats under the input constraint u2 ≤ u1, where ui are the respective dilution rates in the tanks. This constraint brings controllability issues in the study of the optimal strategies. We encounter this difficulty by splitting the state domain into two sub-domains, one with no… (More)

— This work is devoted to maximizing the production of methane in a bioreactor coupling an anaerobic digester and a culture of micro-algae limited by light. The decision parameter is the dilution rate which is chosen as a control, and we enforce periodic constraints in order to repeat the same operation every day. The system is gathered into a… (More)

We address the problem of finding an optimal feedback control for feeding a fed-batch bioreactor with one species and one substrate, from a given initial condition to a given target value in a minimal amount of time. Mortality rate for the biomass and nutrient recycling are taken into account in this work. The optimal synthesis (optimal feeding strategy)… (More)