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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

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)

In this paper, we consider a family of closed planar algebraic curves C which are given in parametrization form via a trigonometric polynomial p. When C is the boundary of a compact convex set, the polynomial p represents the support function of this set. Our aim is to examine properties of the degree of the defining polynomial of this family of curves in… (More)