ANALYTICAL PARAMETERIZATION OF ROTORS AND PROOF OF A GOLDBERG CONJECTURE BY OPTIMAL CONTROL THEORY

Térence Bayen (1)

1 ACSIOM - Analyse, Calcul Scientifique Industriel et Optimisation de Montpellier

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 the diagonals of the n-gon from the parallel sides. Finally, using the extension of Noether's theorem to optimal control (as performed in [D. F. M. Torres, WSEAS Trans. Math., 3 (2004), pp. 620-624]), we show that a minimizer is necessarily a regular rotor, which proves a conjecture formulated in 1957 by Goldberg (see [M. Golberg, Amer. Math. Monthly, 64 (1957), pp. 71-78])

Mots clés: shape optimization; convexity; constant width bodies; rotors; support function; optimal control; Pontryagin maximum principle; switching point; bang control; Noether theorem

Domaines: Optimisation et contrôle [math.OC]

Térence Bayen. ANALYTICAL PARAMETERIZATION OF ROTORS AND PROOF OF A GOLDBERG CONJECTURE BY OPTIMAL CONTROL THEORY. SIAM Journal on Control and Optimization, 2009, 47 (6), pp.3007-3036. ⟨hal-00798653⟩