×

An exponential mapping in the generalized Dido problem. (English. Russian original) Zbl 1079.53050

Sb. Math. 194, No. 9, 1331-1359 (2003); translation from Mat. Sb. 194, No. 9, 63-90 (2003).
The classical Dido problem can be stated as follows. Given two points in the plane connected by a curve \(\gamma_{0}\) and a number \(S\), it is required to connect these points by a shortest curve \(\gamma\) so that the domain in the plane bounded by the curves \(\gamma_{0}\) and \(\gamma\) has area \(S\). The solution of this problem is also well-known: it is an arc or circle or a segment of a straight line connecting the given points.
The paper under review deals with the following generalization of the Dido problem: Apart from two points \((x_{0},y_{0})\), \((x_{1},y_{1}) \in \mathbb R^{2}\), a curve \(\gamma_{0}\) connecting them, a number \(S \in \mathbb R\), also a fixed point \(c=(c_{x},c_{y}) \in \mathbb R^{2}\) are given. It is required to find a shortest curve \(\gamma\) connecting the points \((x_{0},y_{0})\) and \((x_{1},y_{1})\) such that the domain \(D \subset \mathbb R^{2}\) bounded by the curves \(\gamma_{0}\) and \(\gamma\) has the given area \(S\) and center of gravity \(c\). In spite of having a simple and natural statement, the generalized Dido problem is still open.
The goal of this paper is to give a description of the extremals of this problem. For this proposal, the generalized Dido problem is stated as an optimal control problem in a 5-dimensional space with a 2-dimensional control and quadratic cost functional, which is a nilpotent sub-Riemannian problem with the growth vector \((2,3,5)\). Extremals of this problem are parametrized by Jacobian elliptic functions.

MSC:

53C17 Sub-Riemannian geometry
37N35 Dynamical systems in control
70Q05 Control of mechanical systems
93B29 Differential-geometric methods in systems theory (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI