## 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)
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