10 problems

Let $M$ be a smooth manifold of dimension $3$ equipped with a bracket-generating distribution $\Delta$, and for $x\in M$ let $\mathcal{X}^x_{\Delta}$ be the set of endpoints of sin…

Let $\gamma$ be the limiting curve obtained as the $H^1([0,1],M)$-limit of a subsequence of minimising trajectories for the unconstrained optimisation problems described above. Hig…

Let $\mathbb{G}$ be a Carnot group with left-invariant sub-Riemannian structure. A curve $\gamma(t):\mathbb{R}\to\mathbb{G}$ is a metric line if it is a globally minimizing sub-Rie…

Constant-$\theta$ time-optimality conjecture. For symmetric-space controls in $\mathfrak p$ with holonomy targets in $\mathfrak k$, constant-$\theta$ paths are time-optimal.

Constant inverse-velocity conjecture. Local variation of the total time should yield Euler–Lagrange equations implying that

Let ${\mathbb G}$ be a Carnot group and let $W$ be a horizontal half-space. Denote by $S_W$ the semigroup generated by $W$, by $\bar S_W$ its closure, by $\mathfrak w_{\bar S_W}$ t…

Let $M$ be a smooth manifold of dimension $3$ equipped with a bracket-generating distribution $\Delta$, and for $x\in M$ let $\mathcal{X}^x_{\Delta}$ be the set of endpoints of sin…

Alternating-circle conjecture. A curve of minimal length should be a concatenation of $n$ circular arcs, each having curvature $\kappa_1$ or $\kappa_2$.

Generalized Dubins conjecture. Any shortest path in $\hat{\operatorname{L}}_{\kappa_1}^{\kappa_2}(P,Q)$ should be a concatenation of three pieces, each of which is either an arc of…

A swimmer configuration consists of a weighted three-dimensional body, its density, and a finite number of basic shape deformations satisfying the self-propelled constraints. Let…