Singularities of Time-Optimal Trajectories

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $f,g$ be $C^\infty$ vector fields on an $n$-manifold $M$. Controls $u$ are measurable, $|u(t)|\leq1$ almost everywhere; trajectories are Lipschitz solutions of $\dot q=f(q)+ug(q)$ with prescribed endpoints, and time optimality means least travel time. A switching point of an admissible trajectory is a point where the trajectory is not smooth. The local switching complexity $\operatorname{sw}(q)$ is the least finite $k$ for which some neighborhood of $q$ bounds by $k$ the number of switches of every optimal trajectory contained in it; the source also uses the value $\infty$ when every neighborhood contains an optimal trajectory with infinitely many switches. Generic pairs are understood in the smooth space of vector fields.

Statement recorded in the supplied source.

For a generic pair $(f,g)$ in dimension $3$, is each individual time-optimal trajectory piecewise smooth? Is $\operatorname{sw}(q)<\infty$ for every $q\in M$? Separately, in arbitrary dimension with real-analytic $f,g$, whenever two points are joined by a time-optimal trajectory, can they be joined by a time-optimal trajectory with at most countably many switching points? [S2]

Short English statement

Is every individual time-optimal trajectory of a generic three-dimensional system piecewise smooth, and is switching complexity locally finite at every point? Do analytic systems always admit optimal connections with at most countably many switches?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6500001 (AMR-064-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. A. Agrachev, Some open problems, arXiv:1304.2590v2 (2013), Section I, pp. 1--3. https://arxiv.org/pdf/1304.2590.