Sub-Riemannian Geometry and Optimal Control

Sub-Riemannian geometry is a branch of differential geometry in which the motion of a system is constrained to lie within a specified distribution of allowable directions on a manifold. Unlike its Riemannian counterpart, where distance is measured along arbitrary tangent directions, a sub-Riemannian manifold admits movement only along a subset of those directions, reflecting practical restrictions such as non-holonomic constraints in mechanical systems. The central objects of study are horizontal curves—paths tangent at every point to the distribution—and the length minimisers among these curves, known as sub-Riemannian geodesics. Optimal control theory offers a natural framework for analysing such problems, with the Pontryagin maximum principle yielding necessary conditions for minimality by translating the geometric problem into a Hamiltonian system. Together, the interplay of sub-Riemannian structure and control methods has led to significant advances in understanding global connectivity, the structure of cut and conjugate loci, and the synthesis of explicit motion plans. Applications range from robotic path planning and image reconstruction to quantum control and neurogeometry, underlining the field’s broad significance. Recent theoretical progress has deepened insight into the fine structure of singularities, curvature-like invariants, and the precise asymptotics of distance fronts, while computational approaches have begun to tackle practical optimisation in high-dimensional settings.

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Sub-Riemannian Geometry and Optimal Control publication trend

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Technical terms

Sub-Riemannian manifold: A smooth manifold equipped with a smoothly varying distribution of allowed tangent directions and an inner product defined only on that distribution.

Distribution: A choice at each point of a linear subspace of the tangent space, specifying the directions in which motion is permitted.

Horizontal curve: A path on a sub-Riemannian manifold whose tangent vector at every point lies in the distribution.

Geodesic: A horizontal curve that locally minimises length or energy, analogous to a straight line in Euclidean space.

Cut time (or cut locus): The time or locus at which a geodesic ceases to be globally length-minimising.

Optimal control: A mathematical framework for determining control inputs that steer a dynamical system from one state to another while minimising a cost functional, often length or time.

Pontryagin maximum principle: A set of necessary conditions for optimality in control problems, yielding a Hamiltonian system whose extremals correspond to candidate optimal trajectories.

Conjugate point: A point along a geodesic at which the second variation of length fails to be positive, indicating a loss of local optimality.

References

1. Cut time in the sub-Riemannian problem on the Cartan group*. ESAIM Control Optimisation and Calculus of Variations (2022).
2. Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry. Nonlinear Analysis (2024).
3. Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector. Mathematics (2023).