Paper generalizes control contraction metrics to Finsler geometry.
problem Designing nonlinear controllers for complex geometries.
method Generalization of CCMs to Finsler geometry, providing open loop and sampled data controllers.
result Simplified computation of sampled data control without real-time shortest path computation.
3D Ricci flow controls local geometry for a time.
problem Controlling the geometry of a ball in 3D Ricci flow.
method Proving local geometric control persists under Ricci flow.
result Local C/t decay of curvature tensor.
Survey explores geometric aspects of policy optimization in control systems.
problem Understanding the geometric relationships between control design and optimization.
method Geometric perspective on policy optimization, focusing on parameterization and topology.
result Implications of policy geometry on stability and performance of local search algorithms.
Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 s…
New method for creating metrics in complex geometries.
problem Metrizability of parabolic geometries and sub-Riemannian metrics.
method General method for linearizability and classification of cases.
result Natural sub-Riemannian metrics on distributions in geometric control theory.
Develops method to create explicit minimal surfaces with controlled geometry.
problem Creating explicit regular minimal surfaces with specific geometry.
method Applying Björling formula to planar curves with controlled normal fields.
result Explicit construction of minimal surfaces on entire complex plane.
The paper introduces new tests for global controllability in hybrid systems.
problem Global controllability in hybrid systems with discrete events.
method Geometric formulation of hybrid systems and analysis of jump points.
result Hybrid systems can be globally controllable even if continuous systems are not.
We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.
The paper presents the geometry of Lie algebroids and its applications to optimal control. The first part deals with the theory of Lie algebroids, connections on Lie algebroids and dynamical systems defined on Lie algebroids (mainly Lagrangian and Hamiltonian systems). In the second part we use the framework of Lie alg…
A novel controller for wheeled robots handles joystick inputs for smooth steering.
problem Steering control for differential-drive wheeled robots from indirect joystick inputs.
method Developed a geometric controller based on Darboux frame kinematics.
result Smooth trajectories achieved with safety constraints and no desired states.
New insights into cascade feedback linearization of control systems.
problem Obtaining a cascade feedback linearization for invariant control systems.
method Introducing truncated versions of operators from the calculus of variations to prove new theorems.
result Established new geometry and foundational theorems for future work.
The paper studies the geometry of flat manifolds with controlled holonomy.
problem Investigating the geometry of asymptotically flat manifolds with specific properties.
method Analyzes torus fibrations and Hitchin-Thorpe inequalities for Ricci-flat 4-manifolds.
result Proves that certain flat metrics on 4-manifolds are isometric to Euclidean or Taub-NUT.
The paper optimizes policies constrained to Schur stabilizing controllers using a Newton-type algorithm.
problem Optimizing policies under linear constraints in control systems.
method Newton-type algorithm on a manifold of Schur stabilizing controllers with a Riemannian metric.
result Local convergence guarantees for the Newton-type algorithm without relying on exponential mapping or retractions.
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.
Study compares sub-Riemannian curvature to optimal control variational problems.
problem Comparing sub-Riemannian curvature to optimal control variational problems.
method Introducing sub-Riemannian Bakry-Émery curvature and proving sub-Laplacian comparison theorems.
result Established sharp measure contraction property for 3-Sasakian manifolds.
We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.
One of the central difficulties of settling the L2-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of En…
Explains how pre-symplectic structures can be changed.
problem Understanding how pre-symplectic structures can be deformed.
method Uses Dirac geometry to explain the geometric origin of L∞-algebra controlling deformations. result Discovers the geometric origin of the L∞-algebra controlling deformations of pre-symplectic structures. We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.
The paper controls the geometry of surface subgroups in specific Kleinian groups.
problem Understanding the geometry of surface subgroups in specific Kleinian groups.
method Finding surface subgroups that are quasi-conformally conjugate to finite index subgroups of a genus-2 quasi-Fuchsian group.
result The existence of surface subgroups that are K-quasiconformally conjugate to finite index subgroups of a genus-2 quasi-Fuchsian group. Counterexample shows state-constrained optimal control problems can have Young measure gaps.
problem Existence of Young measure gaps in state-constrained optimal control problems.
method Provided a counterexample for smooth controllable systems state-constrained to the unit ball.
result Gap occurs in a regular setting with non-convex Lagrangian density.
Introduces GFC for learning complex dynamical systems with geometric constraints.
problem Challenges in accurately modeling and predicting complex dynamical systems with geometric constraints.
method Geometric Contact Flows (GFC) using Riemannian and Contact geometry as inductive biases.
result Ensemble of contactomorphisms adapt the latent contact Hamiltonian model to target dynamics while preserving desirable properties.
The paper develops a new geometric framework for analyzing optimal control problems.
problem Analyzing second-order conditions in constrained variational problems.
method Constructing Jacobi curves using L-derivatives and proving Morse-type theorems.
result Connecting the negative inertia index of the Hessian to symplectic invariants of Jacobi curves.
We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\ma…
This paper tackles data-efficient nonlinear control in Hamiltonian systems using symplectic geometry.
problem Data-efficient nonlinear control in Hamiltonian systems.
method Combines symplectic geometry, recurrence on energy level sets, and chain policies to solve target reachability problems.
result Data requirements depend on geometric and recurrence properties of the Hamiltonian, not the state dimension.
We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …
GeoIB uses information geometry to control compression in deep learning models.
problem The indirect and biased nature of traditional IB implementations in deep learning.
method GeoIB uses Fisher-Rao and Jacobian-Frobenius terms to control information compression directly.
result GeoIB achieves better trade-off between accuracy and compression than traditional IB methods.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
Wave maps from circle to manifold controllable if homotopy classes match.
problem Global controllability of wave maps from circle to Riemannian manifolds.
method Characterization of controllability via homotopy classes, uniform-time global controllability between steady states, quantitative exponential stability.
result Global controllability is equivalent to homotopy class of data.
Karcher reimagined elliptic functions using geometry.
problem Understanding and controlling elliptic functions.
method Geometrical approach to rewrite elliptic function theory.
result Optimal control over elliptic function behavior and image values.
New method learns dynamics from sparse data using geometric constraints.
problem Learning dynamics from sparse, undersampled data.
method Reformulates inference as a stochastic control problem, using geometry-driven path augmentation.
result Accurately recovers stochastic dynamics from extremely undersampled data.
New equations for geodesics in sub-Riemannian geometry.
problem Finding equations for normal geodesics in sub-Riemannian geometry.
method Developed a new system of equations using a partial connection.
result The new equations split into horizontal and complementary parts.
The Noether theorem is extended to stochastic control problems using contact symmetries.
problem Stochastic optimal control problems.
method Exploiting jet bundles and contact geometry, the authors prove the existence of conserved quantities.
result Optimal control problems admit infinitely many conserved quantities in the form of local martingales.
New method builds hyperbolic spheres with controlled holonomy.
problem Creating hyperbolic spheres with specific holonomy properties.
method Gluing simple building blocks to form hyperbolic cone spheres.
result Any Deroin-Tholozan representation can be realized as cone sphere holonomy.
New trigonometric method for convex sets aids control problems.
problem Optimal control problems with two-dimensional control.
method Proposes a new method to describe convex sets.
result Investigates sub-Finsler problems in various groups.
We show that there are minimal graphs in R^{n+1} whose intersection with the portion of the horizontal hyperplane contained in the unit ball has any prescribed geometry, up to a small deformation. The proof hinges on the construction of minimal graphs that are almost flat but have small oscillations whose geometry we c…
A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar wi…
Foundation models fail to preserve continuous geometry, identified as the Geometric Alignment Tax.
problem Continuous geometry is lost in foundation models due to discrete categorical bottlenecks.
method Controlled ablations on synthetic systems and evaluation of 14 biological models using rate-distortion theory and MINE.
result Replacing cross-entropy with a continuous head reduces geometric distortion by up to 8.5x.
Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on…
High-dimensional geometry makes adversarial examples easier to construct.
problem Adversarial examples in deep neural networks
method Systematic study of input dimensionality
result Adversarial examples become easier to construct as dimensionality increases.
We study control systems invariant under a Lie group with application to the problem of nonlinear trajectory planning. A theory of symmetry reduction of exterior differential systems is employed to demonstrate how symmetry reduction and reconstruction is effective in the explicit, exact construction of planned system t…
We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodi…
A benchmarking framework for studying data geometry.
problem Generalization and approximation error bounds in deep learning.
method Repurposing and extending dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling.
result Near-ground-truth accuracy in curvature, reach, and volume estimation.
We consider decomposition spaces R3/G that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on R3/G constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric spa…
In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, d…
We present a novel approach to the classification of conformally equivariant differential operators on spinors in the case of homogeneous conformal geometry. It is based on the classification of solutions for a vector-valued system of partial differential equations, associated to D-modules for the homogeneo…
The paper extends a geometric model using singular curves.
problem Understanding abnormal extremals in sub-Riemannian geometry.
method Analysis of singular curves and construction of a graded Lie algebra.
result A nilpotent graded Lie algebra is constructed isomorphic to F4. Proves path-connectedness of metrics on 3-manifolds with positive scalar curvature.
problem Proving path-connectedness of metrics on 3-manifolds with positive scalar curvature.
method Uses Ricci flow with surgery and infinite connected sums with geometry control.
result Proves the moduli space of metrics is path-connected.