New conditions for stabilizing systems are shown to be independent but stronger under certain conditions.
problem Necessary conditions for stabilizing feedback control systems.
method Extensions of Brockett's and Coron's conditions in fiber and vector bundle pictures.
result The conditions are independent in the fiber bundle picture but stronger in the vector bundle picture under specific conditions.
Solves a long-standing problem on step-two groups with exact formulas.
problem Long-standing Gaveau--Brockett open problem on step-two groups.
method Combining Varadhan's formulas, heat kernel, and operator convexity.
result Exact formula for Carnot--Carathéodory distance on step-two groups.
We extend Agrachev-Brockett-Jurjdevic's solution to normal sub-Riemannian geodesics.
problem Finding normal sub-Riemannian geodesics in Lie group structures.
method Constructing sub-Riemannian structures from Lie subalgebra filtrations and applying to homogeneous spaces.
result Explicit solutions for normal geodesics in general chains of Lie subgroups.
Characterizes GM-groups via sub-Riemannian geometry properties.
problem Characterizing step-two Carnot groups via sub-Riemannian geometry.
method Sub-Riemannian geometric properties, including squared distance, cut locus, optimal synthesis.
result Characterization of GM-groups and exact expression of d(g)2 for classical cut locus. The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integra…
Contact Lie systems analyze integral curves of Hamiltonian vector fields.
problem Analyzing integral curves of Hamiltonian vector fields.
method Define and study contact Lie systems, including conservative systems.
result Develop Liouville theorems, contact reductions, and Gromov non-squeezing theorems.
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.
Minimum attention improves reinforcement learning performance in high-dimensional dynamics.
problem Improving reinforcement learning performance in high-dimensional nonlinear dynamics.
method Applying minimum attention as a regularization technique in reinforcement learning, including model-based and model-free approaches.
result Minimum attention outperforms state-of-the-art algorithms in few-shot adaptation and variance reduction.
We regard the real symplectic group Sp(2n,R) as a constraint submanifold of the 2n×2n real matrices M2n(R) endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group Gl(2n,R) endowed with the (left) invariant metric. For…