Noise-Induced Navigation in Non-convex Domains and Compact Manifolds

Karthik Elamvazhuthi

Source-linked AI summary

Noise-Induced Navigation in Non-convex Domains and Compact Manifolds

Karthik Elamvazhuthi

arXiv:2610.10949v1Oct 7, 2026math.OCcs.ROcs.SYeess.SY

TL;DR

Global feedback stabilization faces topological obstructions for smooth deterministic laws. The paper introduces noise into reflected closed-loop dynamics and uses local stability with recurrence to establish stochastic stability in the large. Its construction covers Euclidean domains and compact Riemannian manifolds, while non-convex obstacles can still produce metastability.

Problem
Smooth feedback cannot globally stabilize a target on compact domains with obstacles or compact manifolds, motivating alternatives to deterministic global stabilization.
Source: §I · p. 1
Method
The paper designs reflected controlled diffusions with noise, using a stability criterion that combines local Lyapunov stability and positive recurrence.
Source: §I · p. 1
Results
Smooth controls make the target asymptotically stable in probability in the large for reflected diffusions on bounded connected smooth-boundary domains and compact manifolds.
Source: §III · p. 2, §IV · p. 5
Takeaways & Limitations
The construction circumvents deterministic topological obstructions at the cost of probabilistic convergence, but non-convex obstacles can still cause metastability.
Source: A. Stabilization in a Domain with… · p. 6, §VI · p. 8
Extending the strategy to non-holonomic systems remains future work because local stability is harder to attain, even when recurrence follows under a bracket-generating assumption.
Source: §VI · p. 8

Abstract

from arXiv · show

In this note, we study the problem of designing a feedback law that globally steers a system to a prescribed target configuration. Even if the system is fully actuated, topological obstructions generally prevent the existence of globally asymptotically stabilizing continuous feedback laws. We revisit this problem in a stochastic setting by allowing noise to enter through the control channels. Using a criterion for asymptotic stability in the large that combines local Lyapunov stability with positive recurrence, we constructively show that one can construct elementary feedback laws that achieve global asymptotic stability in the large, of the target equilibrium point in connected Euclidean domains with obstacles and manifolds without boundary. For Euclidean domains with obstacles, we also show that the method extends naturally to the problem of finding the minimizer of a strongly convex function with non-convex constraints. Numerical experiments illustrate the effectiveness of the approach for Euclidean domains with circular obstacles and the two dimensional sphere. Additionally, we study the role noise strength when there is a non-convex obstacle, in which case the system might show metastability.

I. INTRODUCTION

The section frames global feedback stabilization as a motion-planning goal obstructed by topology for smooth deterministic laws. It proposes noise-driven reflected diffusions and a stochastic stability criterion, with results extending to Euclidean domains and compact Riemannian manifolds.

I. INTRODUCTION

Topology prevents smooth feedback from globally stabilizing a target on compact domains with obstacles and on compact manifolds.The section presents noise as an alternative to discontinuous feedback or hybrid dynamics.Source: §I · p. 1
The approach uses noise in closed-loop dynamics to help escape undesirable critical points, with domain invariance enforced by reflection.It designs a controlled diffusion whose sample paths converge to the target in a probabilistic sense.Source: §I · p. 1
The stochastic stability framework supports smooth target stabilization in connected Euclidean domains with smooth boundary and extends to compact Riemannian manifolds.The section contrasts this approach with, which requires constructing a navigation function.Source: §I · p. 1
The construction combines local Lyapunov stability near the target with positive recurrence, using uniform ellipticity away from it to drive returns.Near the equilibrium, deterministic stabilizing dynamics dominate; away from it, noise makes the generator uniformly elliptic.Source: §I · p. 1

II. EUCLIDEAN DOMAINS WITH BOUNDARY

This section formulates the control problem for a reflected diffusion on a bounded connected open Euclidean domain with smooth boundary. It asks for controls that make a prescribed equilibrium asymptotically stable in the large.

II. EUCLIDEAN DOMAINS WITH BOUNDARY

The setting is a bounded connected open subset of R^d with smooth boundary and inward unit normal n(x).The process is posed on a filtered probability space with Brownian motion.Source: §II · p. 1
The dynamics use smooth control and noise coefficients in a reflected Stratonovich SDE, with a boundary regulator that increases only when the process reaches the boundary.The reflection term keeps trajectories inside Γ.Source: §II · pp. 1–2
The objective is to choose controls u_i and noise coefficients a_i so a prescribed x_eq in Γ is asymptotically stable in the large.Source: §II · p. 2

III. ANALYSIS

The analysis establishes asymptotic stability in probability in the large for reflected diffusions, then extends the construction to general potentials and strongly convex minimization. Its arguments combine local Lyapunov stability with recurrence away from the target.

III. ANALYSIS

The reflected diffusion has an equilibrium asymptotically stable in probability in the large, with the expected hitting time of a neighborhood used to establish recurrence.The recurrence result combines with uniform stability in probability; the argument applies because the reflected diffusion has the strong Markov property.Source: §III · pp. 2–3
A smooth potential with a positive definite Hessian at the target supports controls ai(x) = 2V(x) and ui(x) = −α ∂xiV(x), extending the stability construction beyond a quadratic Lyapunov function.The potential is zero at the target and positive elsewhere; the same local stability and recurrence arguments apply.Source: §III · p. 3
The general-potential construction also provides the basis for extending the result to compact manifolds.Source: §III · p. 3
For a strongly convex objective with a unique interior minimizer, scaling diffusion by |∇V(x)|² makes it vanish at the minimizer and remain nondegenerate away from it, yielding asymptotic stability in probability in the large.The construction combines local Lyapunov stability near the minimizer with recurrence on the rest of the domain.Source: §III · p. 3

IV. MANIFOLD

The paper extends noise-based global stabilization to connected compact Riemannian manifolds without boundary. Its construction combines local stability near the target with positive recurrence, and it sketches an extension to partially underactuated systems.

IV. MANIFOLD

Smooth feedback coefficients can make the target asymptotically stable in probability in the large on the manifold.The stated manifold setting is connected, compact, smooth, and without boundary.Source: §IV · p. 4
The construction uses ai(x) = 2V(x) and ui(x) = −αAiV(x), making the target an equilibrium when V and dV vanish there.The result assumes the vector fields span the tangent space globally.Source: §IV · p. 4
Uniform local stability and finite expected hitting time of a target neighborhood combine to establish stability in the large.The proof establishes positive recurrence by showing the neighborhood's hitting time has finite expectation.Source: §IV · p. 5
For partially underactuated systems, full rank is required near the target while global bracket generation supports recurrence away from it.The authors leave a rigorous proof of this extension to future work, citing needed hypoelliptic Dirichlet regularity and recurrence results.Source: §IV · pp. 5–6

A. Stabilization in a Domain with Obstacles

Experiments consider reflected-diffusion feedback in a planar domain with circular obstacles and in a domain with concave obstacles. The latter can induce metastability, which can be reduced to some degree by increasing the noise.

A. Stabilization in a Domain with Obstacles

The circular-obstacle experiment uses a reflected-diffusion feedback law in [0, 1]^2 with three obstacles and target (0.15, 0.15).Reflection enforces the state constraint.Source: A. Stabilization in a Domain with… · p. 6
Concave obstacles can produce metastable regions where trajectories spend a long time before escaping toward the target.The target is still described as attractive, despite this long-lived trapping.Source: A. Stabilization in a Domain with… · p. 6
Increasing the noise reduces metastability to some degree in the concave-obstacle experiment.The paper points to Figure 3 for this observation.Source: A. Stabilization in a Domain with… · p. 6

B. Stabilization on the Sphere

The sphere construction targets the north pole using a Lyapunov function and smooth tangent vector fields. Its control coefficients combine deterministic drift with noise, and the experiment displays paths and their x3 coordinates.

B. Stabilization on the Sphere

For S2, the target is the north pole e3=(0,0,1), where the smooth Lyapunov function has its strict minimum.Source: B. Stabilization on the Sphere · p. 6
Projected coordinate vector fields are smooth, tangent to S2, and span each tangent space.Source: B. Stabilization on the Sphere · p. 6
Control coefficients use ui(x) = −α AiV(x) and ai(x) = 2σV(x), with α the drift gain and σ the noise-scale parameter.Source: B. Stabilization on the Sphere · p. 6
The diffusion amplitude vanishes at the north pole.Source: B. Stabilization on the Sphere · p. 6
The sphere experiment shows diffusion paths from different initial conditions and their corresponding x3 coordinates.Source: Figure 4 · p. 8

VI. CONCLUSION

The conclusion establishes stochastic stabilization for the stated compact domains and manifolds, while noting that non-convex obstacles can still cause metastability. It identifies minimizing target-visit times and extending the approach to non-holonomic systems as future directions.

VI. CONCLUSION

Noise and reflection yield smooth stochastic asymptotic stability in the large on compact Euclidean domains with smooth boundary or compact Riemannian manifolds, but metastability may persist with non-convex obstacles.Source: §VI · p. 8
A proposed direction is optimizing control laws to minimize the maximum expected time to visit a target neighborhood via the related Dirichlet problem.Source: §VI · p. 8
Extending the strategy to non-holonomic constraints remains future work; recurrence may follow under bracket generation, but local stability is harder to attain.Source: §VI · p. 8

VII. AI USE

The paper reports that AI was used to develop simulation code and polish the text, while the mathematical discussion and proofs are due to the author.

VII. AI USE

AI assisted with simulation-code development and text polishing; the author produced the mathematical discussion and proofs.Source: §VII · p. 8