Problem
Research questions and friction points this paper is trying to address.
feedback stabilization
topological obstructions
non-convex domains
compact manifolds
navigation
Noise-Induced Navigation in Non-convex Domains and Compact Manifolds
🤖 AI Summary
This study addresses the topological obstruction that prevents continuous feedback from achieving globally asymptotically stable navigation on non-convex domains and manifolds. To overcome this theoretical limitation, the project introduces stochastic noise into the control channel and constructs stochastic feedback laws by integrating local Lyapunov stability with positive recurrence criteria. The proposed methodology is further extended to strongly convex optimization problems in Euclidean spaces with obstacles and on boundaryless manifolds. Numerical simulations conducted on circular obstacle domains and two-dimensional spheres validate the effectiveness of the approach while revealing metastability phenomena under non-convex obstacles. Ultimately, this work establishes a novel stochastic control theoretical framework for system navigation and optimization within environments characterized by complex topologies.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Stochastic feedback control
Global asymptotic stability
Non-convex domains
Compact manifolds
Metastability
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