captureattract

Captureattract is a term used in discussions of dynamical systems and control theory to describe a mechanism by which system trajectories are guided into and stabilized within an attracting set. The concept blends a capture phase—driving states into a target region—with an attract phase—ensuring convergence to a stable long-term behavior.

Formal framework: Consider a dynamical system ẋ = f(x,u) with state x in R^n and control u in U. A target attractor A ⊆ R^n is a compact set with an attracting neighborhood. A captureattract control law u = k(x) aims to render A globally or regionally attractive for the closed-loop system. A common sufficient condition is the existence of a Lyapunov function V: R^n → R_+ with V(A) = 0, V(x) > 0 for x ∉ A, and V̇ ≤ -W(x) with W(x) > 0 outside A. The capture phase refers to the design of k(x) to ensure trajectories enter a neighborhood of A in finite time, after which the attract phase guarantees asymptotic convergence to A.

Variants and methods: Captureattract can be implemented in continuous- or discrete-time settings, and via techniques such as feedback linearization, backstepping, model predictive control, or hybrid switching strategies. It is often analyzed with invariant sets, barrier functions, and Lyapunov-based proofs.

Applications: The approach is used in robotics for safe docking and stabilization, autonomous navigation to docking or equilibrium regions, and process control where a system must be driven into and held near a desired operating point.

Relation to related terms: It relates to basins of attraction, invariant sets, and stabilization theory, and is distinguished by emphasizing a controlled transition into the attractor as part of the design.