Characterization of Region of Exponential Attraction for Substabilization of FASs

This paper addresses the explicit characterization of the region of exponential attraction (RoEA) for sub-fully actuated systems (sub-FASs). By exploiting the eigenstructure of the fully actuated system (FAS) framework, the feasibility conditions of the closed-loop system are formulated as a family of time-dependent linear inequalities with respect to the initial states over an infinite time horizon. The proposed RoEA characterization method transforms the semi-infinite constraints into an equivalent finite set of active constraints, comprising the interval endpoints and, when they exist, the interior stationary points of the associated time-dependent functions. This transformation provides an exact finite-dimensional representation of the RoEA, converting the original infinite-horizon characterization problem into a tractable finite constraint description. In the two-dimensional case, the RoEA admits an explicit piecewise geometric characterization, where the boundary is composed of linear segments and nonlinear curves induced by the active constraints. The framework is further extended to higher-dimensional systems by establishing a finite active-set characterization under mild assumptions. Numerical examples, including a pendulum system and a Lorenz system, demonstrate that the proposed method accurately captures practically meaningful attraction regions while significantly reducing the complexity of RoEA characterization.