Rigorous finite-time stability proof for second-order terminal sliding mode control

Establish a fully rigorous proof of finite-time stability for second-order Terminal Sliding Mode Control systems, explicitly accounting for the phase-plane trajectory geometry near the sliding manifold and avoiding an unverified assumption of smooth entry.

Background

The paper studies continuous sliding mode control for classical second-order systems and shows that trajectories need not enter the designed sliding surface tangentially. Instead, the asymptotic state ratio can bifurcate according to controller parameters and initial conditions, with some trajectories converging along an implicit secondary manifold.

This geometric observation is relevant to finite-time stability analyses of terminal sliding mode controllers because such proofs require precise bounds on the phase-plane trajectory near the sliding manifold. The authors argue that conventional Lyapunov-based arguments may implicitly rely on favorable smooth-entry geometry, whereas the linear sliding mode prototype demonstrates that smooth entry can be non-global and parameter-dependent. The paper does not provide the requested rigorous finite-time stability proof for second-order Terminal Sliding Mode Control systems.