Fixed-Time Adaptive Stabilization of Underactuated Euler–Lagrange Systems with Certified Internal Dynamics

This paper addresses the fixed-time adaptive stabilization problem for a class of underactuated mechanical systems governed by Euler–Lagrange dynamics with matched parametric uncertainty. Unlike conventional adaptive schemes that guarantee only asymptotic convergence and usually assume stable internal dynamics, we develop a framework that (i) drives the actuated coordinates and the parameter-adaptive sliding manifold to the origin in a fixed time whose upper bound is independent of the initial condition, and (ii) supplies an explicit Lyapunov certificate for the zero dynamics induced by underactuation, thereby removing the minimum-phase assumption. The controller couples partial feedback linearization with a recursive fixed-time backstepping design and an online σ-modified adaptation law that preserves the structural properties of Euler–Lagrange systems. A composite Lyapunov function handles the coupled actuated/unactuated dynamics and yields a differential inequality of the form V˙≤−αVp−βVq with 0 1, which certifies fixed-time reaching on the actuated channel. A separate internal-dynamics certificate establishes uniform boundedness of the unactuated coordinate throughout the fixed-time reaching phase and its asymptotic convergence once the actuated manifold is reached, so all closed-loop signals are globally bounded and the full state converges to the origin under matched uncertainty. Numerical studies on the translational oscillator with rotational actuator (TORA) and the cart–pole show the initial-condition-independent settling of the actuated channel and quantify the advantage over an asymptotic adaptive baseline.