Geometric enforcing of sequential convergence in fixed-time stability: An anisotropic vector field approach

This paper addresses the problem of enforcing a strict temporal sequence on the convergence of state components in affine nonlinear systems. Unlike standard finite-time stabilization, where states converge synchronously or in an arbitrary order, we propose a geometric framework that shapes the phase-space trajectories into a prespecified topological hierarchy. We introduce an anisotropic weighted sign operator that renders the closed-loop system fixed-time stable while imposing an invariant order on the hitting times of the coordinate axes. An explicit nonasymptotic settling-time bound and a sufficient disturbance-rejection condition are derived, and the controller is extended to generic affine nonlinear dynamics via input–output feedback linearization. To accommodate unmodelled perturbations, a projected residual actor–critic policy is added on top of the baseline; we prove via a composite Lyapunov argument that, provided that the residual field is confined within a state-dependent descent cone, both fixed-time stability and sequential ordering are preserved. The framework is validated on a three-dimensional integrator benchmark and on a rigid-body spacecraft attitude case study with a 20% inertia mismatch.