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

Xiaotian Liang; Yanbo Wang; Lei Yang; Hui Cao; Shuangsi Xue; Xiaotian Liang; Yanbo Wang; Lei Yang; Hui Cao; Shuangsi Xue

doi:10.3934/math.2026867

AIMS Mathematics

2026, Volume 11, Issue 7: 21385-21411. doi: 10.3934/math.2026867

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

1.
National Key Laboratory for High Energy Pulsed Power, Xi'an Jiaotong University, Xi'an 710048, China
2.
School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China
3.
School of Automation and Information Engineering, Xi'an University of Technology, Xi'an 710048, China

Received: 17 March 2026 Revised: 28 May 2026 Accepted: 24 June 2026 Published: 17 July 2026
MSC : 93D40, 93C10, 93C40

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.
- sequential convergence,
- anisotropic vector field,
- Filippov differential inclusion,
- affine nonlinear systems,
- residual reinforcement learning,
- spacecraft attitude
Citation: Xiaotian Liang, Yanbo Wang, Lei Yang, Hui Cao, Shuangsi Xue. Geometric enforcing of sequential convergence in fixed-time stability: An anisotropic vector field approach[J]. AIMS Mathematics, 2026, 11(7): 21385-21411. doi: 10.3934/math.2026867

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Abstract

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.

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