Robust Safety-Critical Control of Integrator Chains with Mismatched Perturbations via Linear Time-Varying Feedback

In this paper, we propose a novel safety-critical control framework for a chain of integrators subject to both matched and mismatched perturbations. The core of our approach is a linear, time-varying state-feedback design that simultaneously enforces stability and safety constraints. By integrating backstepping techniques with a quadratic programming (QP) formulation, we develop a systematic procedure to guarantee safety under time-varying gains. We provide rigorous theoretical guarantees for the double integrator case, both in the presence and absence of perturbations, and outline general proofs for extending the methodology to higher-order chains of integrators. This proposed framework thus bridges robustness and safety-critical performance, while overcoming the limitations of existing prescribed-time approaches.