Local equilibrium geometry at active-set boundaries in optimization-based feedback systems
Year: 2026
Type: article
Abstract: We study local equilibrium changes in smooth control-affine systems closed by minimum-deviation quadratic-program feedback. At a regular active-set boundary, the optimizer and the closed-loop vector field remain continuous, while the closed-loop Jacobian acquires an explicit rank-one or low-rank jump determined by the plant, quadratic-program metric, and active constraint derivatives. In the single-constraint case, a scalar crossing index distinguishes continuation of one admissible equilibrium branch from creation or annihilation of an admissible equilibrium pair. The singular value of this index yields explicit nullvectors, fold coefficients, and square-root branch scaling through Lyapunov–Schmidt reduction. For simultaneous activation of several previously inactive constraints, smooth active-set extensions generate a finite family of candidate branches, and multiplier and inactive-residual inequalities select their admissible parameter cones. A control-barrier-function quadratic program for obstacle avoidance shows how the index and the Jacobian jump are computed directly from standard safety-filter data. The results provide local branch-classification and stability tests for continuous-time, pointwise quadratic-program feedback under explicit regularity conditions.
Source: Systems & Control Letters
Author: Shen Zeng
Institution: Washington University in St. Louis
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,910
OpenAlex ID: W7213760109