State constrained patchy feedback stabilization

TL;DR

This paper develops a patchy feedback control method for stabilizing a control system within a constrained set, ensuring practical stabilization to a target set with robustness to disturbances.

Contribution

It introduces a novel patchy feedback construction that guarantees practical stabilization under state constraints, extending control theory for discontinuous feedbacks.

Findings

Constructed a patchy feedback for practical stabilization.

Proved the existence of local Carathéodory solutions.

Demonstrated robustness of the feedback to disturbances.

Abstract

We construct a patchy feedback for a general control system on $\R^n$ which realizes practical stabilization to a target set $\Sigma$, when the dynamics is constrained to a given set of states $S$. The main result is that $S$--constrained asymptotically controllability to $\Sigma$ implies the existence of a discontinuous practically stabilizing feedback. Such a feedback can be constructed in "patchy" form, a particular class of piecewise smooth controls which ensure the existence of local Carath\'eodory solutions to any Cauchy problem of the control system and which enjoy good robustness properties with respect to both measurement errors and external disturbances.