Stability of solutions for controlled nonlinear systems under perturbation of state constraints

TL;DR

This paper investigates the stability of solutions for controlled nonlinear systems with unbounded control under perturbations of time-varying state constraints, providing methods to approximate trajectories within constraints.

Contribution

It introduces a technique to find interior trajectories close to a reference, aiding convergence proofs for approximation schemes in control-affine systems.

Findings

Existence of neighboring trajectories within constraints

Approximation can be made arbitrarily close in $L^inity$ and $L^2$

Applicable to control-affine systems

Abstract

This paper tackles the problem of nonlinear systems, with sublinear growth but unbounded control, under perturbation of some time-varying state constraints. It is shown that, given a trajectory to be approximated, one can find a neighboring one that lies in the interior of the constraints, and which can be made arbitrarily close to the reference trajectory both in $L^\infty$-distance and $L^2$-control cost. This result is an important tool to prove the convergence of approximation schemes of state constraints based on interior solutions and is applicable to control-affine systems.