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

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.