  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.
