Approximate and complete controllability of nonlinear systems to a convex target set

The research deals with complete and approximate controllability of the system $\text{(1)}\text{dx}\text{dy}\text{= f(t, x, u)}$, without control restraints to an arbitrary convex target set. First, some characterizations of complete controllability, to the target of (1) and a special case of (1) namely $\text{x}\text{?}\text{= A(t)x + k(t, u)}{}^7$ are given. As a consequence complete controllability is equivalent to null-controllability. Next certain equations are formulated. These are in the same spirit as J. P. Dauer's “A Controllability Technique for Nonlinear Systems” (J. Math. Anal. Appl. oo (1972), 442–451) and are utilized in the main contribution of the paper: Under certain convexity assumption, bounded perturbations of systems which are completely controllable to a fixed target G are completely controllable to G. Without the convexity assumption, but with perturbations satisfying a Lipschitz condition, approximate controllability to G of a perturbed system is equivalent to complete controllability to G of the unperturbed equation.