A controlled closing theorem

In a number of problems in the theory of nonlinear oscillations and the theory of nonlinear control systems, one must investigate the behavior of solutions of differential systems under small changes of right-hand sides. In particular, the following problem is of interest: is it possible to transform a nearly periodic motion (for example, an almost periodic or recurrent motion) into a periodic motion with the use of small admissible changes of the right-hand side? The positive answer to this question was given in the so-called closing lemma proved by C. Pugh. However, in problems of control of oscillations, which have been intensively studied in the recent years, arbitrary variations of the right-hand sides are not allowed; only variations compatible with the actual capabilities of the control are admissible. Therefore, it is of interest to generalize the closing lemma to controlled systems. Some conditions of this type have been stated in the previous works by the authors. They essentially pertain to control theory and provide a criterion for the controllability of a nonlinear system near a recurrent trajectory by controls small in the uniform metric. In the present paper, we give a rigorous statement and a complete proof of this assertion (referred to as the controlled closing theorem).