Minimal time sampled-data controls for the ordinary differential system

In this paper, we design a minimal time optimal control problem for an ordinary differential system with sampled-data controls, and use it to approximate a time optimal control problem for the ordinary differential system with distributed controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the minimal time  sampled-data control problem and the minimal time distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.