Time optimal sampled-data controls for the heat equation

In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls.The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal 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 time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.