  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. Our
design is reasonable from perspective of sampled-data controls. And it might
provide a right way for the numerical approach of a time optimal distributed
control problem, via the corresponding semi-discretized (in time variable) time
optimal control problem.
  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 also prove that such
estimates are optimal in some sense.
