This talk presents certain results on time optimal control problems for heat equations (with

and without potential a(x; t)). Two kinds of time optimal control problems are concerned. One

is to initiate immediately control so as to reach the goal as early as possible, and another is to

reach the goal by a fixed time T while delaying initiation of active control to as late as possible.

The outline of the talk is as follows: (1) Introduction; (2) The existence of optimal controls; (3)

The bang-bang property and uniqueness of the optimal control; (4) The equivalence between

time and norm optimal control controls; (5) Necessary and suffcient conditions on the optimal