  In this paper, we study a certain approximation property for a time optimal
control problem of the heat equation with $L^\infty$-potential. We prove that
the optimal time and the optimal control to the same time optimal control
problem for the heat equation, where the potential has a small perturbation,
are close to those for the original problem. We also verify that for the heat
equation with a small perturbation in the potential, one can construct a new
time optimal control problem, which has the same target as that of the original
problem, but has a different control constraint bound from that of the original
problem, such that the new and the original problems share the same optimal
time, and meanwhile the optimal control of the new problem is close to that of
the original one. The main idea to approach such approximation is an
appropriate use of an equivalence theorem of minimal norm and minimal time
control problems for the heat equations under consideration. This theorem was
first established by G.Wang and E. Zuazua in [20] for the case where the
controlled system is an internally controlled heat equation without the
potential and the target is the origin of the state space.
