  This paper presents the equivalence between minimal time and minimal norm
control problems for internally controlled heat equations. The target is an
arbitrarily fixed bounded, closed and convex set with a nonempty interior in
the state space. This study differs from [G. Wang and E. Zuazua, \textit{On the
equivalence of minimal time and minimal norm controls for internally controlled
heat equations}, SIAM J. Control Optim., 50 (2012), pp. 2938-2958] where the
target set is the origin in the state space. When the target set is the origin
or a ball, centered at the origin, the minimal norm and the minimal time
functions are continuous and strictly decreasing, and they are inverses of each
other. However, when the target is located in other place of the state space,
the minimal norm function may be no longer monotonous and the range of the
minimal time function may not be connected. These cause the main difficulty in
our study. We overcome this difficulty by borrowing some idea from the
classical raising sun lemma (see, for instance, Lemma 3.5 and Figure 5 on Pages
121-122 in [E. M. Stein and R. Shakarchi, \textit{Real Analysis: Measure
Theory, Integration, and Hilbert Spaces}, Princeton University Press, 2005]).
