  This paper investigates the norm and time optimal control problems for
stochastic heat equations. We begin by presenting a characterization of the
norm optimal control, followed by a discussion of its properties. We then
explore the equivalence between the norm optimal control and time optimal
control, and subsequently establish the bang-bang property of the time optimal
control. These problems, to the best of our knowledge, are among the first to
discuss in the stochastic case.
