  This paper studies a kind of minimal time control problems related to the
exact synchronization for a controlled linear system of parabolic equations.
Each problem depends on two parameters: the bound of controls and the initial
state. The purpose of such a problem is to find a control (from a constraint
set) synchronizing components of the corresponding solution vector for the
controlled system in the shortest time. In this paper, we build up a necessary
and sufficient condition for the optimal time and the optimal control; we also
obtain how the existence of optimal controls depends on the above mentioned two
parameters.
