The problem of minimizing the maximum transient energy growth is considered. This problem has importance in some fluid flow control problems and other classes of non-linear systems. Conditions for the existence of static controllers that restrict the maximum transient energy growth to unity are established. An explicit parametrization of all linear controllers ensuring monotonic decrease of the transient energy is derived. It is shown that by means of a Q-parametrization, the problem of minimizing the maximum transient energy growth can be posed as a convex optimization problem that can be solved by means of a Ritz approximation of the free parameter. By considering the transient energy growth at an appropriate sequence of discrete time points, the minimal maximum transient energy growth problem can be posed as a semidefinite problem. The theoretical developments are demonstrated on two numerical problems.