  We study synthesis of controllers for real-time systems, where the objective
is to stay in a given safe set. The problem is solved by obtaining winning
strategies in concurrent two-player \emph{timed automaton games} with safety
objectives. To prevent a player from winning by blocking time, we restrict each
player to strategies that ensure that the player cannot be responsible for
causing a zeno run. We construct winning strategies for the controller which
require access only to (1) the system clocks (thus, controllers which require
their own internal infinitely precise clocks are not necessary), and (2) a
linear (in the number of clocks) number of memory bits. Precisely, we show that
a memory of size $\big(3\cdot|C|+1 + \lg(|C|+1)\big)$ bits suffices for winning
controller strategies for safety objectives, where $C$ is the set of clocks of
the timed automaton game, significantly improving the previous known
exponential bound. We also settle the open question of whether \emph{region}
strategies for controllers require memory for safety objectives by showing with
an example that region strategies do require memory for safety objectives.
