We study supervisory control of timed event graphs in the max-algebra framework. Desirable behavior is assumed to be specified as a set of firing time sequences of transitions. We address a control problem which requires that behavior of the supervised system equal the specification set. A necessary and sufficient condition for the existence of a control input set to solve our problem is presented. We call the condition the strong controllability condition. However, the given specification is not necessarily strongly controllable. We then obtain its supremal strongly controllable subset. Moreover, we derive a necessary and sufficient condition for the supremal strongly controllable subset to be nonempty under the assumption that desirable behavior is specified by the upper time bound sequence.