In an epidemic, individuals can widely differ in the way they spread the infection, for instance depending on their age or on the number of days they have been infected for. The latter allows to take into account the variation of infectiousness as a function of time since infection. In the absence of pharmaceutical interventions such as a vaccine or treatment, non-pharmaceutical interventions (e.g. social distancing) are of great importance to mitigate the pandemic. We propose a model with a double continuous structure by host age and time since infection. By applying optimal control theory to our age-structured model, we identify a solution minimizing deaths and costs associated with the implementation of the control strategy itself. This strategy depends on the age heterogeneity between individuals and consists in a relatively high isolation intensity over the older populations during a hundred days, followed by a steady decrease in a way that depends on the cost associated to a such control. The isolation of the younger population is weaker and occurs only if the cost associated with the control is relatively low. We show that the optimal control strategy strongly outperforms other strategies such as uniform constant control over the whole populations or over its younger fraction. These results bring new facts the debate about age-based control interventions and open promising avenues of research, for instance of age-based contact tracing.