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This work examines the use of reinforcement learning (RL) to optimize cyclic lockdowns, which is one of the methods available for control of the COVID-19 pandemic. The problem is structured as an optimal control system for tracking a reference value, corresponding to the maximum usage level of a critical resource, such as ICU beds. However, instead of using conventional optimal control methods, RL is used to find optimal control policies. A framework was developed to calculate optimal cyclic lockdown timings using an RL-based on-off controller. The RL-based controller is implemented as an RL agent that interacts with an epidemic simulator, implemented as an extended SEIR epidemic model. The RL agent learns a policy function that produces an optimal sequence of open/lockdown decisions such that goals specified in the RL reward function are optimized. Two concurrent goals were used: the first one is a public health goal that minimizes overshoots of ICU bed usage above an ICU bed threshold, and the second one is a socio-economic goal that minimizes the time spent under lockdowns. It is assumed that cyclic lockdowns are considered as a temporary alternative to extended lockdowns when a region faces imminent danger of overpassing resource capacity limits and when imposing an extended lockdown would cause severe social and economic consequences due to lack of necessary economic resources to support its affected population during an extended lockdown.