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Deep reinforcement learning (DRL) has attracted much attention as an approach to solve optimal control problems without mathematical models of systems. On the other hand, in general, constraints may be imposed on optimal control problems. In this study, we consider the optimal control problems with constraints to complete temporal control tasks. We describe the constraints using signal temporal logic (STL), which is useful for time sensitive control tasks since it can specify continuous signals within bounded time intervals. To deal with the STL constraints, we introduce an extended constrained Markov decision process (CMDP), which is called a <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula>-CMDP. We formulate the STL-constrained optimal control problem as the <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula>-CMDP and propose a two-phase constrained DRL algorithm using the Lagrangian relaxation method. Through simulations, we also demonstrate the learning performance of the proposed algorithm.

Deep Reinforcement Learning Under Signal Temporal Logic Constraints Using Lagrangian Relaxation