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In this paper, we will deal with a Linear Quadratic Optimal Control problemwith unknown dynamics. As a modeling assumption, we will suppose that theknowledge that an agent has on the current system is represented by aprobability distribution $\pi$ on the space of matrices. Furthermore, we willassume that such a probability measure is opportunely updated to take intoaccount the increased experience that the agent obtains while exploring theenvironment, approximating with increasing accuracy the underlying dynamics.Under these assumptions, we will show that the optimal control obtained bysolving the "average" Linear Quadratic Optimal Control problem with respect toa certain $\pi$ converges to the optimal control driven related to the LinearQuadratic Optimal Control problem governed by the actual, underlying dynamics.This approach is closely related to model-based Reinforcement Learningalgorithms where prior and posterior probability distributions describing theknowledge on the uncertain system are recursively updated. In the last section,we will show a numerical test that confirms the theoretical results.

Convergence results for an averaged LQR problem with applications to reinforcement learning