This paper presents a framework for online adaptive optimal control of continuous-time linear systems with unknown dynamics. The approach uses approximate and adaptive dynamic programming to learn the optimal control policy and value function in real-time, without prior knowledge of the system matrices. We introduce two algorithms based on policy iteration and value iteration, providing proofs the convergence and stability. Our value iteration method is robust against from exploration noise. The effectiveness of these control strategies is demonstrated through two examples, highlighting their ability to achieve near-optimal performance despite unknown dynamics.

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