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This paper proposes an optimal autonomous search framework, namely Dual Control for Exploration and Exploitation (DCEE), for a target at unknown location in an unknown environment. Source localisation is to find sources of atmospheric hazardous material release in a partially unknown environment. This paper proposes a control theoretic approach to this autonomous search problem. To cope with an unknown target location, at each step, the target location is estimated by Bayesian inference. Then a control action is taken to minimise the error between future robot position and the hypothesised future estimation of the target location. The latter is generated by hypothesised measurements at the corresponding future robot positions (due to the control action) with the current estimation of the target location as a prior. It shows that this approach can take into account both the error between the next robot position and the estimate of the target location, and the uncertainty of the estimate. This approach is further extended to the case with not only an unknown source location, but also an unknown local environment (e.g. wind speed and direction). Different from current information theoretic approaches, this new control theoretic approach achieves the optimal trade-off between exploitation and exploration in a unknown environment with an unknown target by driving the robot moving towards estimated target location while reducing its estimation uncertainty. This scheme is implemented using particle filtering on a mobile robot. Simulation and experimental studies demonstrate promising performance of the proposed approach. The relationships between the proposed approach, informative path planning, dual control, and classic model predictive control are discussed and compared.