In this paper, we present a novel approach for synthesizing provable reach-avoid controllers. Our approach focuses on driving a deterministic discrete-time system, operating in an unknown environment, to safely reach a desired target set. Within the reachability analysis framework, we propose a method based on convex computation of inner-approximations of controlled reach-avoid sets (CRSs). The controlled reach-avoid set represents the set of states from which the system can enter the target set while remaining inside the safe set before the target hitting time. By defining the boundary of the CRS as a barrier, we effectively separate states that can achieve the reach-avoid objective from those that cannot. Thus, the computed inner-approximation provides a viable space for the system to accomplish the reach-avoid objective. Our approach for synthesizing reach-avoid controllers involves three main steps. Firstly, we employ a support vector machine approach to learn a safe set of states in the unknown environment using sensor measurements. Secondly, based on the learned safe set and target set, we compute an inner-approximation of the CRS by solving a convex optimization problem. Finally, we synthesize controllers online to ensure that the system evolves within the computed inner-approximation, ultimately reaching the target set. To illustrate the effectiveness of our proposed method, we demonstrate its application on a Dubin’s car system.