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Reliable control and state estimation of differential drive robots (DDR) operating in dynamic and uncertain environments remains a challenge, particularly when system dynamics are partially unknown and sensor measurements are prone to degradation. This work introduces a unified control and state estimation framework that combines a Lyapunov-based nonlinear controller and Adaptive Neural Networks (ANN) with Extended Kalman Filter (EKF)-based multi-sensor fusion. The proposed controller leverages the universal approximation property of neural networks to model unknown nonlinearities in real time. An online adaptation scheme updates the weights of the radial basis function (RBF), the architecture chosen for the ANN. The learned dynamics are integrated into a feedback linearization (FBL) control law, for which theoretical guarantees of closed-loop stability and asymptotic convergence in a trajectory-tracking task are established through a Lyapunov-like stability analysis. To ensure robust state estimation, the EKF fuses inertial measurement unit (IMU) and odometry from monocular, 2D-LiDAR and wheel encoders. The fused state estimate drives the intelligent controller, ensuring consistent performance even under drift, wheel slip, sensor noise and failure. Gazebo simulations and real-world experiments are done using DDR, demonstrating the effectiveness of the approach in terms of improved velocity tracking performance with reduction in linear and angular velocity errors up to $53.91\%$ and $29.0\%$ in comparison to the baseline FBL.