<![CDATA[This paper proposes an adaptive Lyapunov-based control approach using deep neural networks (DNN) for underactuated nonlinear systems, with case studies on the Furuta pendulum and a wheeled path-following system. This approach combines simultaneous learning of the Lyapunov function V(x) to satisfy the positive-definite condition and the control law u(x) to satisfy negative definiteness of V ̇(x) thus ensuring the asymptotic stability of the system. The proposed model is validated using Python-based simulation. Results show that the proposed method significantly expands the region of attraction (RoA) compared to the linear quadratic regulator (LQR) method. In the Furuta pendulum, the RoA area in the [θ−θ˙] plane increased from 89.04% to 101.14% and in the [α−α˙] plane from 80.28% to 83.79%. Meanwhile, in the wheeled path-following system, the RoA within safety domain increased from 85.28% to 101.69%. Furthermore, robustness tests showed that the controller can maintain tracking performance on a sinusoidal path and reject short disturbances without excessive safety boundary violations. The resulting control signal remained smooth, non-oscillatory, and within the actuator saturation limits, ensuring safe and energy-efficient control. This approach offers a significant contribution by integrating Lyapunov stability theory, deep learning, and online adaptation, resulting a robust and practical for nonlinear underactuated systems.]]>
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