Incorporating Deep Neural Networks with LMI Constraints for Nonlinear Model Predictive Control Optimization

ABSTRACT This paper introduces a deep learning‐enhanced nonlinear model predictive control (MPC) framework, in which the control optimization problem is formulated as the minimization of a cost function subject to linear matrix inequality (LMI) constraints. The resulting constrained optimization is solved via semidefinite programming (SDP), ensuring feasible control inputs with robust stability and performance guarantees. To extend stability analysis beyond local linear approximations, deep neural networks (DNNs) are employed to construct neural Lyapunov functions for nonlinear systems. The framework features a learner‐falsifier loop: a learner proposes candidate control policies and Lyapunov functions, while a falsifier systematically searches for counterexamples that violate stability conditions. This iterative process integrates classical Lyapunov stability theory with deep learning, enabling the optimization of non‐convex cost functions and providing an efficient, automated approach to Lyapunov‐based control design. The method ensures stability of nonlinear systems under disturbances. Numerical validation—through phase portraits and time‐domain trajectory analysis—demonstrates strong agreement between the DNN‐derived Lyapunov functions and the solutions obtained from LMI‐based optimization, confirming the robustness and efficacy of the proposed integrated LMI‐MPC‐DNN methodology.