Neural Network Approximation of Algebraic Riccati Equation Solutions for Optimal Control

Algebraic Riccati Equations (AREs) are fundamental to optimal control design, particularly Linear Quadratic Regulator (LQR), Linear Quadratic Gaussian (LQG), and$H_{\infty}$control. Traditional numerical methods for solving AREs, such as Schur decomposition, eigenvector methods, and Newton's method, while reliable, require computationally intensive iterative algorithms. For large systems or applications requiring real-time control decisions, these computational demands can become impractical. At times, finding a stabilizing solution can be challenging. This paper explores the use of neural networks as a fast, data-driven alternative for approximating ARE solutions. In this work, we trained a neural network to predict the solution matrix$P$directly from system parameters, bypassing traditional iterative solvers. The network architecture consists of four hidden layers with ReLU activation, dropout regularization, and batch normalization. As input, a 30-dimensional vector was used which consisted of the flattened system matrix and five additional stability-related features. Using a motivating aircraft control dynamical system from the literature as our benchmark, we generated a training dataset of 50,000 system-solution pairs through controlled perturbations of the motivating system, maintaining the original sparsity structure while varying numerical values. The trained network achieved a relative error of 6.730% when predicting the ARE solution for the original aircraft system. Closedloop simulations demonstrate that this approximation error does not significantly degrade control performance, with system trajectories closely matching those obtained using traditional ARE solvers. Once trained, the neural network provides near-instantaneous predictions, offering substantial computational advantages over iterative methods. This preliminary study explores the feasibility of neural network-based ARE solvers and provides a foundation for future work extending the approach to higher-dimensional systems and infinite-dimensional systems governed by partial differential equations.