Learning the Riccati solution operator for time-varying LQR via Deep Operator Networks

2026-04-20

AI Summary

This work addresses the high computational burden of traditional finite-horizon linear quadratic regulator (LQR) control, which requires repeated online solution of differential Riccati equations and thus struggles to meet real-time demands. The paper introduces Deep Operator Networks (DeepONets) to the LQR problem for the first time, establishing an offline learning framework that directly maps time-varying system parameters to Riccati solution trajectories, thereby shifting the computational load from online execution to a one-time training phase. The proposed approach features a scalable network architecture tailored for matrix-valued time-varying signals and a progressive training strategy, accompanied by theoretical guarantees on error propagation of the approximate solution and closed-loop stability. Experiments demonstrate that the method achieves high accuracy, strong generalization, and substantial computational speedup across diverse time-varying and time-invariant LQR tasks, making it well-suited for parametric and real-time optimal control applications.

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Abstract

We propose a computational framework for replacing the repeated numerical solution of differential Riccati equations in finite-horizon Linear Quadratic Regulator (LQR) problems by a learned operator surrogate. Instead of solving a nonlinear matrix-valued differential equation for each new system instance, we construct offline an approximation of the associated solution operator mapping time-dependent system parameters to the Riccati trajectory. The resulting model enables fast online evaluation of approximate optimal feedbacks across a wide class of systems, thereby shifting the computational burden from repeated numerical integration to a one-time learning stage. From a theoretical perspective, we establish control-theoretic guarantees for this operator-based approximation. In particular, we derive bounds quantifying how operator approximation errors propagate to feedback performance, trajectory accuracy, and cost suboptimality, and we prove that exponential stability of the closed-loop system is preserved under sufficiently accurate operator approximation. These results provide a framework to assess the reliability of data-driven approximations in optimal control. On the computational side, we design tailored DeepONet architectures for matrix-valued, time-dependent problems and introduce a progressive learning strategy to address scalability with respect to the system dimension. Numerical experiments on both time-invariant and time-varying LQR problems demonstrate that the proposed approach achieves high accuracy and strong generalization across a wide range of system configurations, while delivering substantial computational speedups compared to classical solvers. The method offers an effective and scalable alternative for parametric and real-time optimal control applications.

Problem

• Linear Quadratic Regulator
• Riccati equation
• optimal control
• time-varying systems
• computational efficiency

Innovation

• Deep Operator Networks
• Differential Riccati Equation
• Linear Quadratic Regulator
• Operator Learning
• Exponential Stability