A framework for randomized time-splitting in linear-quadratic optimal control

TL;DR

This paper introduces a randomized time-splitting framework for linear-quadratic optimal control, demonstrating convergence and potential computational savings in large-scale systems through theoretical analysis and numerical experiments.

Contribution

It proposes a novel randomized time-splitting method for linear-quadratic control problems, with proven convergence and efficiency benefits over traditional approaches.

Findings

Convergence of the randomized dynamics and optimal control in expectation.

Validation of convergence rates through numerical experiments.

Potential reduction in computational cost for large-scale systems.

Abstract

Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.