Adaptive Gradient Descent for Optimal Control of Parabolic Equations with Random Parameters

TL;DR

This paper extends the AdaGrad algorithm to optimize control of parabolic PDEs with uncertain parameters, achieving improved convergence and demonstrating its effectiveness in thermal regulation of batteries.

Contribution

It introduces an adaptive gradient method for infinite-dimensional stochastic control problems and proves its convergence under specific conditions.

Findings

Convergence of the algorithm is established for the control of parabolic PDEs.

The method improves convergence rates compared to non-adaptive approaches.

Application to battery thermal regulation demonstrates practical effectiveness.

Abstract

In this paper we extend the adaptive gradient descent (AdaGrad) algorithm to the optimal distributed control of parabolic partial differential equations with uncertain parameters. This stochastic optimization method achieves an improved convergence rate through adaptive scaling of the gradient step size. We prove the convergence of the algorithm for this infinite dimensional problem under suitable regularity, convexity, and finite variance conditions, and relate these to verifiable properties of the underlying system parameters. Finally, we apply our algorithm to the optimal thermal regulation of lithium battery systems under uncertain loads.