A Gradient Method for Risk Averse Control of a PDE-SDE Interconnected System

TL;DR

This paper develops a gradient-based control method for interconnected PDE-SDE systems that minimizes risk, effectively reducing rare but severe undesirable events while maintaining near-optimal average performance.

Contribution

It introduces a novel risk-averse control framework for PDE-SDE systems using a finite-dimensional approximation and gradient optimization, addressing limitations of risk-neutral methods.

Findings

The proposed controller significantly reduces the tail risk of the cost distribution.

Numerical results demonstrate improved system reliability with minimal impact on average performance.

Abstract

In this paper, we design a risk-averse controller for an interconnected system composed of a linear Stochastic Differential Equation (SDE) actuated through a linear parabolic heat equation. These dynamics arise in various applications, such as coupled heat transfer systems and chemical reaction processes that are subject to disturbances. While existing optimal control methods for these systems focus on minimizing average performance, this risk-neutral perspective may allow rare but highly undesirable system behaviors. To account for such events, we instead minimize the cost within a coherent risk measure. Our approach reformulates the coupled dynamics as a stochastic PDE, approximates it by a finite-dimensional SDE system, and applies a gradient-based method to compute a riskaverse feedback controller. Numerical simulations show that the proposed controller substantially reduces the tail of the cost distribution, improving reliability with only a minor reduction in average performance.