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

Gabriel Velho, Jean Auriol, Riccardo Bonalli

TL;DR

This work addresses risk-averse control for interconnected PDE–SDE systems by formulating a risk measure–based objective rather than a purely expected-cost criterion. It develops a three-step pipeline: reformulating the PDE–SDE into an SPDE, projecting to a finite-dimensional SDE via Galerkin methods, and applying a gradient-based risk-averse optimization using CVaR. The approach yields a tractable algorithm that reduces the tail of the cost distribution with only a small sacrifice in average performance, demonstrated on a heat-like PDE–SDE example. The framework provides a foundation for robust control of distributed stochastic systems and motivates future work on theoretical guarantees and nonlinear extensions.

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.

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

TL;DR

This work addresses risk-averse control for interconnected PDE–SDE systems by formulating a risk measure–based objective rather than a purely expected-cost criterion. It develops a three-step pipeline: reformulating the PDE–SDE into an SPDE, projecting to a finite-dimensional SDE via Galerkin methods, and applying a gradient-based risk-averse optimization using CVaR. The approach yields a tractable algorithm that reduces the tail of the cost distribution with only a small sacrifice in average performance, demonstrated on a heat-like PDE–SDE example. The framework provides a foundation for robust control of distributed stochastic systems and motivates future work on theoretical guarantees and nonlinear extensions.

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.

Paper Structure

This paper contains 15 sections, 3 theorems, 31 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Control system
  4. Cost and optimization problem
  5. Method overview
  6. Approximation of the system by a SDE
  7. Recasting into a SPDE
  8. Finite dimensional approximation
  9. Some convergence results
  10. Gradient Method for Finite Dimensional Risk Averse Control
  11. Choice of control space
  12. Recasting into a min--max problem
  13. A gradient ascent-descent algorithm
  14. Numerical results
  15. Conclusion

Key Result

Proposition 2

The augmented state $Z_t$ verifies the following SPDE with where $\gamma_0 \in H^1$ is the Riesz representation of the continuous linear form

Figures (2)

  • Figure 1: Structure of an interconnected parabolic PDE -- SDE
  • Figure 2: Comparison of initial and final distribution of the cost on $10000$ realizations of the trajectories

Theorems & Definitions (4)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4