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Optimal Control of an Interconnected SDE -Parabolic PDE System

Gabriel Velho, Jean Auriol, Islam Boussaada, Riccardo Bonalli

TL;DR

We study control of an interconnected system where a linear SDE with state $X_t$ is actuated by a linear parabolic PDE with state $u(t,x)$ and driven by noise $dW_t$. The problem is reformulated as an SPDE-based Linear Quadratic (LQ) control problem and solved via a spectral discretization of the Laplacian to produce a finite-dimensional optimal feedback. Convergence results show that the approximated Riccati operator $\Pi_N$ and the associated closed-loop dynamics converge to their infinite-dimensional counterparts as the truncation parameter $N\to\infty$, and the cost converges as well. Numerical simulations on a representative example confirm the effectiveness of the finite-dimensional controller in stabilizing the stochastic plant and illustrate the practical viability of the proposed approach for PDE-SDE couplings.

Abstract

In this paper, we design a controller for an interconnected system where a linear Stochastic Differential Equation (SDE) is 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. Our goal is to develop a computational method for approximating the controller that minimizes a quadratic cost associated with the state of the SDE component. To achieve this, we first perform a change of variables to shift the actuation inside the PDE domain and reformulate the system as a linear Stochastic Partial Differential Equation (SPDE). We use a spectral approximation of the Laplacian operator to discretize the coupled dynamics into a finite-dimensional SDE and compute the optimal control for this approximated system. The resulting control serves as an approximation of the optimal control for the original system. We then establish the convergence of the approximated optimal control and the corresponding closed-loop dynamics to their infinite-dimensional counterparts. Numerical simulations are provided to illustrate the effectiveness of our approach.

Optimal Control of an Interconnected SDE -Parabolic PDE System

TL;DR

We study control of an interconnected system where a linear SDE with state is actuated by a linear parabolic PDE with state and driven by noise . The problem is reformulated as an SPDE-based Linear Quadratic (LQ) control problem and solved via a spectral discretization of the Laplacian to produce a finite-dimensional optimal feedback. Convergence results show that the approximated Riccati operator and the associated closed-loop dynamics converge to their infinite-dimensional counterparts as the truncation parameter , and the cost converges as well. Numerical simulations on a representative example confirm the effectiveness of the finite-dimensional controller in stabilizing the stochastic plant and illustrate the practical viability of the proposed approach for PDE-SDE couplings.

Abstract

In this paper, we design a controller for an interconnected system where a linear Stochastic Differential Equation (SDE) is 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. Our goal is to develop a computational method for approximating the controller that minimizes a quadratic cost associated with the state of the SDE component. To achieve this, we first perform a change of variables to shift the actuation inside the PDE domain and reformulate the system as a linear Stochastic Partial Differential Equation (SPDE). We use a spectral approximation of the Laplacian operator to discretize the coupled dynamics into a finite-dimensional SDE and compute the optimal control for this approximated system. The resulting control serves as an approximation of the optimal control for the original system. We then establish the convergence of the approximated optimal control and the corresponding closed-loop dynamics to their infinite-dimensional counterparts. Numerical simulations are provided to illustrate the effectiveness of our approach.

Paper Structure

This paper contains 18 sections, 8 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notations
  4. Control system
  5. Optimal Control Problem
  6. Recasting into an SPDE LQ problem
  7. Change of variable
  8. Well-posedness of the system
  9. Existence of an optimal controller
  10. Discretization of the System
  11. Finite-Dimensional System
  12. Approximation of the optimal feedback control
  13. Convergence Results
  14. Convergence of the Dynamic Operators
  15. Convergence of the Feedback Operator
  16. ...and 3 more sections

Key Result

Proposition 2

Let $V(\cdot)$ be a stochastic process in $L^2_{\mathcal{F}_t}([0,T], \mathbb{R})$. Then, equation eq:PDE_z_after_control_change_variable_V has a unique solution. Moreover, the state $( z , X)$ enjoy the following regularity:

Figures (1)

  • Figure 1: Realization of a closed-loop trajectory with $N=3$

Theorems & Definitions (15)

  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • Proposition 6
  • proof
  • ...and 5 more