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Stabilization and Optimal Control of an Interconnected $n + m$ Hetero-directional Hyperbolic PDE-SDE System

Gabriel Velho, Jean Auriol, Islam Boussaada, Riccardo Bonalli

TL;DR

This work addresses stabilization and variance minimization for a PDE-SDE interconnection where a linear SDE is driven by a linear hyperbolic PDE with $n+m$ transport equations, introducing multiple delays. The authors deploy backstepping to decouple the PDE, transforming the system into an input-delayed SDE with stochastic drift, and then apply Kalman decomposition, Artstein's predictor, and LQ control to achieve mean stability and variance minimization. They establish a structural lower bound on achievable variance and prove that, under standard controllability assumptions, the variance can be driven arbitrarily close to this limit; they also develop a stabilizing feedback that ensures mean convergence to zero with bounded variance. The paper further develops an optimal control framework for variance minimization, derives explicit results in special delay configurations, and demonstrates feasibility via simulations, highlighting robust performance for complex PDE-SDE networks with multiple delays.

Abstract

In this paper, we design a controller for an interconnected system composed of a linear Stochastic Differential Equation (SDE) controlled through a linear hetero-directional hyperbolic Partial Differential Equation (PDE). Our objective is to steer the coupled system to a desired final state on average, while keeping the variance-in-time as small as possible, improving robustness to disturbances. By employing backstepping techniques, we decouple the original PDE, reformulating the system as an input delayed SDE with a stochastic drift. We first establish a controllability result, shading light on lower bounds for the variance. This shows that the system can never improve variance below strict structural limits. Under standard controllability conditions, we then design a controller that drives the mean of the states while keeping the variance bounded. Finally, we analyze the optimal control problem of variance minimization along the entire trajectory. Under additional controllability assumptions, we prove that the optimal control can achieve any variance level above the fundamental structural limit.

Stabilization and Optimal Control of an Interconnected $n + m$ Hetero-directional Hyperbolic PDE-SDE System

TL;DR

This work addresses stabilization and variance minimization for a PDE-SDE interconnection where a linear SDE is driven by a linear hyperbolic PDE with transport equations, introducing multiple delays. The authors deploy backstepping to decouple the PDE, transforming the system into an input-delayed SDE with stochastic drift, and then apply Kalman decomposition, Artstein's predictor, and LQ control to achieve mean stability and variance minimization. They establish a structural lower bound on achievable variance and prove that, under standard controllability assumptions, the variance can be driven arbitrarily close to this limit; they also develop a stabilizing feedback that ensures mean convergence to zero with bounded variance. The paper further develops an optimal control framework for variance minimization, derives explicit results in special delay configurations, and demonstrates feasibility via simulations, highlighting robust performance for complex PDE-SDE networks with multiple delays.

Abstract

In this paper, we design a controller for an interconnected system composed of a linear Stochastic Differential Equation (SDE) controlled through a linear hetero-directional hyperbolic Partial Differential Equation (PDE). Our objective is to steer the coupled system to a desired final state on average, while keeping the variance-in-time as small as possible, improving robustness to disturbances. By employing backstepping techniques, we decouple the original PDE, reformulating the system as an input delayed SDE with a stochastic drift. We first establish a controllability result, shading light on lower bounds for the variance. This shows that the system can never improve variance below strict structural limits. Under standard controllability conditions, we then design a controller that drives the mean of the states while keeping the variance bounded. Finally, we analyze the optimal control problem of variance minimization along the entire trajectory. Under additional controllability assumptions, we prove that the optimal control can achieve any variance level above the fundamental structural limit.

Paper Structure

This paper contains 29 sections, 11 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notations
  4. Equations
  5. Objective
  6. Tracking problem within the PDE
  7. The Backstepping Transformation
  8. Solving the SPDE Through the Characteristic Method
  9. Well-Posedness and Regularity of the Solution
  10. A delayed SDE
  11. Covariance Steering of the SDE
  12. Minimal variance computation
  13. Finite Time Controllability
  14. Stabilization of the PDE+SDE
  15. Optimal Control of the SDE
  16. ...and 14 more sections

Key Result

Lemma 3.1

Let $V_{SDE}$ be in $L^2_\mathcal{F}([0,T] ; \mathbb{R}^m)$ . The input $V_{eff} \in L^2_\mathcal{F}([0,T] ; \mathbb{R}^m)$, recursively defined as with $F_{ij}$ a sequence of functions defined recursively in the Appendix in eq:definition_2_F_multi_Veff and eq:definition_1_F_multi_Veff, yields the following expression for $\beta(t,0)$: where $G_i$ is a deterministic $C^1$ function defined recur

Figures (1)

  • Figure 1: Mean and deviation (square root of the variance) of the two components of the SDE state are depicted. The values were computed through a Monte-Carlo approach.

Theorems & Definitions (31)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Theorem 4.1: Kalman canonical form extended
  • proof
  • Remark 4.2
  • Lemma 4.3
  • ...and 21 more