Optimal control of an interconnected parabolic PDE-SDE system
Abstract
We consider the optimal control of an interconnected system in which a finite-dimensional linear stochastic differential equation (SDE) with multiplicative noise is actuated through a parabolic partial differential equation (PDE) of Sturm-Liouville type. The two subsystems are coupled in a chain structure: the control acts at one boundary of the PDE, while the opposite boundary is bilaterally coupled with the stochastic subsystem. Such dynamics arise, for instance, in district heating networks, where heat is delivered to a building through a thermal emitter, and where the building thermal dynamics are subject to random fluctuations. Our objective is to compute the feedback law that minimizes a quadratic cost associated with the state of the SDE component. We first perform a change of variables that moves the actuation inside the PDE domain, at the price of an additional regularity requirement on the control input, and recast the resulting dynamics as a linear stochastic PDE (SPDE) posed on an augmented Hilbert space. This reformulation turns the original problem into a well-posed linear quadratic (LQ) problem for which the optimal control is given in feedback form through an operator-valued Riccati equation. We then discretize the augmented dynamics by projecting them onto the eigenfunctions of the Sturm-Liouville operator, and compute the optimal control of the resulting finite-dimensional SDE. We prove that the approximated Riccati operator, the associated feedback law, the closed-loop dynamics and the optimal cost all converge to their infinite-dimensional counterparts as the number of retained modes grows. Numerical simulations illustrate the effectiveness of the approach.
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BibTeX
@misc{velho2026optimal,
title = {{Optimal control of an interconnected parabolic PDE-SDE system}},
author = {Gabriel Velho and Jean Auriol and Islam Boussaada and Riccardo Bonalli},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05780150v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal control of an interconnected parabolic PDE-SDE system
AU - Gabriel Velho
AU - Jean Auriol
AU - Islam Boussaada
AU - Riccardo Bonalli
PY - 2026
DA - 2026/10/06
PB - HAL
UR - https://hal.science/hal-05780150v1
AB - We consider the optimal control of an interconnected system in which a finite-dimensional linear stochastic differential equation (SDE) with multiplicative noise is actuated through a parabolic partial differential equation (PDE) of Sturm-Liouville type. The two subsystems are coupled in a chain structure: the control acts at one boundary of the PDE, while the opposite boundary is bilaterally coupled with the stochastic subsystem. Such dynamics arise, for instance, in district heating networks, where heat is delivered to a building through a thermal emitter, and where the building thermal dynamics are subject to random fluctuations. Our objective is to compute the feedback law that minimizes a quadratic cost associated with the state of the SDE component. We first perform a change of variables that moves the actuation inside the PDE domain, at the price of an additional regularity requirement on the control input, and recast the resulting dynamics as a linear stochastic PDE (SPDE) posed on an augmented Hilbert space. This reformulation turns the original problem into a well-posed linear quadratic (LQ) problem for which the optimal control is given in feedback form through an operator-valued Riccati equation. We then discretize the augmented dynamics by projecting them onto the eigenfunctions of the Sturm-Liouville operator, and compute the optimal control of the resulting finite-dimensional SDE. We prove that the approximated Riccati operator, the associated feedback law, the closed-loop dynamics and the optimal cost all converge to their infinite-dimensional counterparts as the number of retained modes grows. Numerical simulations illustrate the effectiveness of the approach.
ER -