Linear-Quadratic optimal control for boundary controlled networks of waves

TL;DR

This paper develops a method to compute optimal boundary controls for networks of wave systems modeled by hyperbolic PDEs, by discretizing infinite-dimensional systems into matrix-based discrete models and solving the LQ problem.

Contribution

It introduces a novel approach to convert infinite-dimensional hyperbolic systems into discrete matrix systems for LQ control, with applications to vibrating strings and heat exchangers.

Findings

Successfully computes boundary controls for wave networks.

Demonstrates method on vibrating strings and heat exchanger.

Provides a practical framework for boundary control of PDE networks.

Abstract

Linear-Quadratic optimal controls are computed for a class of boundary controlled, boundary observed hyperbolic infinite-dimensional systems, which may be viewed as networks of waves. The main results of this manuscript consist in converting the infinite-dimensional continuous-time systems into infinite-dimensional discrete-time systems for which the operators dynamics are matrices, in solving the LQ-optimal control problem in discrete-time and then in interpreting the solution in the continuous-time variables, giving rise to the optimal boundary control input. The results are applied to two examples, a small network of three vibrating strings and a co-current heat-exchanger, for which boundary sensors and actuators are considered.