Dissipativity-based time domain decomposition for optimal control of hyperbolic PDEs

TL;DR

This paper introduces a dissipativity-based time domain decomposition method for optimal control of hyperbolic PDEs, enabling parallelizable solutions with proven convergence.

Contribution

It develops a novel fixed-point iteration framework based on dissipative operators for decoupling and solving coupled PDEs in optimal control problems.

Findings

Method converges for hyperbolic PDEs like wave equations.

Parallelizable approach improves computational efficiency.

Numerical examples demonstrate effectiveness for 2D wave and 3D heat equations.

Abstract

We propose a time domain decomposition approach to optimal control of partial differential equations (PDEs) based on semigroup theoretic methods. We formulate the optimality system consisting of two coupled forward-backward PDEs, the state and adjoint equation, as a sum of dissipative operators, which enables a Peaceman-Rachford-type fixed-point iteration. The iteration steps may be understood and implemented as solutions of many decoupled, and therefore highly parallelizable, time-distributed optimal control problems. We prove the convergence of the state, the control, and the corresponding adjoint state in function space. Due to the general framework of $C_0$-(semi)groups, the results are particularly well applicable, e.g., to hyperbolic equations, such as beam or wave equations. We illustrate the convergence and efficiency of the proposed method by means of two numerical examples subject to a 2D wave equation and a 3D heat equation.