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

2025/07/10 by Bálint Farkas, Farkas, Bálint, Birgit Jacob +5 · 1 citation
Engineering · Mathematics · #46N10 #49M27 #49N10 #65M55 #65Y05 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.07812

openalex publication_date 2025/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 C₀-(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.

Citations

• A Parallel-in-Time Multigrid Preconditioner for Optimal Control
• New time domain decomposition methods for parabolic optimal control problems II: Neumann-Neumann algorithms
• A 2-Level Domain Decomposition Preconditioner for KKT Systems with Heat-Equation Constraints
• Layer-Parallel Training of Deep Residual Neural Networks
• From forms to semigroups
• Splitting Algorithms for the Sum of Two Nonlinear Operators

Cited by

• Iterative solvers for partial differential equations with dissipative structure: Operator preconditioning and optimal control

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