Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems

• Published in: Journal of Complexity Vol. 92; p. 101976
• Main Authors: Löscher, Richard, Reichelt, Michael, Steinbach, Olaf
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.2026
• Subjects: Distributed optimal control; Heat equation; Space-time finite element methods; Space-time finite element methods; Heat equation; Distributed optimal control; primary
• ISSN: 0885-064X
• DOI: 10.1016/j.jco.2025.101976

In this paper we consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The approach presented here is based on the variational formulation of the parabolic evolution equation in anisotropic Sobolev spaces, considering the control in [H0;,01,1/2(Q)]⁎. Since the state equation defines an isomorphism from H0;0,1,1/2(Q) onto [H0;,01,1(Q)]⁎, we can eliminate the control to end up with a minimization problem in H0;0,1,1/2(Q) where the anisotropic Sobolev norm can be realized using a modified Hilbert transformation. In the unconstrained case, the minimizer is the unique solution of a singularly perturbed elliptic equation. In the case of a space-time tensor-product mesh, we can use sparse factorization techniques to construct a solver of almost linear complexity. Numerical examples also include additional state constraints, and a nonlinear state equation.