Issue |
ESAIM: COCV
Volume 31, 2025
|
|
---|---|---|
Article Number | 27 | |
Number of page(s) | 29 | |
DOI | https://doi.org/10.1051/cocv/2025014 | |
Published online | 24 March 2025 |
Perturbations in PDE-constrained optimal control decay exponentially in space
1
Chair of Scientific Computing, School of Business Informatics and Mathematics, University of Mannheim, Germany
2
Faculty of Mathematics, Chemnitz University of Technology, Germany
3
Optimization-based Control Group, Institute of Mathematics, Technische Universität Ilmenau, Germany
* Corresponding author: manuel.schaller@math.tu-chemnitz.de
Received:
22
March
2024
Accepted:
22
January
2025
For linear-quadratic optimal control problems (OCPs) governed by elliptic and parabolic partial differential equations, we investigate the impact of perturbations on optimal solutions. Local perturbations may occur, e.g., due to discretization of the optimality system or disturbed problem data. Whereas these perturbations may exhibit global effects in the uncontrolled case, we prove that the ramifications are exponentially damped in space under stabilizability and detectability conditions. To this end, we prove a bound on the optimality condition’s solution operator that is uniform in the domain size. Then, this uniformity is used in a scaling argument to show the exponential decay of perturbations in space. We numerically validate and illustrate our results by solving OCPs involving Helmholtz, Poisson, and advection-diffusion-reaction equations.
Mathematics Subject Classification: 35Q93 / 49K40 / 93D23
Key words: Sensitivity analysis / exponential stability / optimal control of partial differential equations
© The authors. Published by EDP Sciences, SMAI 2025
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