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PreprintCC BY-SA 4.0arXiv

Optimal Actuator Support Design for Infinite-Horizon Linear-Quadratic Parabolic Control Problems

  • Behzad Azmi
  • Daniel Wachsmuth
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Abstract

We study an optimal actuator design problem for infinite-horizon LQ control problems governed by parabolic equations. The control acts only on a small measurable subregion of the spatial domain, and the objective combines the infinite-horizon control value with a cost for the actuator region. Using the Riccati representation of the value function, we derive a comparison identity for two actuator supports and obtain an exact expansion of the objective. This leads to a topological derivative with respect to local changes in the support. We prove a localized Riccati sensitivity estimate to control the remainder and show that the infinite-horizon term in the derivative can be computed by solving a Lyapunov equation. This yields a computable support-update algorithm with a measure-controlled Armijo search. Under explicit uniform remainder and admissibility assumptions, its first-order stationarity defect tends to zero. Numerical experiments illustrate these results.

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Cite

BibTeXRIS
BibTeX
@misc{azmi2026optimal,
  title = {{Optimal Actuator Support Design for Infinite-Horizon Linear-Quadratic Parabolic Control Problems}},
  author = {Behzad Azmi and Daniel Wachsmuth},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.12553},
  eprint = {2611.12553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.12553},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Optimal Actuator Support Design for Infinite-Horizon Linear-Quadratic Parabolic Control Problems
AU  - Behzad Azmi
AU  - Daniel Wachsmuth
PY  - 2026
DA  - 2026/11/09
PB  - arXiv
DO  - 10.48550/arxiv.2611.12553
UR  - https://arxiv.org/abs/2611.12553
AB  - We study an optimal actuator design problem for infinite-horizon LQ control problems governed by parabolic equations. The control acts only on a small measurable subregion of the spatial domain, and the objective combines the infinite-horizon control value with a cost for the actuator region. Using the Riccati representation of the value function, we derive a comparison identity for two actuator supports and obtain an exact expansion of the objective. This leads to a topological derivative with respect to local changes in the support. We prove a localized Riccati sensitivity estimate to control the remainder and show that the infinite-horizon term in the derivative can be computed by solving a Lyapunov equation. This yields a computable support-update algorithm with a measure-controlled Armijo search. Under explicit uniform remainder and admissibility assumptions, its first-order stationarity defect tends to zero. Numerical experiments illustrate these results.
ER  -