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On quantitative sufficient second-order optimality conditions for elliptic optimal control problems

  • Francisco Fuica
  • Nicolai Jork
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Abstract

In this paper, a general principle for coercivity in optimal control problems is investigated. Utilizing regularity properties of solutions to second-order elliptic PDEs on Lipschitz domains, we provide quantitative sufficient second-order optimality condition for control-constrained semilinear elliptic optimal control problems. The main property of the investigated optimal control problems is the absence of a Tikhonov regularization term on the objective function. For these problems, the second variation is a quadratic form in the linearized state, and its curvature function may vanish or change sign. The quantitative condition we introduce implies coercivity with respect to the L2-norm of the linearized-states. As a consequence, the stability of the sufficient second-order condition under perturbations of the states and the tracking data is shown.

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Cite

BibTeXRIS
BibTeX
@misc{fuica2026quantitative,
  title = {{On quantitative sufficient second-order optimality conditions for elliptic optimal control problems}},
  author = {Francisco Fuica and Nicolai Jork},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.14525},
  eprint = {2608.14525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.14525},
}
RIS
TY  - UNPB
TI  - On quantitative sufficient second-order optimality conditions for elliptic optimal control problems
AU  - Francisco Fuica
AU  - Nicolai Jork
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2608.14525
UR  - https://arxiv.org/abs/2608.14525
AB  - In this paper, a general principle for coercivity in optimal control problems is investigated. Utilizing regularity properties of solutions to second-order elliptic PDEs on Lipschitz domains, we provide quantitative sufficient second-order optimality condition for control-constrained semilinear elliptic optimal control problems. The main property of the investigated optimal control problems is the absence of a Tikhonov regularization term on the objective function. For these problems, the second variation is a quadratic form in the linearized state, and its curvature function may vanish or change sign. The quantitative condition we introduce implies coercivity with respect to the L^2-norm of the linearized-states. As a consequence, the stability of the sufficient second-order condition under perturbations of the states and the tracking data is shown.
ER  -