On the solution stability of parabolic optimal control problems
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915621819645952 |
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| author | Corella, Alberto Domínguez Jork, Nicolai Veliov, Vladimir M. |
| author_facet | Corella, Alberto Domínguez Jork, Nicolai Veliov, Vladimir M. |
| contents | The paper investigates stability properties of solutions of optimal control problems for semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may non-linearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper -- the proved metric subregularity of the mapping associated with the system of first-order necessary optimality conditions. This property also enables error estimates for approximation methods. Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_08925 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the solution stability of parabolic optimal control problems Corella, Alberto Domínguez Jork, Nicolai Veliov, Vladimir M. Optimization and Control 49K20, 35K58, 49K40, 49J40 The paper investigates stability properties of solutions of optimal control problems for semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may non-linearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper -- the proved metric subregularity of the mapping associated with the system of first-order necessary optimality conditions. This property also enables error estimates for approximation methods. Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product. |
| title | On the solution stability of parabolic optimal control problems |
| topic | Optimization and Control 49K20, 35K58, 49K40, 49J40 |
| url | https://arxiv.org/abs/2209.08925 |