Inhaltszusammenfassung

This paper investigates unregularized tracking-type optimal control problems con- strained by the Boussinesq system. In our model, controls appear linearly and distributed in both the equations constituting the Boussinesq system and the objective functional. We establish the existence of weak solutions and the unique existence of strong solutions in $L^p$ for the Boussinesq sys- tem, its linearized form, and the corresponding adjoint system. After demonstrating the existence of an optimal sol...This paper investigates unregularized tracking-type optimal control problems con- strained by the Boussinesq system. In our model, controls appear linearly and distributed in both the equations constituting the Boussinesq system and the objective functional. We establish the existence of weak solutions and the unique existence of strong solutions in $L^p$ for the Boussinesq sys- tem, its linearized form, and the corresponding adjoint system. After demonstrating the existence of an optimal solution, we derive a first-order necessary condition for local optimality. Leveraging recently introduced sufficient optimality conditions based on the joint growth of the first and second variations, we establish the strong metric H\"older subregularity of the optimality mapping. This result enables the analysis of stability for the optimal control and states under various linear and nonlinear perturbations affecting both the Boussinesq system and the objective functional. As an application, we provide a convergence rate for the optimal solutions of the Tikhonov regularized problem as the Tikhonov parameter approaches zero. Additionally, utilizing the stability of the optimal states, we obtain, to the best of our knowledge, the first result on the stability of the second-order sufficient condition in affine PDE-constrained optimization for tracking-type objective functionals when the target is tracked sufficiently well.» weiterlesen» einklappen

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