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On the switching behavior of sparse optimal controls for the one-dimensional heat equation

  • * Corresponding author: Fredi Tröltzsch

    * Corresponding author: Fredi Tröltzsch 
Daniel Wachsmuth was partially supported by the German Research Foundation DFG under project grant Wa 3626/1-1.
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  • An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $ν$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $ν ≥ 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $ν \searrow 0$.

    Mathematics Subject Classification: Primary: 49K20; Secondary: 49J30, 49N10.


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