Schuster M, Gugat M (2024)
Publication Type: Journal article, Original article
Publication year: 2024
DOI: 10.1080/02331934.2024.2373904
The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an L^1-norm control cost with a weight γ>0" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">𝛾>0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared L^L2" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">2-norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the L1" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">L^1-norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong L^L2" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">2-sense.
APA:
Schuster, M., & Gugat, M. (2024). Optimal Neumann control of the wave equation with L1-control cost: the finite-time turnpike property. Optimization. https://doi.org/10.1080/02331934.2024.2373904
MLA:
Schuster, Michael, and Martin Gugat. "Optimal Neumann control of the wave equation with L1-control cost: the finite-time turnpike property." Optimization (2024).
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