Here, J: U x Z → ℜ stands for the objective functional depending on the state variables u ∈ U and the control variables z ∈ Z. The equation corresponds to the PDE (linear/nonlinear), and U_{ad} ⊂ U, Z_{ad} ⊂ Z refer to the sets of admissible states and control variables, respectively. The variable z could be an optimal control (optimal control problems) or a shape parameter (shape optimization problems).
The numerical solution to PDE constrained optimization problems involves a series of theoretical and practical challenges:
[1] | Frontiers in PDE-constrained Optimization. The IMA Volumes in Mathematics and its Applications, Springer 2018. |
National Science Foundation (NSF) DMS - 2110263 (together with Pablo Stinga)
Nonlocal School on Fractional Equations.
June 2022-May 2023.