TY - GEN

T1 - Different approaches for Dirichlet and Neumann boundary optimal control

AU - Bornia, Giorgio

AU - Ratnavale, Saikanth

N1 - Funding Information:
The use of restricted lifting functions as an alternative approach for Dirichlet and Neumann boundary optimal control problems has been presented. Numerical results show that this method can be effective for minimizing target functionals with a number of degrees of freedom that is comparable to standard boundary control approaches. The use of reduced methods of gradient-based or Hessian-based type for the numerical solution of these optimal control problems, together with the addition of inequality constraints of either control or state type, are subject of current research. Acknowledgements. This work is partly supported by the National Science Foundation grant DMS-1412796.
Funding Information:
This work is partly supported by the National Science Foundation grant DMS-1412796.
Publisher Copyright:
© 2018 Author(s).

PY - 2018/7/10

Y1 - 2018/7/10

N2 - In this paper we consider different methods for formulating boundary optimal control problems of either Dirichlet or Neumann type. On one hand, standard approaches typically have the drawback of searching for the optimal controls in proper sub-spaces of natural trace spaces. Therefore, we consider alternative formulations with lifting functions that have several advantages. First of all, boundary controls can be determined on natural trace spaces, without additional regularity as required by standard approaches. This also leads to numerical discretizations that have the same rate of convergence for the unknowns defined on the domain and for those on the boundary. A potential drawback of the method consists in the use of control functions defined over the problem domain instead of on its boundary, thus increasing the number of degrees of freedom of the problem. This drawback can be compensated by using lifting functions with restricted support, whose boundary contains the control boundary under interest. Numerical results solving the optimality systems in an all-at-once approach show that it is possible to use restricted functions which do not substantially change the minimum value of the target functional with respect to the case of lifting functions with non-restricted support.

AB - In this paper we consider different methods for formulating boundary optimal control problems of either Dirichlet or Neumann type. On one hand, standard approaches typically have the drawback of searching for the optimal controls in proper sub-spaces of natural trace spaces. Therefore, we consider alternative formulations with lifting functions that have several advantages. First of all, boundary controls can be determined on natural trace spaces, without additional regularity as required by standard approaches. This also leads to numerical discretizations that have the same rate of convergence for the unknowns defined on the domain and for those on the boundary. A potential drawback of the method consists in the use of control functions defined over the problem domain instead of on its boundary, thus increasing the number of degrees of freedom of the problem. This drawback can be compensated by using lifting functions with restricted support, whose boundary contains the control boundary under interest. Numerical results solving the optimality systems in an all-at-once approach show that it is possible to use restricted functions which do not substantially change the minimum value of the target functional with respect to the case of lifting functions with non-restricted support.

UR - http://www.scopus.com/inward/record.url?scp=85049971181&partnerID=8YFLogxK

U2 - 10.1063/1.5043899

DO - 10.1063/1.5043899

M3 - Conference contribution

AN - SCOPUS:85049971181

T3 - AIP Conference Proceedings

BT - International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017

A2 - Tsitouras, Charalambos

A2 - Simos, Theodore

A2 - Simos, Theodore

A2 - Simos, Theodore

A2 - Simos, Theodore

A2 - Simos, Theodore

PB - American Institute of Physics Inc.

Y2 - 25 September 2017 through 30 September 2017

ER -