Novel Finite Element Methods for Elliptic Distributed Optimal Control Problems

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Optimal control problems with elliptic partial differential equation constraints appear in many optimal design processes in engineering and science. In these problems the state (output) is connected to the control (input) through an elliptic partial differential equation, and the objective is to find the control that will produce a desired state in an optimal fashion. The proposed research is on the design, analysis and efficient implementation of novel numerical methods for such problems, with applications to mechanical engineering, electrical engineering and materials science.Traditional numerical approaches for these optimal control problems treat the control as the primary unknown. The resulting finite element methods only involve low order elements. The convergence analysis, where the error estimates for the control, the state and the adjoint state are intertwined, is substantially more complicated than the convergence analysis for elliptic boundary value problems. In contrast, the approach in the proposed research treats the state as the primary unknown by reformulating the optimal control problems as variational inequalities for the state. A new analytical framework developed recently by the PI and the Co-PI shows that the convergence analysis for these elliptic variational inequalities can be obtained by using the same tools for the convergence analysis for elliptic boundary value problems. Consequently many finite element methods originally intended for elliptic boundary value problems can also be applied to the optimal control problems constrained by elliptic partial differential equations. The goal of the proposed research is to apply this new insight to design novel finite element methods for optimal control problems with general cost functionals, problems with semi-linear second order and fourth order elliptic partial differential equation constraints, problems for electromagnetics and problems with rough coefficients that appear in materials science.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.