Project 02: Mixed Dirichlet-Neumann Boundary Control - NMOPT — Numerical Methods for Optimal Control

Goal¶

Study a boundary-control problem with two control components: a Dirichlet control on one part of the boundary and a Neumann control on another part.

This project combines trace-based control with weak boundary forcing.

Mathematical Formulation¶

Let

\partial\Omega=\Gamma_D\cup\Gamma_N\cup\Gamma_0, \qquad \Gamma_i\cap\Gamma_j=\emptyset \text{ for } i\ne j.

The control is

u=(u_D,u_N),

with

u_D\in H^{1/2}(\Gamma_D), \qquad u_N\in L^2(\Gamma_N).

Given $y_d\in L^2(\Omega)$ , solve

\min_{(y,u_D,u_N)} \frac12\|y-y_d\|_{L^2(\Omega)}^2 + \frac{\alpha_D}{2}\|u_D-u_{D,\mathrm{ref}}\|_{H^{1/2}(\Gamma_D)}^2 + \frac{\alpha_N}{2}\|u_N-u_{N,\mathrm{ref}}\|_{L^2(\Gamma_N)}^2

subject to

\begin{cases} -\Delta y=f & \text{in }\Omega,\\ y=u_D & \text{on }\Gamma_D,\\ \partial_n y=u_N & \text{on }\Gamma_N,\\ y=0 & \text{on }\Gamma_0. \end{cases}

Using a lifting $R_Du_D$ , write $y=w+R_Du_D$ with $w=0$ on $\Gamma_D\cup\Gamma_0$ . The weak state equation becomes

\int_\Omega \nabla w\cdot\nabla v\,dx = \int_\Omega f v\,dx + \int_{\Gamma_N} u_N v\,ds - \int_\Omega \nabla(R_Du_D)\cdot\nabla v\,dx.

The adjoint equation is

\begin{cases} -\Delta p=y-y_d & \text{in }\Omega,\\ p=0 & \text{on }\Gamma_D\cup\Gamma_0,\\ \partial_n p=0 & \text{on }\Gamma_N. \end{cases}

The formal optimality conditions are

\alpha_N(u_N-u_{N,\mathrm{ref}})-p=0 \qquad\text{on }\Gamma_N,

and

\alpha_D(u_D-u_{D,\mathrm{ref}})-\partial_n p=0 \qquad\text{on }\Gamma_D,

with the second identity understood in the dual trace space.

Implementation Hints¶

Build on Project 01.
Use separate DoF sets or component vectors for $u_D$ and $u_N$ .
Test three cases: only Dirichlet control, only Neumann control, and both.
Use different regularization parameters to see which boundary component is more efficient.

Relevant Examples¶

Course material: jupyterbook/lectures/lecture14.md.
Course code: codes/dealii/execs/laplacian.cc.
deal.II: step-7 for boundary integrals; step-4 and step-6 for elliptic finite elements.

Deliverables¶

A derivation of the adjoint and the two boundary gradients.
A 2D implementation with labeled boundary parts.
Plots of $u_D$ , $u_N$ , $y$ , $p$ , and $y-y_d$ .
A comparison of the effect of $\alpha_D$ and $\alpha_N$ .