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Project 01: Dirichlet Boundary Control for Poisson’s Equation

University of Pisa

Goal

Extend the elliptic distributed-control codes to a boundary-control problem where the control is the Dirichlet value imposed on part or all of the boundary.

This is the first project where the control does not act as a volume source. The main challenge is the correct treatment of traces, liftings, and boundary degrees of freedom.

Mathematical Formulation

Let Ω⊂Rd\Omega\subset\mathbb R^d be a bounded Lipschitz domain and let Γ=∂Ω\Gamma=\partial\Omega. Given yd∈L2(Ω)y_d\in L^2(\Omega), α>0\alpha>0, and a reference control uref∈H1/2(Γ)u_{\mathrm{ref}}\in H^{1/2}(\Gamma), solve

min⁡(y,u)J(y,u):=12∥y−yd∥L2(Ω)2+α2∥u−uref∥H1/2(Γ)2\min_{(y,u)} J(y,u) := \frac12\|y-y_d\|_{L^2(\Omega)}^2 + \frac\alpha2\|u-u_{\mathrm{ref}}\|_{H^{1/2}(\Gamma)}^2
(1)

subject to

{−Δy=fin Ω,y=uon Γ.\begin{cases} -\Delta y=f & \text{in }\Omega,\\ y=u & \text{on }\Gamma. \end{cases}
(2)

A simpler numerical variant replaces the H1/2(Γ)H^{1/2}(\Gamma) norm by a discrete boundary mass norm:

α2∥u−uref∥L2(Γ)2.\frac\alpha2\|u-u_{\mathrm{ref}}\|_{L^2(\Gamma)}^2.
(3)

This is easier to implement but should be discussed as a variational crime.

Using a lifting Ru∈H1(Ω)R u\in H^1(\Omega) with γ(Ru)=u\gamma(Ru)=u, write

y=w+Ru,w∈H01(Ω),y=w+Ru, \qquad w\in H_0^1(\Omega),
(4)

and solve

∫Ω∇w⋅∇v dx=∫Ωfv dx−∫Ω∇(Ru)⋅∇v dx∀v∈H01(Ω).\int_\Omega \nabla w\cdot\nabla v\,dx = \int_\Omega f v\,dx - \int_\Omega \nabla(Ru)\cdot\nabla v\,dx \qquad \forall v\in H_0^1(\Omega).
(5)

The adjoint satisfies

{−Δp=y−ydin Ω,p=0on Γ.\begin{cases} -\Delta p=y-y_d & \text{in }\Omega,\\ p=0 & \text{on }\Gamma. \end{cases}
(6)

The boundary gradient can be derived through the lifting or through a transposition argument. In the formal smooth case,

α(u−uref)−∂np=0on Γ.\alpha(u-u_{\mathrm{ref}})-\partial_n p=0 \qquad \text{on }\Gamma.
(7)

Implementation Hints

Relevant Examples

Deliverables

Lectures
Implementing SQH for a Poisson Control Problem
Projects
Project 02: Mixed Dirichlet-Neumann Boundary Control