Project 04: Parabolic Boundary Control - NMOPT — Numerical Methods for Optimal Control

Goal¶

Bridge the elliptic boundary-control projects and the parabolic control laboratories by controlling boundary data in a heat equation.

This is an intermediate project before moving to fluid and inverse problems.

Mathematical Formulation¶

Let $Q=\Omega\times(0,T)$ and $\Sigma=\partial\Omega\times(0,T)$ . Find a boundary control $u$ minimizing

J(y,u) = \frac12\int_0^T \|y(t)-y_d(t)\|_{L^2(\Omega)}^2\,dt + \frac\beta2\|y(T)-y_T\|_{L^2(\Omega)}^2 + \frac\alpha2\int_0^T \|u(t)\|_{L^2(\Gamma)}^2\,dt

subject to

\begin{cases} y_t-\mu\Delta y+\sigma y=f & \text{in }Q,\\ y=u & \text{on }\Sigma,\\ y(0)=y_0 & \text{in }\Omega. \end{cases}

A Neumann-control variant replaces $y=u$ by

\partial_n y=u \qquad\text{on }\Sigma.

The adjoint equation runs backward in time:

\begin{cases} -p_t-\mu\Delta p+\sigma p=y-y_d & \text{in }Q,\\ p=0 & \text{on }\Sigma,\\ p(T)=\beta(y(T)-y_T) & \text{in }\Omega. \end{cases}

For Neumann control the formal gradient is

g=\alpha u-p|_{\Gamma}.

For Dirichlet control the gradient involves $\partial_n p$ on the boundary.

Implementation Hints¶

Start from ParabolicProjectedGradient.
Replace volume control mass matrices with boundary mass matrices.
Store one boundary control vector per time level.
Verify the gradient by finite differences before optimizing.

Relevant Examples¶

Course code: codes/dealii/include/parabolic_projected_gradient.h, codes/dealii/source/parabolic_projected_gradient.cc.
Course lectures: lecture12.md, lecture13.md, lecture14.md.
deal.II: step-26 for time-dependent heat equation structure.

Deliverables¶

Forward and backward time loops with boundary control.
Reduced-gradient check.
A small tracking experiment with moving boundary actuation.
Optional box constraints on $u$ .