Project 06: Box-Constrained Stokes Control - NMOPT — Numerical Methods for Optimal Control

Goal¶

Extend Project 05 by adding pointwise constraints on the distributed force control.

This project is a useful bridge between the scalar box-constrained Poisson code and vector-valued fluid control.

Mathematical Formulation¶

Use the Stokes state equation from Project 05 and impose

u_a(x)\le u(x)\le u_b(x) \qquad\text{a.e. in }\Omega,

componentwise. The admissible set is

U_{ad} = \{v\in L^2(\Omega)^d: u_a\le v\le u_b\}.

The optimality condition is

(\alpha u-p,v-u)_{L^2(\Omega)^d}\ge 0 \qquad \forall v\in U_{ad}.

Equivalently,

u = P_{[u_a,u_b]} \left(\frac1\alpha p\right),

where the projection is componentwise.

The projected-gradient residual is

r_s(u) = \frac1s \left[ u- P_{[u_a,u_b]}(u-s(\alpha u-p)) \right].

Implementation Hints¶

First implement projected gradient.
Then implement PDAS componentwise: lower active, upper active, and inactive regions for each component.
Output active-set indicators for each velocity component.
Compare projected gradient and PDAS iteration counts.

Relevant Examples¶

Course code: codes/dealii/execs/laplacian_box_constraints.cc, codes/dealii/execs/kkt_box_constraints.cc.
Course lectures: lecture09.md, lecture11.md.
deal.II: step-22 for Stokes.

Deliverables¶

Projected-gradient Stokes control solver.
Optional PDAS implementation.
Comparison of saturation patterns for different $\alpha$ .
A short report on componentwise vs vector-norm control constraints.