Project 08: Box-Constrained Navier-Stokes Boundary Control - NMOPT — Numerical Methods for Optimal Control

Goal¶

Add box constraints to the mixed boundary-control Navier-Stokes problem of Project 07.

This is the nonlinear analogue of Project 03.

Mathematical Formulation¶

Use the Navier-Stokes system from Project 07 and impose pointwise constraints

u_D^a\le u_D\le u_D^b \qquad\text{on }\Gamma_D,

u_N^a\le u_N\le u_N^b \qquad\text{on }\Gamma_N.

The admissible set is

U_{ad} = \{(u_D,u_N): u_D^a\le u_D\le u_D^b,\; u_N^a\le u_N\le u_N^b\}.

The first-order condition is the variational inequality

\langle g_D,\widetilde u_D-u_D\rangle_{\Gamma_D} + (g_N,\widetilde u_N-u_N)_{\Gamma_N} \ge 0 \qquad \forall(\widetilde u_D,\widetilde u_N)\in U_{ad},

where $g_D$ and $g_N$ are the boundary reduced gradients from Project 07.

A projected-gradient step is

u_D^{k+1} = P_{[u_D^a,u_D^b]} \left(u_D^k-s_Dg_D^k\right),

u_N^{k+1} = P_{[u_N^a,u_N^b]} \left(u_N^k-s_Ng_N^k\right).

Implementation Hints¶

Reuse the Project 07 state and adjoint solvers.
Start with projected gradient and Armijo line search.
Add active-set output on the boundary.
Test several initial guesses to observe nonconvex behavior.

Relevant Examples¶

Course code: codes/dealii/execs/laplacian_box_constraints.cc.
Course lectures: lecture09.md, lecture13.md, lecture15.md.
deal.II: step-22 for fluid block structure; Navier-Stokes tutorials for the nonlinear solver.

Deliverables¶

A constrained Navier-Stokes control implementation.
Active-set plots for boundary controls.
Comparison with unconstrained Project 07.
A discussion of local minima and globalization.