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Project 03: Box-Constrained Mixed Boundary Control

University of Pisa

Goal

Add pointwise bounds to the mixed Dirichlet-Neumann boundary-control problem of Project 02.

This project is the natural boundary-control analogue of the projected gradient and PDAS laboratories.

Mathematical Formulation

Use the state equation from Project 02 and impose

uDa≤uD≤uDba.e. on ΓD,u_D^a\le u_D\le u_D^b \qquad\text{a.e. on }\Gamma_D,
(1)

and

uNa≤uN≤uNba.e. on ΓN.u_N^a\le u_N\le u_N^b \qquad\text{a.e. on }\Gamma_N.
(2)

The admissible set is

Uad={(uD,uN):uDa≤uD≤uDb,  uNa≤uN≤uNb}.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\}.
(3)

The cost functional is

J(y,uD,uN)=12∥y−yd∥L2(Ω)2+αD2∥uD−uD,ref∥2+αN2∥uN−uN,ref∥2.J(y,u_D,u_N) = \frac12\|y-y_d\|_{L^2(\Omega)}^2 + \frac{\alpha_D}{2}\|u_D-u_{D,\mathrm{ref}}\|^2 + \frac{\alpha_N}{2}\|u_N-u_{N,\mathrm{ref}}\|^2.
(4)

The variational inequality is

⟨gD,u~D−uD⟩ΓD+(gN,u~N−uN)L2(ΓN)≥0∀(u~D,u~N)∈Uad,\langle g_D,\widetilde u_D-u_D\rangle_{\Gamma_D} + (g_N,\widetilde u_N-u_N)_{L^2(\Gamma_N)} \ge 0 \qquad \forall(\widetilde u_D,\widetilde u_N)\in U_{ad},
(5)

where formally

gD=αD(uD−uD,ref)−∂np,gN=αN(uN−uN,ref)−p∣ΓN.g_D=\alpha_D(u_D-u_{D,\mathrm{ref}})-\partial_n p, \qquad g_N=\alpha_N(u_N-u_{N,\mathrm{ref}})-p|_{\Gamma_N}.
(6)

For an L2L^2 boundary discretization, a projected-gradient update is

uDk+1=P[uDa,uDb](uDk−sDgDk),u_D^{k+1} = P_{[u_D^a,u_D^b]} \left(u_D^k-s_D g_D^k\right),
(7)
uNk+1=P[uNa,uNb](uNk−sNgNk).u_N^{k+1} = P_{[u_N^a,u_N^b]} \left(u_N^k-s_N g_N^k\right).
(8)

Implementation Hints

Relevant Examples

Deliverables

Projects
Project 02: Mixed Dirichlet-Neumann Boundary Control
Projects
Project 04: Parabolic Boundary Control