On optimal design of control problems for Stokes
Abstract
We study optimal shape design for a Stokes boundary-control problem with an internal no-slip obstacle and a fixed outer boundary carrying impermeability, Navier friction and a controlled tangential load. A coercive-form turnpike theorem gives horizon-uniform estimates, a continuous optimal-control representative and an exact endpoint decomposition of the optimal value. For compact Piola-transported families, flux compactness and recovery controls prove continuity of dynamic values without a common strong KKT inverse. For fixed spatial force and tracking data and arbitrary normal heights in Hm(Γ0), m≥5, in dimensions two and three, we derive the stationary reduced derivative and Hessian. A current-domain Piola argument and localized Green identities give a Hadamard density under additional collar regularity. A normal-graph Sobolev metric and a quadratic reference-shape penalty define an infinite-dimensional GFDS. The Riesz-transformed reduced Stokes gradient K is locally Lipschitz and compact. Local existence and uniqueness follow from local Lipschitz regularity of the full regularized field Gη(h)=K(h)+η h; compactness of K enters the long-time analysis. For a positive penalty, a strict energy barrier gives global existence and precompactness. Under this barrier, polynomial spatial data give convergence to one critical shape through an analytic Fredholm gradient inequality. If the penalty also exceeds the uniform reduced-Hessian bound on the safe ball, its minimizer is unique and interior, and the flow converges exponentially to it. The exact-volume projected flow has corresponding confinement and analytic convergence results for feasible initial data satisfying the barrier. Normalized dynamic values converge uniformly to stationary values; with a fixed positive running reference penalty, their design minimizers converge strongly in the height space. The retained spherical flow uses a distinct material-data convention. The shape-flow theorem is reference-chart dependent, not an intrinsic L2 or Willmore Stokes flow theorem.
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BibTeX
@misc{sokolowski2026optimal,
title = {{On optimal design of control problems for Stokes}},
author = {Jan Sokołowski},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05775538v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On optimal design of control problems for Stokes
AU - Jan Sokołowski
PY - 2026
DA - 2026/10/02
PB - HAL
UR - https://hal.science/hal-05775538v1
AB - We study optimal shape design for a Stokes boundary-control problem with an internal no-slip obstacle and a fixed outer boundary carrying impermeability, Navier friction and a controlled tangential load. A coercive-form turnpike theorem gives horizon-uniform estimates, a continuous optimal-control representative and an exact endpoint decomposition of the optimal value. For compact Piola-transported families, flux compactness and recovery controls prove continuity of dynamic values without a common strong KKT inverse. For fixed spatial force and tracking data and arbitrary normal heights in H^m(Γ_0), m≥5, in dimensions two and three, we derive the stationary reduced derivative and Hessian. A current-domain Piola argument and localized Green identities give a Hadamard density under additional collar regularity. A normal-graph Sobolev metric and a quadratic reference-shape penalty define an infinite-dimensional GFDS. The Riesz-transformed reduced Stokes gradient K is locally Lipschitz and compact. Local existence and uniqueness follow from local Lipschitz regularity of the full regularized field G_η(h)=K(h)+η h; compactness of K enters the long-time analysis. For a positive penalty, a strict energy barrier gives global existence and precompactness. Under this barrier, polynomial spatial data give convergence to one critical shape through an analytic Fredholm gradient inequality. If the penalty also exceeds the uniform reduced-Hessian bound on the safe ball, its minimizer is unique and interior, and the flow converges exponentially to it. The exact-volume projected flow has corresponding confinement and analytic convergence results for feasible initial data satisfying the barrier. Normalized dynamic values converge uniformly to stationary values; with a fixed positive running reference penalty, their design minimizers converge strongly in the height space. The retained spherical flow uses a distinct material-data convention. The shape-flow theorem is reference-chart dependent, not an intrinsic L^2 or Willmore Stokes flow theorem.
ER -