Optimal control of the Navier-Stokes equations via pressure boundary conditions

Boris Vexler; Jakob Wagner

doi:10.3934/mcrf.2025053

Article Contents

2026, Volume 18: 79-121. Doi: 10.3934/mcrf.2025053

This volume Previous Article Collective behavior in discontinuous second-order consensus model with collision avoidance Next Article Energy decay of a linear isotropic thermoelastic system with damping subject only to full Wentzell boundary

Optimal control of the Navier-Stokes equations via pressure boundary conditions

Boris Vexler and
Jakob Wagner^, ,

Technical University of Munich, School for Computation, Information and Technology, Chair of Optimal Control, Germany

^*Corresponding author: Jakob Wagner
^*Corresponding author: Jakob Wagner

Received: January 2025

Revised: June 2025

Early access: September 9, 2025

Published online: September 9, 2025

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Abstract

In this work, we studied an optimal control problem subject to the unsteady Navier-Stokes equations, where the control enters via an inhomogeneous Neumann/Do-Nothing boundary condition. Despite the Navier-Stokes equations with these boundary conditions not being well-posed for large times and/or data, we obtained well-posedness of the optimal control problem by choosing a proper tracking type term. Moreover, we derived first- and second-order optimality conditions. In order to discuss the regularity of the optimal control, state, and adjoint state, we presented new results on $ L^2(I;H^2( \Omega)) $ regularity of solutions to a Stokes problem with mixed inhomogeneous boundary conditions.

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Mathematics Subject Classification: 35Q30, 76D05, 49J20, 49K20.

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Figure 1. Domain $ \Omega $ satisfying Assumption 2.1

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Figure 2. Domain $ \Omega $ depending on the parameters $ r, R, L $

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Figure 3. Two reasons for blowup of numerical solutions: large initial data or large boundary data

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Figure 4. Optimal controls for different choices of $ \mathbf{u}_d $ and $ \alpha $

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Figure 5. Results of an optimal control problem with data close to blowup

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Figure 6. Solution of the optimal control problem with desired state that features a sinusoidal dependence of time

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Figure 7. Extension of $ \Omega $, which admits an extension of divergence-free vector fields from $ \Omega $ to the larger domain

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