Differential Stability of Control-Constrained Optimal Control Problems for the Navier-Stokes Equations

ABSTRACT

Distributed optimal control problems for the time-dependent and the stationary Navier-Stokes equations subjected to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functions of perturbation parameters such as the Reynolds number, the desired trajectory, or the initial conditions. The derivative is characterized as the solution of an auxiliary linear-quadratic optimal control problem. Thus, it can be computed at relatively low cost. Taylor expansions of the minimum value function are provided as well.

Keywords:

• Differential stability
• Navier-Stokes equations
• Optimal control

Mathematics Subject Classification:

• 35Q30
• 49K40

ACKNOWLEDGMENTS

The third author acknowledges support of the Sonder-forschungsbereich 609 Elektromagnetische Strömungskontrolle in Metallurgie, Kristallzüchtung und Elektrochemie, located at the Technische Universität Dresden, and supported by the German Research Foundation.