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Tröltzsch, Fredi; Wachsmuth, Daniel

Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. (English) Zbl 1111.49017

ESAIM, Control Optim. Calc. Var. 12, 93-119 (2006).
The authors establish sufficient optimality conditions for a class of optimal control problems governed by Navier-Stokes equations. In the steady state case the problem is: minimize the quadratic cost functional \(J(u):=\int_\Omega | y(u,x)-y_d(x)| ^2\,dx+\gamma/2 \int_\Omega | u(x)| ^2\,dx\) subject to the Navier-Stokes equations \(-\nu\triangle y+(y\cdot \nabla)y+\nabla p=u\) in \(\Omega\), \(\operatorname{div}y=0\) in \(\Omega\), \(y=0\) on \(\partial\Omega\), where \(\Omega\subset {\mathbb R}^n\) is bounded. The controls satisfy the unilateral conditions \(u_a(x)\leq u(x)\leq u_b(x)\) on \(\Omega\). Under a smallness assumption the authors show that an optimal control which satisfies the necessary condition of first order defines a (local) minimizer providing the second derivative of the associated Lagrange function is positive for all nonzero functions from \(L^2(\Omega)\) which are zero on the active sets of the control in consideration. The proof relies on the smallness assumption (A2), the quadratic structure of the cost functional and on the definition of the admissible set of controls.
Reviewer: Erich Miersemann (Leipzig)

Cited in 46 Documents

MSC:

49K20 Optimality conditions for problems involving partial differential equations
35Q30 Navier-Stokes equations
76D55 Flow control and optimization for incompressible viscous fluids

Keywords:

optimal control; Navier-Stokes equations; second order optimality criterion
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References:

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