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

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.

### MSC:

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

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