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Maximum principle in the boundary control problem for flow of a viscous fluid. (English. Russian original) Zbl 0808.76079

Sib. Math. J. 34, No. 6, 1171-1179 (1993); translation from Sib. Mat. Zh. 34, No. 6, 189-197 (1993).
We consider the control problem for the Navier-Stokes evolution equations that describe the flow of a viscous incompressible fluid in an unbounded domain \(\Omega\subset \mathbb{R}^ d\), \(d=2,3\). This problem can be interpreted as a problem of minimizing the flow through the segment \(\Gamma\subset \partial\Omega\) under restrictions on the flow head on the boundary. The singular optimality system for the considered problem is obtained in the form of an analogue of the Pontryagin maximum principle; moreover, for two-dimensional flows we impose no additional conditions like smallness or regularity.

MSC:

76M30 Variational methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
49J20 Existence theories for optimal control problems involving partial differential equations
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