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Optimal controls of 3-dimensional Navier–Stokes equations with state constraints. (English) Zbl 1022.93026
The author considers the controlled Navier-Stokes system \begin{align*}{ {\partial y \over \partial t} - \gamma \Delta y + y \cdot \nabla y + \nabla p &= D_0 u + f_0 \ \text{in} \ \Omega \cr \nabla \cdot y = 0 \ \text{in} \ \Omega , \quad y &= 0 \ \text{on} \ \Gamma }\end{align*} where $$\Omega$$ is a 3-dimensional bounded domain with boundary $$\Gamma.$$ The function $$u(\cdot) \in L^2(0, T; U)$$ $$(U$$ a Hilbert space) is the control and $$D_0 : U \to L^2(\Omega)^3$$ is a linear bounded operator. The cost functional is $L(y, u) = \int_0^T \big(g(t, y(t)) + h(u(t))\big) dt .$ The author derives versions of Pontryagin’s maximum principle (with attention paid to whether the multiplier is nonzero) for the optimal problem with three types of state constraints: integral, two-point boundary (in the time variable) and periodic.

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 49K20 Optimality conditions for problems involving partial differential equations 76D05 Navier-Stokes equations for incompressible viscous fluids 93C10 Nonlinear systems in control theory
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