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Periodic optimal control of the Boussinesq equation. (English) Zbl 1039.49026
The author considers an optimal control problem for a Boussinesq problem which is posed in \(\Omega \times \left( 0,{\mathbb R}\right) \), where \(\Omega \) is a smooth, bounded and open subset of \({\mathbb R}^{2}\). He introduces two control variables : one (named \(u\)) for the velocity field \(v\), which is the solution of a Navier-Stokes equation, and one (named \(u_{1}\)) for the temperature field \(\theta \), which is the solution of a classical heat equation. The two equations are coupled in the classical way for the Boussinesq problem. The velocity field and the temperature satisfy homogeneous Dirichlet boundary conditions on the boundary of \(\Omega \) and are supposed to be periodic in time. In the Navier-Stokes equation, the control variable \(u\) is multiplied by a function \(m\) which belongs to \( L^{\infty }\left( \Omega \right) \) and which has a support contained in a fixed and open subset \(\omega \) of \(\Omega \). Under some natural hypotheses on the cost functional, the author first proves the existence of a solution \( \left( v^{\ast },\theta ^{\ast },u^{\ast },u_{1}^{\ast }\right) \) of this optimal control problem in appropriate functional spaces. The proof essentially relies on estimates obtained on a minimizing sequence of this problem, the imposed hypotheses ensuring the nice properties of the underlying operators. In the last part of the paper, the author imposes some further conditions on a term \(h(u) \) which appears in the cost functional in order to characterize the optimal solution in terms of two adjoint variables which are proved to exist within this restricted context. The function \(h\) is now supposed to be convex and to have a quadratic growth. The author thus uses the Moreau-Yosida approximation of this convex function \(h\) and let the corresponding extra small parameter go to 0. He proves appropriate estimates in order to prove the existence of the two adjoint variables.

MSC:
49K20 Optimality conditions for problems involving partial differential equations
49J20 Existence theories for optimal control problems involving partial differential equations
76D55 Flow control and optimization for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
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