A mathematical analysis of an optimal control problem for a generalized Boussinesq model for viscous incompressible flows
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A. B. San Martin A.M. Moya adjoint equation admissible controls Banach spaces Boussinesq approximation Boussinesq equations Boussinesq model Branddo Catuogno Climent-Ezquerra cone of decreasing consider corresponding DC(J defined denotes Dirichlet boundary conditions dual cone Dubovitskii and Milyutin Elliptic equation in 4.7 existence results Fernandez and W.A. Fernandez-Cara Filidor Frechet differentiable functional spaces Geometric Algebra global existence Guillen-Gonzalez Helmholtz decomposition hypotheses Hi)-(Hio IMECC-UNICAMP integrate J.L. Boldrini Jn Jn JoTJn Lemma Lorca and Boldrini Luiz A. B. San M.A. Rojas-Medar Mathematical Analysis minimum principle Neumann boundary conditions nonlinear norm optimal control problem optimal solution order optimality conditions partial differential equations Pontriagyn positive constant proceed as usual Proposition 2.3 Regular time-reproductive S.A. Lorca satisfy second equation set of admissible Sobolev Spaces spectral Faedo-Galerkin method strong solutions suitable Theorem thermal conductivity usual to obtain V.V. Fernandez Vilca Viscous Incompressible Flows W.A. Rodrigues Jr Wg x Wc Wicu Wu x Wg xWgxU