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We introduce reduced order methods as an efficient strategy to solveparametrized non-linear and time dependent optimal flow control problemsgoverned by partial differential equations. Indeed, the optimal controlproblems require a huge computational effort in order to be solved, most of allin physical and/or geometrical parametrized settings. Reduced order methods area reliable and suitable approach, increasingly gaining popularity, to achieverapid and accurate optimal solutions in several fields, such as in biomedicaland environmental sciences. In this work, we employ a POD-Galerkin reductionapproach over a parametrized optimality system, derived from theKarush-Kuhn-Tucker conditions. The methodology presented is tested on twoboundary control problems, governed respectively by (i) time dependent Stokesequations and (ii) steady non-linear Navier-Stokes equations.

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Reduced order methods for parametrized non-linear and time dependent  optimal flow control problems, towards applications in biomedical and  environmental sciences