We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.

1 aStrazzullo, Maria1 aZainib, Zakia1 aBallarin, F.1 aRozza, Gianluigi1 aVermolen, Fred, J1 aVuik, Cornelis uhttps://www.springerprofessional.de/en/reduced-order-methods-for-parametrized-non-linear-and-time-depen/1912267600698nas a2200157 4500008004100000245015800041210006900199300001200268490000800280100001500288700002200303700002400325700002100349700001800370856015200388 2021 eng d00aA weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences0 aweighted PODreduction approach for parametrized PDEconstrained o a261-2760 v1021 aCarere, G.1 aStrazzullo, Maria1 aBallarin, Francesco1 aRozza, Gianluigi1 aStevenson, R. uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85117948561&doi=10.1016%2fj.camwa.2021.10.020&partnerID=40&md5=cb57d59a6975a35315b2cf5d0e3a600101629nas a2200121 4500008004100000245014500041210006900186520106800255100002201323700001701345700002101362856012401383 2020 eng d00aPOD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations0 aPODGalerkin Model Order Reduction for Parametrized Nonlinear Tim3 aIn this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

1 aStrazzullo, Maria1 aBallarin, F.1 aRozza, Gianluigi uhttps://math.sissa.it/publication/pod-galerkin-model-order-reduction-parametrized-nonlinear-time-dependent-optimal-flow01835nas a2200133 4500008004100000245014300041210007100184490000700255520124500262100002201507700001701529700002101546856013401567 2020 eng d00aPOD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation0 aPOD–Galerkin Model Order Reduction for Parametrized Time Depende0 v833 aIn this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

1 aStrazzullo, Maria1 aBallarin, F.1 aRozza, Gianluigi uhttps://math.sissa.it/publication/pod%E2%80%93galerkin-model-order-reduction-parametrized-time-dependent-linear-quadratic-optimal00505nas a2200145 4500008004100000245011100041210006900152300001600221490000700237100002200244700001700266700001600283700002100299856003900320 2018 eng d00aModel Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering0 aModel Reduction for Parametrized Optimal Control Problems in Env aB1055-B10790 v401 aStrazzullo, Maria1 aBallarin, F.1 aMosetti, R.1 aRozza, Gianluigi uhttps://doi.org/10.1137/17M1150591