In 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.

ER - TY - JOUR T1 - POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation JF - Journal of Scientific Computing Y1 - 2020 A1 - Maria Strazzullo A1 - Francesco Ballarin A1 - Gianluigi Rozza AB -In 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.

VL - 83 ER - TY - JOUR T1 - POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation JF - JOURNAL OF SCIENTIFIC COMPUTING Y1 - 2020 A1 - Maria Strazzullo A1 - Ballarin, F. A1 - Rozza, G. VL - 83 UR - https://arxiv.org/abs/1909.09631 ER - TY - CONF T1 - Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences T2 - ENUMATH2019 proceedings Y1 - 2020 A1 - Maria Strazzullo A1 - Zakia Zainib A1 - Francesco Ballarin A1 - Gianluigi Rozza AB -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, optimal control problems require a huge computational effort in order to be solved, most of all in a physical and/or geometrical parametrized setting. Reduced order methods are a reliably 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 exploit POD-Galerkin reduction over a parametrized optimality system, derived from Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (i) time dependent Stokes equations and (ii) steady non-linear Navier-Stokes equations.

JF - ENUMATH2019 proceedings PB - Springer UR - https://arxiv.org/abs/1912.07886 ER - TY - JOUR T1 - Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering JF - SIAM JOURNAL ON SCIENTIFIC COMPUTING Y1 - 2018 A1 - Maria Strazzullo A1 - Ballarin, Francesco A1 - Mosetti, Renzo A1 - Rozza, Gianluigi VL - 40 UR - https://arxiv.org/abs/1710.01640 ER - TY - JOUR T1 - Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering JF - SIAM Journal on Scientific Computing Y1 - 2018 A1 - Maria Strazzullo A1 - Francesco Ballarin A1 - Mosetti, R. A1 - Gianluigi Rozza VL - 40 UR - https://doi.org/10.1137/17M1150591 ER -