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This paper presents a projection-based reduced order modelling (ROM)framework for unsteady parametrized optimal control problems (OCP$_{(\mu)}$s)arising from cardiovascular (CV) applications. In real-life scenarios,accurately defining outflow boundary conditions in patient-specific modelsposes significant challenges due to complex vascular morphologies,physiological conditions, and high computational demands. These challenges makeit difficult to compute realistic and reliable CV hemodynamics by incorporatingclinical data such as 4D magnetic resonance imaging. To address thesechallenges, we focus on controlling the outflow boundary conditions to optimizeCV flow dynamics and minimize the discrepancy between target and computed flowvelocity profiles. The fluid flow is governed by unsteady Navier--Stokesequations with physical parametric dependence, i.e. the Reynolds number.Numerical solutions of OCP$_{(\mu)}$s require substantial computationalresources, highlighting the need for robust and efficient ROMs to performreal-time and many-query simulations. Here, we aim at investigating theperformance of a projection-based reduction technique that relies on theoffline-online paradigm, enabling significant computational cost savings. TheGalerkin finite element method is used to compute the high-fidelity solutionsin the offline phase. We implemented a nested-proper orthogonal decomposition(nested-POD) for fast simulation of OCP$_{(\mu)}$s that encompasses two stages:temporal compression for reducing dimensionality in time, followed byparametric-space compression on the precomputed POD modes. We tested theefficacy of the methodology on vascular models, namely an idealized bifurcationgeometry and a patient-specific coronary artery bypass graft, incorporatingstress control at the outflow boundary, observing consistent speed-up withrespect to high-fidelity strategies.

Projection-based Reduced Order Modelling for Unsteady Parametrized Optimal Control Problems in 3D Cardiovascular Flows