Optimal Control in Multiscale Interface Couplings of Partial and Ordinary Differential Equations

This project advances the understanding of how fluid flow and mechanical deformation interplay within porous media (such as biological tissues) that are connected to larger fluid networks like the circulatory system. These complex systems are essential in biomedical and engineering applications, including organ perfusion modeling, the design of bioartificial organs, and strategies to prevent tissue damage under mechanical stress. By combining advanced mathematical analysis with multiscale modeling, the project aims to improve prediction and control of coupled fluid-structure behavior, and will integrate research and education, therefore also contributing to the development of the future STEM workforce. The research focuses on optimal control and well-posedness in multiscale interface systems that couple fluid flow through deformable, porous media and lumped hydraulic networks. The approach integrates detailed three-dimensional models of local tissue behavior with simplified time-dependent models representing systematic circulation. Poro-(visco)-elastic systems are used to describe porous media flows under a range of physical scenarios. Coupling partial differential equations with ordinary differential equations enables accurate modeling of local and global interactions, including flow distribution and pressure propagation. The project develops and analyzes novel mathematical strategies -- leveraging tools such as semigroup theory, Rothe's method, fixed-point techniques, and convex and non-convex optimization -- to address control and stability in these multiscale environments. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.