The research program is focused on analysis and control in multi-scale interface coupling between partial and ordinary differential equations with direct biological applications to the perfusion of a tissue - a local phenomenon - studied in correlation with the global features of the surrounding blood circulation. The proposed research will deepen the understanding of eye physiology. A porous tissue called lamina cribrosa (LC) within the optic nerve head is thought to be the primary site of injury in glaucoma, which is the leading cause of irreversible blindness worldwide. The LC and related blood vessels will be modeled via a multi-scale interface coupling to investigate how LC perfusion depends on intraocular pressure (IOP) and blood pressure (BP). The proposed sensitivity and control studies will help elucidate the disease mechanisms in glaucoma and advance the development of novel therapeutic approaches coregulating BP and IOP that could potentially help many patients.

Overview: This program will advance the theory of optimal control problems subject to moving boundary fluid-structure interactions and fluid-solid mixtures, by developing the theoretical framework for optimality conditions and sensitivity analysis, necessary for all parameter estimations and derivative-based optimization algorithms. The research will have wide applicability, from the design of small-scale unmanned aircrafts and morphing aircraft wings to blood flow in arteries, and fluid flow in biological tissues. The work will make pertinent contributions to the study of the lamina cribrosa (LC) in the eye, which is thought to be the primary site of injury in glaucoma. The objectives of this project are (1) to provide the groundwork for sensitivity analysis and optimality conditions for both moving boundary fluid-structure interactions and poro-visco-elastic models, and (2) to perform sensitivity analysis on the LC model with respect to important biological parameters, and develop and address relevant control problems with respect to the development of glaucoma. The research will bridge applied and computational mathematics, and use methods from analysis and control theory of PDEs, micro-local analysis, tangential calculus, and sensitivity and shape analysis. The research results will be discussed in the courses taught by the PI in order to expose students to current research topics in applied mathematics and attract them to this field. All PI������������������s current graduate students will work on the proposed projects. The PI will recruit undergraduates from NCSU Women in Science and Engineering program for research experiences. The results will be disseminated to a broad audience during the PI������������������s well established, annual program Math Doesn������������������t Bug Me at the NC Museum of Natural Sciences������������������ BugFest in Raleigh. The PI������������������s outreach program GAMMA (Girls in Applied Math, Modeling, and Analysis) Day at NC State will showcase the PI������������������s research, and demonstrate the use and impact of applied math, in order to build interest in math research and related careers among women at the high school level. Intellectual Merit: Most studies of optimal control problems subject to fluid-structure interactions focus on the case of static common interface. The PI������������������s work will bridge control theory and shape analysis techniques, and the results will break new ground in the theory of optimal control problems associated with moving boundary fluid-elasticity interactions and pave the way for efficient optimization algorithms. The research will include the first investigation of fluid-elasticity interactions that take into account the geometry of the common interface. The work will provide new well-posedness and differentiability results for nonlinear poro-visco-elasticity systems, and advance the available theory for sensitivity of nonlinear dynamical systems to Banach space parameters. Performing sensitivity analysis on the LC with respect to important parameters will further the understanding of the cause and progression of glaucoma, and enable novel means for preventing or treating glaucoma. The projects will launch new research in the field, as the results will represent the theoretical foundation for parameter estimation and derivative-based optimization algorithms in moving boundary fluid-elasticity interactions and fluid-mixture problems. The approach and the techniques introduced will be used to investigate inverse and control problems in many other (moving boundary) coupled physical and biological systems. Broader Impacts: The research focused on the LC will advance the understanding of the key factors in the development of glaucoma and will be applicable to other biological fluid-solid mixtures such as cartilages, bones, and engineered tissues. While the main focus of this project is on glaucoma, alterations in retinal hemodynamics are also associated to hypertension, diabetes, multiple sclerosis, Alzheimer������������������s, and Parkinson������������������s disease. This project will help lead to a better understanding of these diseases and their potential cures, thereby largely benefiting society in general. The PI will use her work and results to cultivate science literacy among a broad audience and diversify the next generation of STEM students through her exhibit Math Doesn������������������t Bug Me at the NC Museum of Natural Sciences������������������ BugFest in Raleigh, and her outreach program GAMMA Day at NC State.

The 40th Southeastern-Atlantic Regional Conference on Differential Equations (SEARCDE) will be held at NC State University, Raleigh, NC, on November 12-13, 2022. The conference promotes research and education in differential equations and their applications by bringing together leading experts and young researchers in this field. It provides a unique experience for the Southeastern-Atlantic region due to its research focus, size and location, as well as its strong commitment to building regional research networks and mentoring the next generation of mathematicians who work in differential equations. This year SEARCDE will be focused on the following themes: (i) Applications of differential equations, especially to problems in life sciences, (ii) Mathematical control and related topics, and (iii) Applied analysis of differential equations. The confirmed plenary speakers for SEARCDE 2020 are: (1) Piermarco Cannarsa (University of Rome Tor Vergata), (2) Giovanna Guidoboni (University of Missouri), (3) Irena Lasiecka (University of Memphis), and (4) Daniel (University of California Berkeley). The 40th SEARCDE will also celebrate and honor the life and contributions of H.T. Banks, a prominent researcher in differential equations and their applications and professor at NC State, who passed away this past December. SEARCDE is committed to inclusion and diversity. This year the conference will be held ``in cooperation with the Association for Women in Mathematics (AWM)".

Intellectual Merit: ? The proposal addresses the problem of minimizing turbulence inside fluid flow in the case of free boundary interaction between a viscous fluid and a moving and deforming elastic body. The project is motivated by the issue of reducing turbulence inside the blood flow in stenosed and stented arteries. ? The project tackles the difficult, but important issue of optimal control in a free boundary fluid-elasticity interaction, where the common interface in the coupling is not assumed static, as in most of the previous literature on this subject. This will lead to the construction of quasilinear theory arising in Navier-Stokes equations coupled with waves. ? The results will break new ground in the theory of strong shape derivatives for hyperbolic problems with non-smooth Neumann boundary conditions, which is challenging due to the failure of the Lopatinski condition. ? The theory will provide well-posedness analysis for the first linear model of fluid-elasticity interaction that takes into account the common interface and its curvatures, which are critical for a correct physical interpretation of the coupling. ? The PI is well qualified to carry out this research. Her previous work on sensitivity analysis for a free boundary nonlinear coupling of viscous fluid flowing inside elasticity, and nonlinear waves with Neumann boundary conditions provides a solid foundation for this project. Methods to be employed include: shape and tangential calculus, the extractor and Fourier techniques, min-max, optimal control strategies, nonlinear semigroups. Broader Impacts: ? The project will launch new research in the field. The approach and the techniques introduced for the minimization of drag in the fluid-elasticity interaction can be adjusted and used to investigate inverse or control problems in many other free boundary coupled physical systems, and with different types of controls. ? Controlling turbulence in fluid flow is relevant in medical research (reducing turbulence in blood flow), as well as engineering design (reducing the drag of a car inside the atmospheric air flow, aeroelastic stability of structures, etc.). ? The proposed project lies at the interface of control theory of PDEs, well-posedness analysis for fluid-structure interactions, sensitivity and shape differentiability analysis. Therefore, the strategies, techniques, and results will be of interest to a broad mathematical and engineering audience. ? The PI has proven track record in advising students and will continue these efforts. The PI plans to attract students to the field and her research, through her courses in applied math, and talks in the Applied Math Club. She has already identified graduate students who are interested in her projects (Kristina Martin, Nathan Tryon), and will run special topics seminars for them. ? The PI has partnered with the NC Museum of Natural Sciences in an effort to create a ?Math Day at the Museum?, in order to promote and present her research to a broad audience, showcase students? research through posters and presentations, and encourage the participation of women and minorities in the study of math. During the events, the PI will give public presentations on her research, develop and set up demonstrations of real world phenomena illustrating applications of PDEs and their control, and promote women and minorities in math, by involving several student groups from NCSU: Assoc. for Women in Math, Women in Science and Engineering, and Society of African American Physical and Mathematical Sciences.

Interaction between a fluid and a structure via a common interface is a basic coupling mechanism in continuum mechanics. There are essentially two dierent scenarios: one in which the elastic solid is fully immersed in a fluid (like a submarine under water, or adherence/detachment of leukocytes in blood flow), and the other one is when the fluid is flowing inside a body, like the blood flowing in an artery. There has been a lot of interest in both models over the years, and mathematical and numerical subtleties are common to both cases. Historically, the problem of rigid body motion in both compressible and incompressible flows was first considered. This entailed coupling the Navier-Stokes equations with systems of ordinary dierential equations. More recently, there have been works that treat the motion of an elastic body in an incompressible flow, by coupling the Navier-Stokes equation with hyperbolic elasticity equations on xed domains: for linear models [1, 19], and for nonlinear models [2, 3, 23, 24, 27]. In any real-life problem of interaction between a fluid and an elastic structure, the appropriate case is that of a \free boundary" [19], i.e. a moving fluid-solid interface, which is generally not known, and must be found as part of the solution process. The case of free boundary fluid-structure interaction was considered recently for incompressible flow in [14, 15, 23, 24], and for the compressible flow case in [11]. Mathematically, the interaction is described by a partial dierential equations (PDEs) system that couples the parabolic and hyperbolic phases, where the key issue is that the traces of the elastic (wave/hyperbolic) component at the energy level are not dened via the standard trace theory. The loss of regularity induced by the hyperbolic component left the basic question of existence open until recently [14, 15, 23, 24] (for the case of a linear or quasi-linear elastic body flowing within a fluid). The local existence and uniqueness of solution were obtained by analyzing the fluid part in a hyperbolic-type functional framework that, at the price of increased compatibility assessments between the initial and boundary conditions, allows the increased regularization of the elastic behavior to be discarded. These results have not yet been extended to the conguration of a fluid inside an elastic envelope, which is of great interest in the modeling and analysis of the cardiovascular system. This proposal will focus on the dynamical case of fluid moving inside a compliant elastic body. A specic example of this is represented by blood flow in human arteries. However, the theory and the techniques also apply to the counterpart case of an elastic body floating within a fluid. The project represents the foundation of the PI's long term plan to investigate control problems and perform sensitivity analysis in fluid-structure interactions. When linearizing any coupling of fluid and structure, such as those occurring in arteries, submarines, and bridges, a major challenge arises in just how to linearize the free boundary. Linear models for the coupling present in the literature were obtained by linearizing each equation separately, then coupling the two linearizations. However, such an approach misses important coupling terms on the boundary, which would be present if the system were linearized as a whole.