Control Properties of Partial Differential Equations: Data-Driven and Operator Theoretic Approach

The control theory for partial differential equations (PDEs) has many applications in engineering, biological, and physical sciences, including soft robotics, building energy systems, vehicle autonomy, smart manufacturing, and advanced material technologies. Control design for these applications is often formulated as a mathematical optimal control problem. However, deriving accurate physics-based PDE models is challenging due to the complex, high-dimensional nature of the underlying dynamics. This research will develop a new framework for data-driven control of systems governed by linear and nonlinear PDEs, using sensor technology to collect high-resolution spatiotemporal data from controlled processes, such as soft robotic systems observed via distributed sensors. It will also establish analytical foundations for controlling multi-flow dynamics interacting with elastic bodies, which describe phenomena such as aircraft design, ocean-air interactions, and atmospheric events that include freezing rain or ice accretion during flight. This work aims to strengthen the scientific foundation for the controllability of complex PDE systems and train undergraduate and graduate students through an interdisciplinary program bridging mathematics and engineering. Students will gain a strong background and opportunities for research at the intersection of theory and application, including data science, control theory, dynamical systems, optimization, and robotics. This project juxtaposes applied mathematics and engineering methodologies to address several interconnected directions at the interface of optimal control and partial differential equations. The investigators will first construct a framework for the data-driven optimal control of nonlinear PDE systems of thermoelasticity. This framework is predicated on a Koopman operator theoretic approach. A frequency domain methodology will then be developed to study the well-posedness and boundary null controllability of coupled structure-multiflow interaction PDE systems, based on the analysis of the resolvent of the corresponding semigroup generator. Then, the stability and instability properties of composite structured PDE systems will be analyzed via the spectral properties of the associated semigroup generator of the dynamics. Subsequently, a new methodology for the challenging question of ascertaining the stability of fluid-structure interactive PDE systems, which are posed on geometries that are not assumed to be smooth and share the same dimensionality, will be established. 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.