Research Projects

The proposed research addresses fundamental problems in the control theory of nonlinear partial differential equations. Specifically, it considers two key models: the 1D semilinear heat equation with superlinear nonlinearity and the 1D compressible Navier-Stokes equations. The motivation arises from both theoretical challenges and practical applications where actuation is spatially constrained or limited to a few modes. This study addresses the crucial challenge of controlling strongly nonlinear PDE systems; characterized by infinite-dimensional dynamics and complex modal interactions, using only partial or low-dimensional information. Unlike well-studied linear or weakly nonlinear systems with full control, the stabilization and controllability of superlinear or compressible fluid models under limited control are still largely unexplored. This project aims to bridge that gap by developing innovative low-mode control strategies capable of regulating global behavior in such nonlinear systems. The scientific objectives of the project are twofold: • To design a localized feedback control law that guarantees global exponential stabilization of the 1D semilinear heat equation with superlinear nonlinearity, under homogeneous Dirichlet boundary conditions. • To establish approximate controllability of the 1D compressible Navier-Stokes system, using finite-dimensional control acting only on the velocity component. The main hypotheses/models to be tested include: • That spectral decomposition and low-mode feedback control are sufficient to stabilize semilinear heat equations, even in the presence of strongly nonlinear (e.g., polynomial) source terms. • That a combination of modal projection, observability analysis, and compactness methods can be used to drive the compressible Navier-Stokes system approximately to a desired state using limited internal actuation. The main experiments and theoretical components include: • Construction of a Lyapunov functional capturing both low and high-frequency energy components for the heat equation, followed by rigorous stability analysis. • Use of fixed-point arguments and continuation techniques to extend linear controllability to the full nonlinear system. • Investigation of the minimum number of controlled modes needed and the influence of viscosity and amplitude constraints on control performance. The significance of this research lies in its potential to advance the theoretical foundation of PDE control under realistic constraints and to bridge the gap between infinite-dimensional systems and finite-dimensional actuation strategies. If successful, the project would open pathways to the efficient control of thermal and fluid systems in engineering, climate modeling, and biomedical applications, where full-state access is often infeasible. Moreover, it will contribute fundamentally to nonlinear analysis, semigroup theory, and control design in infinite-dimensional settings.