This project focuses on control and inverse problems arising in complex systems governed by Partial Differential Equations (PDEs). We address both theoretical and numerical aspects, including the analysis of existence, uniqueness, multiplicity, stability, and continuous dependence of solutions, as well as the development of approximation schemes, convergence analysis, and numerical validations.
The systems under consideration encompass a broad range of applications, including parabolic and hyperbolic PDEs, variants of the Navier-Stokes equations, tumor growth models (with and without chemotaxis), and free-boundary problems. These models are motivated by phenomena in science and engineering, where accurate modeling and data integration are essential.
A key component of the project is the integration of machine learning techniques to enhance the analysis, simulation, and solution of control and inverse problems. By combining tools from mathematical analysis, numerical methods, and artificial intelligence, we aim to develop robust methodologies that bridge theory and application.