Diving Into Reliable Numerical Observability and Stabilization of the One-dimensional Wave Equation

In this project, a one-dimensional wave equation, which is a partial differential equation (PDE) describing vibrations on a string, is considered. It is known that the PDE model is exactly observable and exponentially stabilizable. The main goal of this project is to construct a numerical approximation technique, so-called the direct filtering technique, to prove that the Finite Difference and Finite Element space-discretized 1-D wave equations (i) with homogeneous Dirichlet boundary conditions are uniformly observable, (ii) with controlled boundary conditions are uniformly exponentially stable, as the approximation parameters tend to zero. It is crucial to develop reliable numerical approximation techniques for the controlled systems modeled by wave equations so that engineers can design reliable and robust controllers and sensors