Executive Summary : | In this project our aim is twofold. First, we focus on global stabilization problems and related finite element analysis using a feedback control approach e.g., Lyapunov type/interpolant operator. The remaining project is related to numerical approximation to Hamilton- Jacobi-Bellman (HJB) and Hamilton-Jacobi-Isaacs (HJI) type equations with an application to closed-loop control in the sense of stabilization to different Ordinary differential Equations (ODEs) and Partial differential Equations (PDEs). Open-loop control depends only on time, which has various drawbacks. If the initial state changes, then optimal control must be recomputed again from the beginning. Moreover, open-loop control is not designed to handle modeling errors or exogenous disturbances. On the other hand, feedback control has several advantages and is essential in many scenarios e.g., robotics and flight control. In the literature mainly local stabilization result is available. Even for one dimensional problem there are very few results on global stabilization i.e., convergence of the unsteady solution to its steady state solution which motivate us to start this project. We will investigate global stabilization for various time dependent problems starting with Sobolev type equation applying Neumann boundary feedback control via Lyapunov method. For conforming finite element method, it will be more interesting to see whether we can compute the error analysis of the state and control variables. Then, we go for the Navier- Stokes and Kelvin -Voigt systems. Further, we are interested in stabilization problems with distributed control via interpolant operator technique. Also, with a linear Dirichlet control law, we are planning to stabilize locally nonlinear systems for example two-dimensional Burgers’ type equation. Regarding the last part of the project, applying dynamic programming techniques one can further reduce the optimal control problem in terms of the value function. With regularization of the value function, we are interested in studying efficient solution technique so that we can solve highly nonlinear system via some known Newton’s type method and obtain the control as a by-product. We are also planning to apply machine learning techniques to the aforementioned stabilization problems. |