Computational Optimal Control Methods Of Several Spatial-temporal Processes In Fusion Plasmas

Controlled magnetic confinement fusion is becoming the most important field in the area of new energy development and research, and may be the ultimate solution to solve the energy crisis challenge. Magnetic confinement is considered as the most promising approach to harvest nuclear fusion energy. So far, Tokamak is regarded as an ideal de-vice to realize magnetic confinement of fusion fuel. There are many complex processes arising in plasma discharges which can be mathematically modeled as nonlinear partial differential equations (PDEs). Many extremely challenging modeling and control prob-lems related to Tokamak operations must be solved before it becomes a viable energy utility. The focus of this dissertation is to solve PDE-constrained optimal control prob-lems arising in the spatial-temporal evolution processes of fusion plasma, and mainly achieved the following results:1. An effective approach is proposed to solve the dynamic optimization problem of attaining the best possible current spatial profile during the ramp-up phase of the Tokamak. The proposed approach first uses the Galerkin method to reduce the di-mension of the original system model, and obtains a finite-dimensional ordinary differential equation (ODE) model based on the original magnetic diffusion PDE. Then, we combine the control parameterization method with a novel time-scaling transformation to obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques such as se-quential quadratic programming (SQP). The approach can not only optimize the control variables but also choose the best switching times of control variables, which improve the result more accurate. We also show that the gradients of the objective function and constraints with respect to the decision variables can be computed explicitly by solving an auxiliary dynamic system governing the state sensitivity matrix.2. A direct optimization algorithm framework based on PDE model is proposed to solve the optimal evolution trajectories of attaining a specific desired current pro-file during the ramp-up phase of plasma discharging. By using the MATLAB programming language and combining the SQP algorithm as well as PDE solver PDEPDE, a more user-friendly and efficient graphical user interface (GUI) is de-signed. The optimization problem can be solved successfully directly under the given conditions of system input parameters in the GUI. The designed algorithm framework is very convenient to handle the control problem of fusion plasmas, and the proposed framework of combining existing PDE and numerical optimiza-tion solvers to solve PDE-constrained optimization problem has the prospective to target challenging advanced control problems arising in more general spatial-temporal evolution processes.3. A new gradient-based optimization approach guaranteeing the stability of the closed-loop system is proposed to design the optimal state feedback for the unsta-ble spatial-temporal process arising in fusion plasmas evolution. The approach parameterizes the state feedback kernel directly as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, and avoids solving non-standard and complex Riccati-type equa-tions or Klein-Gorden type PDEs. The gradients of the system cost functional and constraints are computed analytically by solving a so-called "costate" PDE in a standard form. The optimized kernel is also theoretically proved for ensuring the feasible solutions and stability of the closed-loop system.