Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Real-time plasma state monitoring and supervisory control on TCV

In ITER and DEMO, various control objectives related to plasma control must be simultaneously achieved by the plasma control system (PCS), in both normal operation as well as off-normal conditions. The PCS must act on off-normal events and deviations from the target scenario, since certain sequences (chains) of events can precede disruptions. It is important that these decisions are made while maintaining a coherent prioritization between the real-time control tasks to ensure high-performance operation. In this paper, a generic architecture for task-based integrated plasma control is proposed. The architecture is characterized by the separation of state estimation, event detection, decisions and task execution among different algorithms, with standardized signal interfaces. Central to the architecture are a plasma state monitor and supervisory controller. In the plasma state monitor, discrete events in the continuous-valued plasma state are modeled using finite state machines. This provides a high-level representation of the plasma state. The supervisory controller coordinates the execution of multiple plasma control tasks by assigning task priorities, based on the finite states of the plasma and the pulse schedule. These algorithms were implemented on the TCV digital control system and integrated with actuator resource management and existing state estimation algorithms and controllers. The plasma state monitor on TCV can track a multitude of plasma events, related to plasma current, rotating and locked neoclassical tearing modes, and position displacements. In TCV experiments on simultaneous control of plasma pressure, safety factor profile and NTMs using electron cyclotron heating (ECH) and current drive (ECCD), the supervisory controller assigns priorities to the relevant control tasks. The tasks are then executed by feedback controllers and actuator allocation management. This work forms a significant step forward in the ongoing integration of control capabilities in experiments on TCV, in support of tokamak reactor operation

Overview of progress in European medium sized tokamaks towards an integrated plasma-edge/wall solution

Integrating the plasma core performance with an edge and scrape-off layer (SOL) that leads to tolerable heat and particle loads on the wall is a major challenge. The new European medium size tokamak task force (EU-MST) coordinates research on ASDEX Upgrade (AUG), MAST and TCV. This multi-machine approach within EU-MST, covering a wide parameter range, is instrumental to progress in the field, as ITER and DEMO core/pedestal and SOL parameters are not achievable simultaneously in present day devices. A two prong approach is adopted. On the one hand, scenarios with tolerable transient heat and particle loads, including active edge localised mode (ELM) control are developed. On the other hand, divertor solutions including advanced magnetic configurations are studied. Considerable progress has been made on both approaches, in particular in the fields of: ELM control with resonant magnetic perturbations (RMP), small ELM regimes, detachment onset and control, as well as filamentary scrape-off-layer transport. For example full ELM suppression has now been achieved on AUG at low collisionality with n  =  2 RMP maintaining good confinement ${{H}_{\text{H}\left(98,\text{y}2\right)}}\approx 0.95$ . Advances have been made with respect to detachment onset and control. Studies in advanced divertor configurations (Snowflake, Super-X and X-point target divertor) shed new light on SOL physics. Cross field filamentary transport has been characterised in a wide parameter regime on AUG, MAST and TCV progressing the theoretical and experimental understanding crucial for predicting first wall loads in ITER and DEMO. Conditions in the SOL also play a crucial role for ELM stability and access to small ELM regimes

Limiting Imbeddings -- The Case Of Missing Derivatives

We consider fractional Sobolev spaces with dominating mixed derivatives and prove some generalizations of Trudinger's limiting imbedding theorem