Controls and their Large Scale Computation for Long Time Dynamics of Navier-Stokes Equations and Burgers' Equation

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9626154 Yan The investigator studies incompressible flows governed by the Navier-Stokes equations or Burgers' equation. Some special issues are studied in depth. Various types of control objectives (minimizing velocity tracking, drag, vorticity) and control parameters (distributed control and boundary control) are studied. Long-time dynamics of controlled flows is the main subject of this project. The long-time behavior of global-in-time and piecewise-in-time optimal controls are studied in different settings. Numerical schemes for computing the control problems are designed and their performance is analyzed. Effectiveness of control is analyzed for both continuous and discrete systems; the latter is particularly close to the design of practical control. Numerical schemes are implemented to evaluate the effectiveness of the control problems. The numerical results both reinforce the theoretical results and reveal more quantitative information of the controlled dynamics. Improving the design and control of modern aircraft can be achieved through optimal design of wings, engines, etc, or through the active control of air flowing over wings and into engines. The project studies issues basic to active control. The investigator develops new mathematical and computational tools that provide good engineering insight and practical design criteria for controlling fluid motions in aerodynamic and hydrodynamic systems. This requires high performance computational resources an order of magnitude greater than typical modeling for uncontrolled flows. Direct applications can also be found in the design of submarine body shape and propelling systems.