The present proposal is focused on the study of control-theoretic issues for systems of strongly coupled PDEs, where the (active, passive) control action is exercised in the transmission conditions at the interface between two media. They include: stabilization and long-time behavior/theory of attractors, controllability, optimality, min-max game theory, inverse reconstruction from boundary measurements of a single system component, etc. A preliminary question is well-posedness of the overall dynamically interactive system, typically consisting of components of different type, e.g., parabolic versus hyperbolic. Emphasis is given to problems in mathematical and control-theoretic areas which are rather unexplored, such as: gas flow-structure interaction in aeroelasticity; elastic structure-flow interaction with moving interface; structural acoustic interaction with a non-shallow shell as active curved wall and a nonlinear acoustic Westervalt-Kuznetesov chamber (based on Fourier Law of Heat Flux) or Jordan-Moore-Gibson-Thompson chamber (based on Maxwell-Cattaneo Law), both arising in High Intensity Ultrasound focusing.

Mathematical control theory injects in the study of differential equations the active viewpoint of an external agent or manipulator, who is not satisfied with just passively observing or contemplating the evolution of a natural phenomenon; but instead seeks to intervene as to influence and modify its dynamical behavior, in order to achieve a preassigned outcome. Celestial mechanics offers an established example of the passive, non-controlled viewpoint: The astronomer may calculate orbits, predict time and location of eclipses, but cannot alter these phenomena. In contrast, governmental action on social or economic issues---such as reducing unemployment or stimulating growth---illustrates the active, control-based strategy. Analysis versus synthesis. The present project applies the active control approach of synthesis to study complex multi-faceted phenomena or combined units---ubiquitous in the physical world---consisting of systems of strongly coupled partial differential equations, where the controlling authority has access only to the interface between two different media. To suppress turbulence or flutter in fluid-structure or air/gas flow-structure interactions. Or to achieve noise reduction in an acoustic chamber, such as the interior of an aircraft. Or to obtain high intensity ultrasound focusing in medical applications, ranging from lithotripsy or thermo-therapy to ultrasound cleaning and welding.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Pedro Embid
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University of Virginia
United States
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