Mathematical Control Theory and Analysis of Partial Differential Equations Coupled Across a Boundary Interface

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In one component of this project, the investigators will study equations which mathematically describe the flow of gas as it streams within a walled cavity and interacts with a vibrating portion of the cavity's boundary. Within this mathematical framework, the investigators will attempt to determine if the gas flow and cavity wall displacement can be favorably altered or "controlled" by means of the deliberate placement of forces which act on the cavity wall. The accomplishment of this program objective could lend insight into understanding and controlling the gas flows which are associated with high-speed aircraft and natural gas pipelines. Furthermore, a mathematical model which has recently been derived to describe a structure-fluid interaction in which an elastic body is wholly immersed within a given fluid will be studied. This new mathematical model differs from previous structure-fluid models, in that numerical simulations of solutions to the new model suggest that the associated structural displacements, as well as the fluid flow field, elicit a much more quiescent behavior than previously thought. The investigators intend to mathematically verify this observed "stability" for such structure-fluid dynamics. A successful conclusion to this work could potentially enhance the understanding of cellular dynamics: such structure-fluid interaction models have been invoked to describe a nuclear body, within a cell, as it interacts with the surrounding viscous cytoplasm. Moreover, a mathematical model which describes acoustic wave flow within a structural chamber, with one of the chamber walls being flexible, will be studied. In particular, it is intended to determine if the acoustic waves in this structural model will die out as time evolves. Such a conclusion of wave stability in this mathematical setting would have implications in real-world control engineering: With a view of promoting a quiescent acoustic field within the interior of an aircraft cabin, the stability results of the project could conceivably be taken into account in aircraft fuselage design. In the course of fulfilling the project objectives, graduate students, including first generation university students and women, will receive training in the necessary qualitative analysis and numerical computation. The gas flow-cavity model described above consists of the 3D compressible Stokes or Navier-Stokes flow equations coupled to an elastic plate equation which describes the displacements of the flexible portion of the cavity wall. This PDE system is subjected to cavity boundary control terms. The investigators will determine the optimal regularity of solutions to such controlled (gas) flow-structure PDE models under the influence of boundary control terms of prescribed smoothness. Subsequently, they intend to solve the associated null controllability problem with respect to the appropriate space of boundary control functions. Moreover, in the aforementioned structure-fluid interaction, the structure is composed of a "thick" and "thin" layer: the thick layer is described by a 3D wave equation, while the the thin layer is described by a 2D wave equation and constitutes the boundary interface between the thick wave equation and the 3D fluid component equation. The investigators intend to establish that classical solutions of this 3D wave-2D wave-3D heat PDE system manifest at least a polynomial rate of decay, which would be in line with what has been observed numerically. In addition, the rational decay problem for the structural acoustic interaction, in which an acoustic wave equation within a bounded domain interacts with a dissipative elastic equation across a boundary interface, will be considered. Moreover, an algorithm which will confirm that the polynomial rates obtained are optimal will be developed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.