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Boundary Feedback Control Design for Classes of Mixed-Type Partial Differential Equations


The work in this dissertation summarizes some advancement in the theory of boundary controller design for coupled partial differential equations (PDEs), including a new interpretation of designing bilateral boundary controllers as equivalent coupled PDEs via a method dubbed “folding.” In particular, systems of purely hyperbolic type, purely parabolic type, and mixed (hyperbolic-parabolic) type are all explored in the context of boundary control. The work centers around the method of infinite-dimensional backstepping consisting of an non-trivially invertible spatial transformation mapping to a system with desireable properties, such stability of equilibria and convergence speed. A companion kernel PDE must be solved to properly define the transform. The transformations across the different classes of PDEs are similiar (and in certain cases, identical); however, curious behavior arises from a heterogeneous mixed type PDE, in which the companion kernel PDE becomes non-standard. The thesis studies some preliminary work into mixed type PDEs in an attempt to recover a more general backstepping design for linear PDE.

In the purely hyperbolic work, a special case of a underactuated hyperbolic system is considered. This is in opposition to pre-existing literature, which assumed a fully actuated system. The classical backstepping boundary controller is modified for the underactuated hyperbolic case, admitting a two-tiered trasnformation approach in which the backstepping controller is augmented by a predictor-based controller to achieve a finite-time stability for the trivial solution of the system. In the purely parabolic work, the notion of the folding approach is introduced as an alternative design method to pre-existing bilateral boundary control design work. The folding approach admits additional design parameters for the control designer, allowing the controls to be biased for differing performance indexes. A complimentary state estimator is designed, which allows for collocated point measurement at any arbitrary point in the domain independent of the control design. The two are combined to achieve an output-feedback control result.

Several results are given for mixed type PDE systems of hyperbolic-parabolic type as well. A first result invovles a scalar system with coupling on the boundary and the interior. The interior coupling necessitates more advanced techniques in the analysis of the companion kernel PDE, particularly in showing well-posedness. These ideas are also applied to other higher-order coupled systems of hyperbolic-parabolic type, including delay compensation for systems of parabolic PDEs, and delay compensation for bilateral controller design of parabolic PDEs.

Both the notion of bilateral boundary control and mixed type PDE systems arise in short- wavelength light generation. In the state-of-the-art light generation at the extreme ultraviolet (EUV) wavelengths, instabilities in the light generation process potentially arise due to coupled plasma interactions with the generating process. Phenomena such as ion-acoustic waves, free- electron plasma diffusion, magnetohydrodynamics, and thermofluidics arise due to the interacting plasma, introducing potential modes of instability. This instability neceessitates the introduction of feedback control, which can be introduced via two controllers on either boundary of the process domain.

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