Publication Date
8-2024
Advisor(s) - Committee Chair
Ahmet Ozer, Ferhan Atici, Dominic Lanphier, Mark Robinson
Degree Program
Department of Mathematics
Degree Type
Master of Science
Abstract
This thesis proposes novel “uniform observability”-preserving space-discretized filtered Standard Finite Difference (SFD)-based model reductions for the Partial Differential Equation (PDE) models of three-layer laminates with fully clamped boundary conditions: (i) Mead-Marcus-type and (ii) Rao-Nakra-type. Three-layer laminates consist of relatively stiff outer elastic/piezoelectric layers constraining a viscoelastic compliant layer, allowing transverse shear. These two PDE models primarily describe the overall bending profile of the laminate and the shear of the constrained layer. Model (ii) is a more refined version of model (i), retaining the longitudinal displacements (stretching/compression) of the outer layers. The primary goal is to design a single boundary sensor for model (i) and three boundary sensors for model (ii), placed at the tip of the beam, to reconstruct the overall dynamics.
The uniform observability for the PDEs of (i) and (ii) with clamped boundary conditions were previously proved in [35, 38]. However, the standard Finite Difference model reductions, which are finite-dimensional, fail to maintain the same observability uniformly as the Finite Difference discretization parameter (mesh size) tends to zero. This discrepancy arises because the model reduction process generates spurious “high-frequency eigenvalues,” preventing the sensor from distinguishing between distinct vibrational modes. Another way to explain this discrepancy is that the uniform gap between these high-frequency eigenvalues tends to zero as the discretization parameter decreases.
A widely accepted remedy to overcome this issue is the “direct Fourier filtering” method, which manually eliminates these high-frequency eigenvalues, preserves a uniform gap among the eigenvalues, and therefore retains uniform observability. While this filtering method works well for single PDEs in the literature, its implementation is challenging for coupled PDEs like those considered in this thesis due to the strong coupled dynamics of the equations. Additionally, dealing with fully clamped boundary conditions is much harder than dealing with simply-supported or cantilevered boundary conditions.
First, it is shown that a “time-independent” spectral problem for the three-layer Mead- Marcus laminate model (i) is uniformly observable as the Finite Difference discretization parameter tends to zero. The proof is based on the careful implementation of the “discrete multipliers” technique together with “direct Fourier filtering”. Next, the “time-dependent” full model (i) is considered. It is also shown that the uniform boundary observability is retained, guaranteeing a small observation time. Finally, through a careful application of Haraux’s theorem, the same uniform observability result is shown to hold for any arbitrarily small observation time, closely resembling its PDE counterpart.
For model (ii), applying the same methodology of “discrete multipliers” and “direct Fourier filtering” used for model (i), uniform spectral observability is achieved, assuming identical wave speeds for all three layers of model (ii). The minimal observation time is also provided, closely resembling its PDE counterpart.
This thesis marks the first attempt in the literature to prove uniform observability for the Finite Difference model reductions of the fully-clamped Mead-Marcus and Rao-Nakra type laminates as the discretization parameter tends to zero. Proving uniform observability of the Finite Difference model reductions in the finite-dimensional setting is essential to establish exact controllability and exponential stabilization of these models for real-world implementations. These contributions open up new research opportunities in the field.
Disciplines
Applied Mathematics | Control Theory | Dynamic Systems | Numerical Analysis and Computation | Partial Differential Equations | Physical Sciences and Mathematics
Recommended Citation
Haider, Md Zulfiqur, "Uniform Observability Results For Three-Layer Smart Laminate Models and Their Finite Difference- Based Model Reductions" (2024). Masters Theses & Specialist Projects. Paper 3750.
https://digitalcommons.wku.edu/theses/3750
Included in
Control Theory Commons, Dynamic Systems Commons, Numerical Analysis and Computation Commons, Partial Differential Equations Commons