Publication Date

Spring 2021

Advisor(s) - Committee Chair

Dr. Ahmet Ozkan Ozer (Director), Dr. Mikhail Khenner, Dr. Mark Robinson, Dr. Richard Schugart

Degree Program

Department of Mathematics

Degree Type

Master of Science


*see note below

In control theory, the time it takes to receive a signal after it is sent is referred to as the observation time. For certain types of materials, the observation time to receive a wave signal differs depending on a variety of factors, such as material density, flexibility, speed of the wave propagation, etc. Suppose we have a strongly coupled system of two wave equations describing the longitudinal vibrations on a piezoelectric beam of length L. These two wave equations have non-identical wave propagation speeds c1 and c2. First, we prove the exact observability inequality with the optimal observation time satisfying T > T1 = 2L min (c1,c2) by adopting two different techniques: the multipliers method (non-spectral) and non-harmonic Fourier series (spectral). Next, we discretize the spacial variable for the system via central Finite Differences. We find that for this particular discretization, the minimal observability time approaches infinity as the discretization parameter h goes to zero, and therefore, the discretized equations lack uniform observability unlike the original equations. This is simply due to the blind use of Finite Differences which generates spurious high frequency vibrational modes. To resolve this issue, a filtering technique, known as the direct Fourier filtering, is adopted, and an observability inequality is proved with a (sub-optimal) observation time T > T1 > T2 as the discretization parameter tends to zero. These results show that filtered finite differences can be safely applied to the system of piezoelectric beam equations in designing stabilizing controllers.

*because of formatting limitations some formulas in this field may not appear accurately. Please see the actual thesis for definitive versions.


Applied Mathematics | Control Theory | Numerical Analysis and Computation | Partial Differential Equations