Controllability of hyperbolic and degenerate parabolic equations in one dimension
In this thesis, we study the controllability problem for two systems of partial differential equations. We will first consider the wave equation with variable coefficients and potential in one dimension, $u_{tt}  (a(x)u_x)_x + pu = 0$, with control function $v(t)$ acting on the boundary. We consider a class of functions corresponding to a special weight function that contains the variable coefficient $a(x)$. From here, we derive a global Carleman estimate for this system, and establish the controllability property. We then later extend the class of admissible functions $a(x)$ for which the controllability property holds true. We then study the controllability problem for the degenerate heat equation in one dimension. For $0\leq \alpha <1$, on $(0,1) \times (0,T)$, we consider $w_t  (x^{\alpha}w_x)_x = f$. This equation is degenerate because the diffusion coefficient $x^{\alpha}$ is positive in the interior of the domain and vanishes at the boundary. We consider this problem under the Robin boundary conditions. Again, we derive a Carleman estimate for this system, taking into account the new boundary terms that arise from the Robin conditions.
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 Copyright Status
 In Copyright
 Material Type

Theses
 Authors

Bohn, Jonathan Matthew
 Thesis Advisors

Zhou, Zhengfang
 Committee Members

Abbas, Casim
Promislow, Keith
Yan, Baisheng
 Date
 2018
 Program of Study

Mathematics  Doctor of Philosophy
 Degree Level

Doctoral
 Language

English
 Pages
 viii, 78 pages
 ISBN

0438752708
9780438752702