Doctor of Philosophy
In this work, we introduce a differential equation model for a bacterial infection in a radially symmetric chronic wound. The goal of the current work is the following: show the model produces a unique solution in a suitable setting and then determine whether one can drive the system to the origin using controllers contained in a small subdomain of the wound. The model consists of three parabolic-like partial differential equations along with a single vector-valued ordinary differential equation. Of the critical features of the model, one of the most challenging is the nonlinear coupling describing chemotactic attraction between the neutrophils and a chemoattractant distributed throughout the wound. Lower order nonlinearities, describing various coupling phenomena, present further mathematical difficulties including the potential for our system to become singular. For the problem of well-posedness, we invoke a fixed point argument. First, we show that the system has a solution for predetermined finite time provided the initial conditions are taken sufficiently small, using maximal regularity results from the linear heat equation. We then show the model has a maximum-principle type result, namely that nonnegative initial conditions will give rise to nonnegative solutions for the aforementioned time duration. Finally, we show that the model has local-in-time existence for (potentially large) initial conditions provided these conditions are nonnegative. In this case, the corresponding solutions are also nonnegative. For the problem of null-controllability, we first study the control of the corresponding linear problem. Using a corollary of the Closed Graph Theorem, we derive an appropriate dual system to the linear control problem as well as the so-called observability inequality. Establishing the controllability inequality is equivalent to the controllability problem, and hence becomes our new goal. To this end, we introduce the Carleman-type estimates for the linear heat equation. These estimates are used first to establish the observability estimate using controllers on all the equations. Then, another round of estimation using the coupling in the linear equations allows us to show the observability inequality can be established for fewer controllers. Thus, we get an optimal controllability result for our system in terms of the number of required controllers. We then return to the control of the nonlinear system. For this, we introduce the state-to-control map. By showing this map is (locally) a homeomorphism through the inverse function theorem, we can deduce that the nonlinear system is locally null-controllable as well.
Dissertation or thesis originally submitted to ProQuest
Guffey, Stephen, "Local Null-Controllability of a Chemotactic Model for a Bacterial Infection in a Radially-Symmetric, Chronic Wound" (2021). Electronic Theses and Dissertations. 2904.