## Differential/Difference Equations

Advisor: Professors Allan Peterson

Prerequisite: A first course in Differential Equations.

We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite.

## Dynamics of Interacting Populations

Advisors: Professors Bo Deng and Glenn Ledder

,Prerequisite: a thorough understanding of the material in an introductory course in differential equations and a basic course in linear algebra. An elementary analysis course, considerable experience with computational software (Maple, Matlab, or Mathematica), and/or some background in population biology or ecology is preferred.

Simple differential equation models for two interacting species and for interaction of a host species with a disease organism have been known since the 1920's. Recent advances in mathematical techniques and computer software have brought about the opportunity to study more complicated systems of interacting populations. There are a lot of unsolved problems in the dynamics of interacting populations that are accessible to undergraduates armed with some basic tools in dynamical systems and mathematical modeling.

We will develop and study a mathematical model for a problem of current interest in population dynamics. This could be a problem with several interacting species in a food chain or a problem in which the predator-prey dynamics is influenced by a disease of either the predator or the prey. The REU students will be studying mathematical questions, such as whether or not there is a stable equilibrium state, a stable limit cycle, or chaos, and modeling questions, such as whether or not we can tell from current data what the final state of the system will be and how rapidly the system will approach that state.

A key feature of population dynamics is the importance of processes having very different time scales. These differences allow for the use of singular perturbation methods to identify simple components within complex models whose union gives approximately the behavior of the complex system. At the beginning of the project, we will provide tutorials in dynamical systems, (phase plane analysis and singular orbit analysis) and mathematical modeling (common predator-prey and epidemiology models and scaling). The students will apply these skills to some previously solved problems and then use them to study a new problem.

## Sampled-Data Control

Advisor: Professor Richard Rebarber,

Prerequisites: Introductory courses in differential equations, partial differential equations, and linear algebra. Some complex analysis and some computer programming would also be helpful, but are not required.

Project Description: This project is in the area called control theory, which is a topic in both mathematics and engineering. In a control problem there is an equation which gives the relationship between two functions of time: *w(t)*, which we call the *state*, and *u(t)*, which we call the *control*. In the type of control problem we consider, called a *stabilization* problem, the job is to choose the control *u(t)* to make the state *w(t)* decay to 0 as t ! 1. This kind of problem would arise in applications where the goal is to stop a structure (or an electrical circuit) from vibrating.

If we can observe *w(t)* at all times *t* > 0, we might be able to do this job by choosing *u(t)* to be a *linear continuous-time feedback of w(t)*. However, in many real-world applications we cannot observe *w(t)* for all times, but just at discrete times 0, *h*, 2*h*, 3*h*, . . . ., where *h* is called the *sampling time*. In this project we try to determine how to use this sampled data (instead of the continuous-time data) to choose the control *u(t)* to do the job. One way of doing this is to find a continuous-time feedback control which does the job, then apply a *sample-and-hold* process to this feedback to obtain a new controller which only uses the sampled data; we call this new feedback a *sampled-data controller*. We consider the following question: *If the continuous-time feedback causes the state to be stable, will the sampled-data controller also cause the state to be stable, provided we use small enough sampling times h?*

It seems reasonable that the answer to this question should be "yes", because we would expect that if we sample-and-hold fast enough, then the sampled data controller would look like the continuous time controller; roughly speaking, this is why compact discs and mp3's sound good as long as the sampling time is fast enough. In fact, the answer to this question is indeed yes when the relationship between the state and the controller is given by a linear ordinary differential equation.

When the state and control are related by a linear partial differential equation, the answer is "sometimes yes, sometimes no, often we have no idea." In this project we look at a case where we have no idea: when *w* satisfies a one-dimensional wave equation, and *u* is a boundary control. In this case we can easily find a stabilizing continuous-time feedback (essentially the common-sense solution of pushing one end of the wave up when you feel it go down, and pushing down when you feel it go up). We then apply a sample-and-hold to this feedback, and try to determine whether the sampled-data controller also helps stabilize the state, or possibly makes it worse.