Ph.D. Baylor University, 2015
B.S. The University of Puget Sound, 2010
In 1988, Stefan Hilger introduced time scales calculus, which unifies difference equations and differential equations by showing they are special classes of dynamic equations on time scales. A time scale is an arbitrary closed subset of the real numbers. If the real numbers are used as the time scale, the theory yields calculus and differential equations. If the integers are the time scale, the theory yields difference calculus and difference equations. The power of the theory lies in the arbitrary choice of the time scale; one could choose a mixture of discrete points and closed intervals for the time scale, and the theory would describe how to analyze dynamic equations defined on the time scale.
My research focuses on applying time scales calculus to engineering problems where system updates occur non-uniformly in time. This is a common situation for systems that span long distances, such as the Texas power grid, as well as systems that are controlled by laggy networks. The collaboration with the engineers raises interesting mathematical questions, whose answers drive engineering questions, and so on.
A very basic but very interesting question involves the nature of the first order dynamic equation
The solution of the above dynamic equation is called the time scales exponential function since when the time scale is the real numbers, the solution is the exponential function
My research focuses on the stability theory of the time scales exponential function; we discover conditions when solutions stay bounded or tend to zero in the long run. Researchers define stability in many different ways, most of which turn out to be the same for this first-order problem on the reals or the integers, but which become distinct on a general time scale.
The set of all values λ in the complex plane that make the solution of the first order equation in question exponentially stable , that is, bounded by a decaying exponential function, has been known for 11 years. This stability region, however, is in general difficult to calculate. We have recently discovered an expression for the best circular approximation to the stability region at the origin. Under mild conditions, this circle is a subset of the region of uniform exponential stability, a different type of stability. The stability region and the best circular approximation at the origin can be seen below for a pulse time scale , a repeating pattern of an interval of length a followed by a gap of length b.
My research also concerns time scales that are generated randomly. In particular, we assume the distance between adjacent time scale points is a sequence of i.i.d. random variables. In this case, we can find an analogue of the region of exponential stability, called the region of almost sure exponential stability. If the eigenvalue is in this region, a solution of the dynamic equation is not exponentially stable with zero probability. In a recently submitted paper, we show the best circular approximation of this region at the origin is the region of mean-square exponential stability, which is a qualitatively “nicer” form of stability.
Once we understand the behavior of this simple problem, we are able to extend our results to first order linear systems of ordinary differential equations, which are widely used to model physical systems. When we know the qualitative behavior of these systems, we are able to design ways to control the physical system and make it behave to our liking. We can use the same techniques to discover values of the physical system that we are unable to measure directly, a process called designing an observer. Knowing how these systems behave on arbitrary time scales helps us design for situations where the timing of inputs to the system are not precisely uniform.
I am on junior leave for the Fall 2018 semester.