Nonlinear Dynamical Control Theory
During my second semester of graduate school I wrote a termpaper on nonlinear dynamical control theory (link to paper here). This work was by no means a serious research project, but it served as an introduction to this field and was my first introduction to modern differential geometry and integrability. Specifically, I studied how to control a simple mathematical model of a car using Lie brackets. These Lie brackets replicate the parallel parking procedure that drivers use everyday to essentially drive their cars sideways, i.e. parallel park their car. Although Lie brackets provide an existience proof that cars can actually be parallel parked (controllability) they do not provide any information about how a car may be optimally parked. I am interested in the question of how systems can not only be controlled but how they can be optimally controlled. For example, in the parallel parking problem two quantities that could be optimized include the distance travelled by the car or the amount of fuel consumed during parking.
Potential Student Research:
I am very interested in working with a student that is interested in optimal control theory. In particular I believe that a natural project is to study how to optimize a car's parallel parking procedure. A simple model of the motion of the car is:
The functions \(u_1(t)\) and \(u_2(t)\) correspond to how the driver controls the speed of the car with the gas pedal and the orientation of the car with the steering wheel. In this project we are asking what functions \(u_1\) and \(u_2\) can be selected that best optimize the parking procedure of the car. Specifcally, two natural payoff functionals to be minimized over solutions to the above differential equations that move the car from point \(a\) to point \(b\) are distance and fuel consumption:
For interested students, a wonderful introduction to optimal control theory was written by Craig Evans and can be found here: http://math.berkeley.edu/~evans/control.course.pdf.