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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: \begin{cases}\dot{x} =u_1(t)\cos(\theta)\\ \dot{y}=u_1(t)\sin(\theta)\\ \dot{\theta}=u_2(t), \end{cases} 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: \begin{cases} D[u_1,u_2]=\int_{a}^{b}\sqrt{\dot{x}^2+\dot{y}^2}dt\\ F[u_1,u_2]=\int_{a}^{b}u^2_1(t)dt. \end{cases} 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. |