I am working on methods for control design in nonlinear stochastic systems. In other words, given some sort of robot or other mechanical system, how can I do some amount of precomputation to then allow the system to solve a wide range of tasks in real time?

The general strategy for solving this problem is to patch together many locally valid control policies into a global control policy. This involves verifying that a given feedback law will accomplish a given task in a given region, which usually reduces to solving a semi-definite program.

However, these semi-definite programs can be quite large, so I am also studying methods for approximately solving large semi-definite programs.

Finally, I am working on a method for verifying that a control policy will accomplish its goal even in the presence of noise (so far most of our work has focused on deterministic systems; I am adapting it to systems with randomness).

I chose these problems because they seemed like the most interesting and important problems that were being studied by the research group I work with. Similar to paulfc, I am working on them partially due to inertia; I will probably not continue to work on them after I go to graduate school. However, I do believe that the problem of nonlinear control is extremely important. I just believe that there are other problems that are even more important.