Help with a project in Control Theory

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I have to develop a project in a Control Theory course along the semester. The problem is that we just started the semester so I don't know any about control theory haaha, so I post this here to ask you for ideas. The project has no clear requirements, but past projects include solving a concrete problem, investigation about stochastic and discrete extensions of some of the tools seen along the year and applications on game theory.

Also, in general, I like functional analysis and algebra.

The contents that we will see along the semester are:

-Linear systems, controllability, observability, stability, etc

-Quadratic linear problem.

-Pontryagin maximum principle

-Hamilton-Jacobi-Bellman equations

If you need more information please say it.

PD: sorry for my english, it is not my native language.

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