0x202 Optimal Control
Dynamics
The general functino to represent a smooth system is
\[\dot{x} = f(x,u)\]
where \(x\) is the state and \(u\) is the state. In a mechanical system \(x\) can be \((q, v)\) where \(q\) is configuration (might not be a vector) and \(v\) is velocity.
For example of an pendulum, \(x = (\theta, \dot\theta) \in S^1 \times R\) where \(S^1\) is a circle. this is a cylinder
Definitino (control affine systems) Many systems have the following specific structure (can be put into this form)
\[\dot{x} = f_0(x) + B(x) u\]
where \(f_0(x)\) is a drift and \(B(x)\) is the input Jacobian.