We’ve seen

Dynamic Programming to solve Cost-to-go $J^* (x)$

• on a mesh
• neural value iteration (RL DQN)

LQR

• higher dimensions
• linear systems

TODO: Sums Of Squares (SOS)/Lyapunov

Formulation

$\dot{x} = f(x, u) \ \forall{t = [0, t_f]}$ - does not have to be linear

$\displaystyle\min_{x(\cdot), y(\cdot)} \int^{t_f}_{t_0}\ell (x(t), u(t)) dt$

Discrete case

$\displaystyle\min_{x[\cdot], y[\cdot]} \int^{t_f}_{t_0}\ell (x(t), u(t)) dt\\\text{s.t.} \ x[n + 1] = A x[n] + B u[n]$

Note that the dynamics and constraints are linear

However, our costs are quadratic and convex

$\ell(x, u) = x^TQx + u^TRu$

Therefore, this is the quadratic program problem