Associative iLQR

Implementation of the Parallel Linear Quadratic Regulator (PLQR) algorithm

1. Initialisation: compute elements \(a=\{A, b, C, η, J\}\) do for all in parallel i.e. vmap;

2. Parallel backward scan: initialise with all elements & apply associative operator note association operator should be vmap. Scan will return \(V_{k}(x_{k})=\{V, v\}\);

3. Compute optimal control: \(u_k = -K_kx_k + K^{v}_{k} v_{k+1} - K_k^{c} c_{k}\). \(K`s have closed form solutions, so calculate :math:`u_k\) in parallel vmap.