[\dotx(t) = v(t)] [\dotv(t) = u(t) - g]
[PA + A'P - PBR^-1B'P + Q = 0]
Using LQR theory, we can derive the optimal control: Dynamic Programming And Optimal Control Solution Manual
where (P) is the solution to the Riccati equation:
| (t) | (x) | (y) | (V(t, x, y)) | | --- | --- | --- | --- | | 0 | 10,000 | 0 | 12,000 | | 0 | 0 | 10,000 | 11,500 | | 1 | 10,000 | 0 | 14,400 | | 1 | 0 | 10,000 | 13,225 | [\dotx(t) = v(t)] [\dotv(t) = u(t) - g]
[V(t, x, y) = \max_x', y' R_A(x') + R_B(y') + V(t+1, x', y')]
[\dotx(t) = (A - BR^-1B'P)x(t)]
Using optimal control theory, we can model the system dynamics as: