[\dotx(t) = v(t)] [\dotv(t) = u(t) - g]
Using LQR theory, we can derive the optimal control:
The optimal closed-loop system is:
[J(u) = x(T)]
[u^*(t) = g + \fracv_0 - gTTt]
where (P) is the solution to the Riccati equation:
The optimal trajectory is:
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[V(t, x, y) = \max_x', y' R_A(x') + R_B(y') + V(t+1, x', y')]
Using optimal control theory, we can model the system dynamics as: Dynamic Programming And Optimal Control Solution Manual
[u^*(t) = -R^-1B'Px(t)]
[PA + A'P - PBR^-1B'P + Q = 0]