Time-optimal control

Remarks

For linear systems $x' = A x + B u$ with $U = [-1, 1]^m$ the theory is complete. If $x_1$ can be reached from $x_0$ at some time, then a time-optimal control exists, because the reachable sets are compact and convex and depend continuously on time. By Pontryagin's maximum principle a time-optimal control satisfies $u_j(t) = \sgn\big(b_j^{T} p(t)\big)$ wherever $b_j^{T} p(t) \neq 0$, where $b_j$ is the $j$-th column of $B$ and $p' = -A^{T} p$. If each pair $(A, b_j)$ satisfies the Kalman rank condition, then the time-optimal control is unique and bang–bang with finitely many switches. If moreover all eigenvalues of $A$ are real, each component $u_j$ switches at most $n - 1$ times. Example: for the double integrator $x_1' = x_2$, $x_2' = u$, $\abs{u} \le 1$, steered to the origin, the optimal control is $u = -1$ above the switching curve $x_1 = -\frac12 x_2 \abs{x_2}$ and $u = 1$ below it. On the curve itself it is the constant that drives the state along the curve into the origin, and every optimal control switches at most once.