Time optimality

Interesting interpretations of the minimum principle exist for the case of optimizing the time to reach the goal [424,903]. In this case, $l(x,u) = 1$ in (15.26), and the cost term can be ignored. For the remaining portion, let $\lambda$ be defined as

$$\lambda_i = -\frac{\partial G^*}{\partial x_i} ,$$ (15.40)

instead of using (15.25). In this case, the Hamiltonian can be expressed as

$$H(x,u,\lambda) = \sum_{i=1}^n \lambda_i f_i(x,u) = \left\langle -\frac{\partial G^*}{\partial x},f(x,u) \right\rangle ,$$ (15.41)

which is an inner product between $f(x,u)$ and the negative gradient of $G^*$. Using (15.40), the Hamiltonian should be maximized instead of minimized (this is equivalent to Pontryagin's original formulation [801]). An inner product of two vectors increases as their directions become closer to parallel. Optimizing (15.41) amounts to selecting $u$ so that ${\dot x}$ is as close as possible to the direction of steepest descent of $G^*$. This is nicely interpreted by considering how the boundary of the reachable set $R(x_0,{\cal U},t)$ propagates through $X$. By definition, the points on $\partial R(x_0,{\cal U},t)$ must correspond to time-optimal trajectories. Furthermore, $\partial R(x_0,{\cal U},t)$ can be interpreted as a propagating wavefront that is perpendicular to $-\partial G^*/\partial x$. The minimum principle simply indicates that $u$ should be chosen so that ${\dot x}$ points into the propagating boundary, as close to being orthogonal as possible [424].