A girl begins swimming from a point $(0,0)$ on the bank of a straight river. She swims at a constant speed $v$ relative to the water. The speed of the downstream current at a distance $y$ from the shore is $c(y)$. Hence her trajectory is described by

$$\dot{x}=v \cos \theta+c(y), \quad \dot{y}=v \sin \theta$$

where $\theta$ is the angle at which she swims relative to the direction of the current.

She desires to reach a downstream point $(1,0)$ on the same bank as she starts, as quickly as possible. Construct the Hamiltonian for this problem, and describe how Pontryagin's maximum principle can be used to give necessary conditions that must hold on an optimal trajectory. Given that $c(y)$ is positive, increasing and differentiable in $y$, show that on an optimal trajectory

$$\frac{d}{d t} \tan (\theta(t))=-c^{\prime}(y(t))$$