State Pontryagin's Maximum Principle (PMP).

In a given lake the tonnage of fish, $x$, obeys

$$d x / d t=0.001(50-x) x-u, \quad 0<x \leqslant 50$$

where $u$ is the rate at which fish are extracted. It is desired to maximize

$$\int_{0}^{\infty} u(t) e^{-0.03 t} d t$$

choosing $u(t)$ under the constraints $0 \leqslant u(t) \leqslant 1.4$, and $u(t)=0$ if $x(t)=0$. Assume the PMP with an appropriate Hamiltonian $H(x, u, t, \lambda)$. Now define $G(x, u, t, \eta)=$ $e^{0.03 t} H(x, u, t, \lambda)$ and $\eta(t)=e^{0.03 t} \lambda(t)$. Show that there exists $\eta(t), 0 \leqslant t$ such that on the optimal trajectory $u$ maximizes

$$G(x, u, t, \eta)=\eta[0.001(50-x) x-u]+u$$

and

$$d \eta / d t=0.002(x-10) \eta$$

Suppose that $x(0)=20$ and that under an optimal policy it is not optimal to extract all the fish. Argue that $\eta(0) \geqslant 1$ is impossible and describe qualitatively what must happen under the optimal policy.