= Solution
The <Pontryagin maximum principle> addresses optimization of an integral running reward, possibly with terminal reward, subject to a controlled differential equation, admissible control constraints and prescribed endpoint conditions. For a normal maximization problem $\dot x=f(t,x,u)$ with running reward $L$, the <Hamiltonian> is $H=L+\lambda f$; the <optimal control> maximizes it pointwise, and the adjoint satisfies $\dot\lambda=-H_x$ with the appropriate terminal transversality condition. State constraints require additional care when active.
Assume $p>0$ and $x(0)>0$. Here
$$
\boxed{H(x,u,\lambda)=(p-\lambda)u-u^2/x.}
$$
The terminal stock is free and has no salvage reward, so an optimum with positive terminal stock has $\lambda(T)=0$. For $x>0$, maximization over $u\ge0$ gives $u=x(p-\lambda)/2$ whenever $p-\lambda>0$. Then
$$
\dot\lambda=-u^2/x^2=-\tfrac14(p-\lambda)^2.
$$
Set $v=p-\lambda$, so $\dot v=v^2/4$ and $v(T)=p$. Integrating gives
$$
\boxed{\lambda(t)=p-\frac1{1/p+(T-t)/4},\qquad
\dot x=-\frac{2px}{4+p(T-t)}.}
$$
The resulting $p-\lambda$ is positive throughout, validating the interior control formula. Integrating the state equation yields
$$
\boxed{x(t)=x(0)\left(\frac{4+p(T-t)}{4+pT}\right)^2,\qquad
x(T)=\frac{16x(0)}{(4+pT)^2}.}
$$
The state remains positive, so its nonnegativity constraint is inactive and the terminal condition is consistent. The reward $pu-u^2/x$ is jointly concave in $(x,u)$ and the dynamics are linear, giving sufficiency of this solution. Alternatively the value function $V(t,x)=\lambda(t)x$ satisfies the <Hamilton-Jacobi-Bellman equation> with this maximizing control. At zero price the <optimal control> is zero, obtained by a continuous $p\downarrow0$ limit; zero initial stock also gives zero extraction.
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