 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 $x(0)>0$. HereThe 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 $u\ge 0$ gives $u=x(p-\lambda )/2$ whenever $p-\lambda >0$. ThenSet $v=p-\lambda$, so $\dot{v}=v^2/4$ and $v(T)=p$. Integrating givesThe resulting $p-\lambda$ is positive throughout, validating the interior control formula. Integrating the state equation yieldsThe 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 $p\downarrow 0$ limit; zero initial stock also gives zero extraction.