इष्टतम नियंत्रण

The manager maximizes profit ${\displaystyle \Pi }$:

${\displaystyle \Pi =\sum _{t=0}^{T-1}\left[pu_{t}-{\frac {u_{t}^{2}}{x_{t}}}\right]}$

subject to the law of evolution for the state variable ${\displaystyle x_{t}}$

${\displaystyle x_{t+1}-x_{t}=-u_{t}\!}$

Form the Hamiltonian and differentiate:

${\displaystyle H=pu_{t}-{\frac {u_{t}^{2}}{x_{t}}}-\lambda _{t+1}u_{t}}$

${\displaystyle {\frac {\partial H}{\partial u_{t}}}=p-\lambda _{t+1}-2{\frac {u_{t}}{x_{t}}}=0}$

${\displaystyle \lambda _{t+1}-\lambda _{t}=-{\frac {\partial H}{\partial x_{t}}}=-\left({\frac {u_{t}}{x_{t}}}\right)^{2}}$

As the mine owner does not value the ore remaining at time ${\displaystyle T}$,

${\displaystyle \lambda _{T}=0\!}$

Using the above equations, it is easy to solve for the ${\displaystyle x_{t}}$ and ${\displaystyle \lambda _{t}}$ series

${\displaystyle \lambda _{t}=\lambda _{t+1}+{\frac {(p-\lambda _{t+1})^{2}}{4}}}$

${\displaystyle x_{t+1}=x_{t}{\frac {2-p+\lambda _{t+1}}{2}}}$

and using the initial and turn-T conditions, the ${\displaystyle x_{t}}$ series can be solved explicitly, giving ${\displaystyle u_{t}}$.

The manager maximizes profit ${\displaystyle \Pi }$:

${\displaystyle \Pi =\int _{0}^{T}\left[pu(t)-{\frac {u(t)^{2}}{x(t)}}\right]dt}$

subject to the law of evolution for the state variable ${\displaystyle x(t)}$

${\displaystyle {\dot {x}}(t)=-u(t)}$

Form the Hamiltonian and differentiate:

${\displaystyle H=pu(t)-{\frac {u(t)^{2}}{x(t)}}-\lambda (t)u(t)}$

${\displaystyle {\frac {\partial H}{\partial u}}=p-\lambda (t)-2{\frac {u(t)}{x(t)}}=0}$

${\displaystyle {\dot {\lambda }}(t)=-{\frac {\partial H}{\partial x}}=-\left({\frac {u(t)}{x(t)}}\right)^{2}}$

As the mine owner does not value the ore remaining at time ${\displaystyle T}$,

${\displaystyle \lambda (T)=0}$

Using the above equations, it is easy to solve for the differential equations governing ${\displaystyle u(t)}$ and ${\displaystyle \lambda (t)}$

${\displaystyle {\dot {\lambda }}(t)=-{\frac {(p-\lambda (t))^{2}}{4}}}$

${\displaystyle u(t)=x(t){\frac {p-\lambda (t)}{2}}}$

and using the initial and turn-T conditions, the functions can be solved to yield

${\displaystyle x(t)={\frac {(4-pt+pT)^{2}}{(4+pT)^{2}}}x_{0}}$