The following is a problem which arises in the economics of natural resources, but I’m posting it here as my question is essentially mathematical:
Output $Y$ is produced according to a function $Y = F(K,rS)$ where $K$ is man-made capital, $S$ is a non-renewable natural resource, and $r$ is the proportionate rate of extraction and use of the natural resource ($K,S,r \geq 0$). Part of output is consumed ($C$), and the remainder adds to man-made capital. Given initial stocks of capital $K_0$ and resource $S_0$, what is the maximum constant level of consumption that can be sustained forever, and what are the associated time paths of $K$ and $S$?
Trying to write this as an optimal control problem, it is simple enough to identify $r_t$ as a control variable, $K_t$ and $S_t$ as state variables, and two state equations:
$$\frac{dK_t}{dt} = F(K_t,r_tS_t) – C_t$$ $$\frac{dS_t}{d_t} = - r_tS_t$$
The difficult part is to specify a suitable objective function.
Option 1: Take C to be constant and maximise C:
$$\max_r C$$
This does not fit the standard form of an optimal control problem as the objective function has no time dimension.
Option 2: Maximise the sum to infinity of $C$, adding an additional state equation $dC_t/dt = 0$ to ensure constancy of $C$: $$\max_r \int_0^{\infty} C_t dt$$
This has the consequence that if $C > 0$ then the value of the integral will be infinite.
Option 3: Treat $C$ as if it were given and minimise the sum to infinity of $r_tS_t$.
$$\min_r \int_0^{\infty} r_tS_t dt$$
Then (in outline) having identified conditions in terms of $C$ that solve this minimisation problem, find the value of $C$ such that the value of the above integral equals $S_0$. This approach (which is used in Solow (1974) p 35) is the only one I have found workable.
Question: Can the problem be solved by a more direct approach along the lines of Options 1 or 2 above?