Consider the fishing optimal control problem defined by

Question

$\dot{P}_t = aP_t - bP_t x_t,$

where $P_t$ is the fish population at time $t$ , and $x_t$ is the fishing intensity or catch ( $a$ and $b$ are constants). If $r$ is the discount rate, the objective function is given by

$V(x) = \int_0^\infty e^{-rt} u(x_t) \, dt,$

where the utility of consumption is $c_t = x_t$ .
a) State the transversality condition.

b) Find the optimal consumption $x_t$ , if $u(x_t) = \ln(x_t)$ .

11 Jan 2025

Answer :

Word Count : 561

## Optimal Control Problem Analysis for Fishing Resource Management

Given the problem formulation:

\[
\dot{P}_t = aP_t - bP_t x_t
\]

Where:
- $ P_t $ is the fish population at time $ t $
- $ x_t $ is the fishing intensity or catch rate
- Constants $ a > 0 $ and $ b > 0 $ represent the natural growth rate of the fish population and the catch efficiency, respectively

The objective function is:

\[
V(x) = \int_0^\infty e^{-rt} u(x_t) \, dt
\]

with utility function $ u(x_t) = \ln(x_t) $ and consumption \( c_t = x_t _______ __________ _________ ____ _______ ______.
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