| Abstract: |
| We consider semi-linear elliptic equations of the following form:
$\begin{equation*}
\left\{
\begin{aligned}
-\Delta u &= \lambda[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:\lambda f_h(u) &&\rm{in} ~ \Omega,\
\frac{\partial u}{\partial \eta}&+qu = 0 &&\rm{on} ~ \partial\Omega,
\end{aligned}
\right.
\end{equation*}$
where, $h\in U=\{h\in L^2(\Omega): 0\leq h(x)\leq H\}.$ We prove the existence and uniqueness of the positive solution for large $\lambda.$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional
$J(h)=\int_{\Omega}h(x)u_h(x)~{\rm{d}}x-\int_{\Omega}(B_1+B_2 h(x))h(x)~{\rm{d}}x$
over $U$, where $u_h$ is the unique positive solution of the above problem associated with $h$, $B_1>0$ is the cost per unit effort when the level of effort is low and $B_2>0$ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system. |
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