Optimization of discounted income for a structured population exposed to harvesting

A structured population the individuals of which are divided into $n$ age or typical groups $x_1,\ldots,x_n$ is considered. We assume that at any time moment $k,$ $k=0,1,2\ldots$ the size of the population $x(k)$ is determined by the normal autonomous system of difference equations $x(k+1)=F\bigl(x(k)\bigr)$, where $F(x)={\rm col}\bigl(f_1(x),\ldots,f_n(x)\bigr)$ are given vector functions with real non-negative components $f_i(x),$ $i=1,\ldots,n.$ We investigate the case when it is possible to influence the population size by means of harvesting. The model of the exploited population under discussion has the form
$$ x(k+1)=F\bigl((1-u(k))x(k)\bigr),$$
where $u(k)=\bigl(u_1(k),\dots,u_n(k)\bigr)\in[0,1]^n$ is a control vector, which can be varied to achieve the best result of harvesting the resource. We assume that the cost of a conventional unit of each of $n$ classes is constant and equals to $C_i\geqslant 0,$ $i=1,\ldots,n.$ To determine the cost of the resource obtained as the result of harvesting, the discounted income function is introduced into consideration. It has the form
$$ H_\alpha\bigl(\overline u,x(0)\bigr)={\sum\limits_{j=0}^{\infty}}\sum\limits_{i=1}^{n}C_ix_i(j)u_i(j)e^{-\alpha j}, $$
where $\alpha>0$ is the discount coefficient. The problem of constructing controls on finite and infinite time intervals at which the discounted income from the extraction of a renewable resource reaches the maximal value is solved. As a corollary, the results on the construction of the optimal harvesting mode for a homogeneous population are obtained (that is, for $n =1$).