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
$x(k+1)=F\left(\right(1-u\left(k\right) \left)x\right(k) ),$
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

$H_{\alpha }(u¯,x(0\left)\right)=\sum _{j=0}^{\infty } {\sum _{i=1}^n}_{(}{C}_i{x}_i\left(j\right)u_i\left(j\right)e^{(}-\alpha j),$

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 $n =1$).