## Optimal control for a nonlinear age-structured population dynamics model.(English)Zbl 1016.92026

Let $$u^{\nu}$$ be the solution of the following age-structured population dynamics: \begin{aligned}\partial_tu+\partial_au-k\Delta_xu+\mu(x,t,a,P(x,t))u=f-\nu u &\quad\text{in }Q=\Omega\times(0,T)\times(0,A),\\ (\partial u/\partial \eta)(x,t,a)=0 &\quad\text{on }\Sigma=\partial\Omega\times(0,T)\times(0,A),\\ u(x,t,0)=\int_{0}^{A}\beta(x,t,a,P(x,t))u(x,t,a) da &\quad\text{in }\Omega\times(0,T),\\ u(x,0,a)=u_0(x,a)&\quad\text{in }\Omega\times(0,A), \end{aligned}\tag{1} where $$P(x,t)=\int_0^Au(x,t,a) da$$ is the total population at time $$t$$ and location $$x$$; $$A$$ is the maximal age of an individual; $$\beta(x,t,a,P(x,t))\geq 0$$ is the natural fertility rate; $$\mu(x,t,a,P(x,t))\geq 0$$ is the natural death rate of age $$a$$ at time $$t$$ and location $$x$$; $$f(x,t,a)\geq 0$$ is external supply of individuals; $$\nu(x,t,a)$$ expresses the spatial harvesting effort; the constant $$k>0$$ and $$\Delta$$ is the gradient vector with respect to the spatial variable $$x$$. Let $S=\{\nu\in L^2(\Omega):\zeta_1(x,t,a)\nu(x,t,a)\leq\zeta_2(x,t,a)\;\text{a.e.},\;(x,t,a)\in Q\},$ where $$\zeta_1,\zeta_2\in L^{\infty}(Q)$$, $$0\leq \zeta_1(x,t,a)\leq \zeta_2(x,t,a)$$ a.e. in $$Q$$. The problem dealt with in this paper is to find the harvesting effort $$\nu\in S$$ in order to obtain the best harvest, i.e., to maximize, over all $$\nu\in S$$, the value of $$\int_Q\nu(x,t,a)g(x,t,a)u^{\nu}(x,t,a) dx dt da$$, where $$g$$ is a given bounded function.
The following results are obtained: the existence, uniqueness and compactness of solutions of (1), existence of an optimal control pair $$(\nu^*,u^{\nu^*})$$, and necessary optimality conditions.

### MSC:

 92D25 Population dynamics (general) 49J20 Existence theories for optimal control problems involving partial differential equations 49K20 Optimality conditions for problems involving partial differential equations 49N90 Applications of optimal control and differential games

### Keywords:

optimal control; age-structured population dynamics
Full Text: