Optimal control for the degenerate elliptic logistic equation
The paper considers the problem of maximizing the functional \[ I(z,u)= \int_\Omega [\lambda h(u(x)) z(x)- k(u(x))] dx \] subject to the state equation \[ \Delta z(x)+ (a(x)- u(x)) z^\alpha(x)- e(x) z^\beta(x)= 0\quad\text{in }\Omega,\;z= 0\quad\text{on }\partial\Omega.\tag{1} \] Here \(\Omega\subset\mathbb{R}^n\) is a bounded regular domain and \(u\), \(0< u(x)\leq a(x)\) a.e. \(x\in\Omega\), plays the role of control. The existence and dependence on \(u\) of positive solutions of (1) are investigated. After that the existence, uniqueness and smoothness properties of the optimal control are derived.
scientific article; zbMATH DE number 1761093