Optimal control of a nonlinear PDE governed by fractional Laplacian
The following optimal control problem is considered: $([(-\Delta)_{\omega}]^{\beta}z)(x) = f(x,z(x),u(x))$, $x\in\Omega$ a. e., $\min J (z, u) = \int_{\Omega}$ $f_0(x,z(x),u(x))dx$ where $u(x)\in M$ for $x\in\Omega$ a.e., $\Omega\subset\mathbb{R}^N$, $N\geq 1$, is a bounded set, $\beta >0$, $M\subset\mathbb{R}^m$ and $[(-\Delta)_{\omega}]^{\beta}$ denotes a weak fractional Laplace operator of order $\beta$ with zero Dirichlet boundary values on the boundary of $\Omega$. The aim of the paper is to investigate the existence of optimal solutions to the system. The first section is devoted to preliminaries where one mainly explains the notion of weak Dirichlet-Laplace operator of fractional order. The result on the existence of optimal solutions is presented in the second section. If $M$ is a compact set, then, under five quite restrictive assumptions on $f$, $f_0$ and $J$, one proves the existence of an optimal solution $(z^*,u^*)\in\mathrm{dom}([(-\Delta)_{\omega}]^{\beta}))\times \mathcal{U}_M$ to the stated optimal control problem, $\mathcal{U}_M$ denoting the set of controls. An example is shown in the third section.
scientific article; zbMATH DE number 7498412
scientific article