Optimal control of a system governed by an elliptic partial differential equation
The paper considers the problem \hskip 17mm \(\max_{x\in\Omega} y(x)\to \min\), \hskip 17mm \(\text{div}(A(x)\nabla y)= f\) in \(\Omega\), \hskip 17mm \(y= g_1\) on \(\Gamma_1\), \({\partial y\over\partial\nu}= g_2\) on \(\Gamma_2\), \(y= u\) on \(\Gamma_3\), \hskip 17mm \(u\in H^{1/2}(\Gamma_3)\), \(y(x)\geq \psi(x)\) in \(\overline\Omega\), where \(\Omega\subset \mathbb{R}^n\) is a bounded polynomial domain, \(\Gamma_1\), \(\Gamma_2\), \(\Gamma_3\) are disjoint parts of the boundary \(\partial\Omega\), \(A\) is a positive definite matrix function and the functions \(g_1\), \(g_2\), \(f\), \(\psi\) are given, \(\psi(x)\leq g_1(x)\) on \(\Gamma_1\). The authors discuss the existence of optimal controls and the discretization of the problem via the finite element method. It is suggested to reformulate the original problem as a linear programming problem \[ \min\{\alpha\in R\mid u(x)\leq \alpha\text{ on }\Gamma_3, y= Tu+\varphi\text{ in }H^1(\Omega),\;y(x)\geq\psi(x)\text{ in }\overline\Omega,\;u\in H^{1/2}(\Gamma_3)\}, \] where \(T\) is the solver for the boundary value problem. The discrete and the standard maximum principle for solutions of elliptic equations are used.
scientific article; zbMATH DE number 1419816