Boundary control problems for quasi-linear elliptic equations: A Pontryagin's principle
The control system is \[ \begin{aligned} -\text{div }a(x,\nabla y(x)) &+a_0(x,y(x))=0\qquad (x\in\Omega),\\ a(x,\nabla y(x))\cdot\nu(x) &=b(x,y(x),u(x))\qquad (x\in\Gamma),\end{aligned} \] where \(\Omega\subseteq\mathbb{R}^n\) is a bounded domain with boundary \(\Gamma\) and \(\nu(x)\) is the outer normal to \(\Gamma\) at \(x\); the control \(u(x)\) is defined on \(\Gamma\) and takes values in a metric space \(\mathbb{K}\). Motivation comes from steady laminar flow of non-Newtonian fluids, reaction-diffusion problems, magnetostatics and glaciology. The cost functional is \[ J(u)= \int_\Omega L(x,y_u(x))dx+ \int_\Omega l(x,y_u(x),u(x))d\sigma, \] (\(y_u\) the solution corresponding to \(u\)) and the state constraints \[ \int_\Omega g_j(x,y_u(x))dx\leq \delta_j\qquad(1\leq j\leq m). \] The results are two versions of Pontryagin's maximum principle, the second under an additional assumption involving stability of the minimum with respect to small perturbations of the state; this assumption is shown to be generic. Previous papers (by the author and others) dealt with distributed controls; in the boundary control case, variations of the state due to perturbations of the control are much more difficult to elucidate. One of the essential tools is a result involving perturbations of the control in certain measurable sets; this is a generalization (due to the author) of a unidimensional type of perturbation due to X. Li and Y. Yao (see references in the paper) and has proved its worth in the treatment of control problems with state constraints, both for steady-state and for evolution equations.
scientific article; zbMATH DE number 883411
scientific article