Boundary control problems for quasi-linear elliptic equations: A Pontryagin's principle
type: Item
instance of: scholarly article
title: Boundary control problems for quasi-linear elliptic equations: A Pontryagin's principle
zbMATH Open document ID: 0868.49017
DOI: 10.1007/BF01204705
author: Eduardo Casas
published in: Applied Mathematics and Optimization
publication date: 1997-08-18T00:00:00Z
review text: 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.
reviewed by: Hector O. Fattorini
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 883411
zbMATH Keywords: cost functional
zbMATH Keywords: Pontryagin's maximum principle
zbMATH Keywords: boundary control
MaRDI profile type: Publication
cites work: Nonlinear partial differential equations in engineering
cites work: Q4077008
cites work: Un principe de Pontryagine pour le contrôle des systèmes semilinéaires elliptiques. (A Pontryagin principle for control of semilinear elliptic systems)
cites work: An Extension of Pontryagin’s Principle for State-Constrained Optimal Control of Semilinear Elliptic Equations and Variational Inequalities
cites work: Q4015593
cites work: Q4315454
cites work: A Green's formula for quasilinear elliptic operators
cites work: Q3988918
cites work: Q4036790
cites work: Distributed control of systems governed by a general class of quasilinear elliptic equations
cites work: Dealing with Integral State Constraints in Boundary Control Problems of Quasilinear Elliptic Equations
cites work: Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations
cites work: A New Approach to Lagrange Multipliers
cites work: Positivity of weak solutions of non-uniformly elliptic equations
cites work: Nonconvex minimization problems
cites work: Q4767831
cites work: Optimal control problems for distributed parameter systems in Banach spaces
cites work: Necessary conditions for infinite-dimensional control problems
cites work: Q3725066
cites work: Necessary Conditions for Optimal Control of Distributed Parameter Systems
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: Boundary value problems for some degenerate-elliptic operators
cites work: Q4932610
cites work: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus
cites work: Q5607484
cites work: Q4765475
cites work: Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints
description: scientific article; zbMATH DE number 883411