Optimality conditions for state-constrained Dirichlet boundary control problems
type: Item
instance of: scholarly article
title: Optimality conditions for state-constrained Dirichlet boundary control problems
zbMATH Open document ID: 0951.49029
DOI: 10.1023/A:1021886227276
published in: Journal of Optimization Theory and Applications
publication date: 2000-07-06T00:00:00Z
review text: The system is \[ \begin{aligned}{\partial y(t, x) \over \partial t} &= Ay(t, x) + \Phi(t, x, y(t, x)),\quad (t, x) \in (0, T) \times \Omega, \\ y(0, x) &= y_0(x), \qquad x \in \Omega,\end{aligned} \] \((\Omega\) an \(n\)-dimensional domain with boundary \(\Gamma,\) \(A\) a second order elliptic operator). The control \(v(t, x)\) acts on the Dirichlet boundary condition, \[ y(t, x) = v(t, x), (t, x) \in (0, T) \times \Gamma. \] There are pointwise control and state constraints, the latter of the form \[ a(t, x) \leq y(t, x) \leq b(t, x). \] The functional undergoing minimization is \[ J(y, v) = \int_{(0, T) \times \Omega} F(t, x, y(t, x)) dt dx + \int_{(0, T) \times \Gamma} G(t, x, v(t, x)) dt d\sigma. \] As pointed out in this paper, the main difficulty when control is applied on the Dirichlet boundary condition is that the state \(y(t, x)\) is not continuous up to the boundary of \((0, T) \times \Omega,\) which makes it necessary to use multipliers in the dual of the space \(BC((0, T) \times \Omega)\) of bounded continuous functions in \((0, T) \times \Omega,\) or in the dual of \(L^\infty((0, T) \times \Omega).\) The elements of these duals are finitely additive measures. The authors avoid the complexities associated with finitely additive measures by showing that if the functions \(a(t, x)\) and \(b(t, x)\) in the state constraint are continuous up to the boundary, the multipliers can be replaced by bounded Radon measures. Necessary optimality conditions are obtained under Zowe-Kurcyusz type regularity conditions. The optimality conditions are shown to be sufficient if the cost functional is convex and the equation linear.
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 93C20
zbMATH DE Number: 1347769
zbMATH Keywords: Dirichlet boundary control
zbMATH Keywords: semilinear parabolic equations
zbMATH Keywords: pointwise state constraints
zbMATH Keywords: Lagrange multiplier theorems
zbMATH Keywords: Stone-Čech compactification
author: Nadir Arada
author: J-P Raymond
reviewed by: Hector O. Fattorini
MaRDI profile type: Publication
cites work: Necessary Optimality Conditions for Control Problems and the Stone--Cech Compactification
cites work: Optimal control problems for nonlinear parabolic boundary control systems: The Dirichlet boundary condition
cites work: Minimax control of parabolic systems with Dirichlet boundary conditions and state constraints
cites work: Regularity and stability for the mathematical programming problem in Banach spaces
cites work: A lagrange multiplier theorem for control problems with state constraints
cites work: Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints
full work available at URL: a:1021886227276
OpenAlex ID: W64850344
description: scientific article; zbMATH DE number 1347769